Transient Analysis of an Electrostatically Tunable Multiferroic Inductor

A Thesis Presented

by

Carl Joseph Hansen

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

in

Electrical and Computer Engineering

Northeastern University Boston, Massachusetts

August 2013

ii

ABSTRACT

This thesis examines the dynamic behavior of an electric-field tunable

multiferroic inductor, a four-terminal device that is comprised of a two-terminal variable

inductor with an inductance controlled by a two-terminal voltage input. Although the

inductor features a wide tuning range and high quality factor, little information is

available regarding the inductor’s transient characteristics in response to a fast changing

control voltage. Therefore, its role in certain high-speed electronic applications is

currently undetermined.

To reduce the aforementioned knowledge gap, the time-varying tuning response

of the multiferroic tunable inductor is explored in this thesis to establish its relevance to

high-speed electronics. Specifically, it is demonstrated that the multiferroic tunable

inductor can achieve a newly selected inductance value within a settling time of ten

microseconds. To accomplish this, the following steps are performed: First, a

mathematical model is developed that provides insight into the tunable inductor’s

physical behavior. Next, techniques for achieving a high-speed tuning response are

studied, including the derivation of measurable parameters, design and analysis of proper

electronic driver circuits, and measurement and processing techniques. Finally, a

laboratory measurement setup is designed and constructed to test the tunable inductor and

prove the stated hypothesis.

Results show that electrostatically tunable multiferroic inductors are well suited

for high-speed applications, and that tuning speeds of several microseconds are

achievable with appropriate driver electronics. These results bode well for electric-field

tunable multiferroic inductors and their use in radio frequency applications.

iii

TABLE OF CONTENTS

LIST OF FIGURES v

LIST OF TABLES vii

1 INTRODUCTION 1

1.1 Motivation 2

1.1.1 Tunable Inductors 2

1.1.2 High-Performance Tunable Inductors 8

1.2 Objective 11

1.3 Outline 12

2 LITERATURE REVIEW 13

2.1 Background 13

2.2 Performance 16

2.2.1 Tuning Range 16

2.2.2 Quality Factor 18

2.2.3 Tuning Speed 19

2.3 Modeling 21

2.3.1 Piezoelectric 21

2.3.2 Ferromagnetic 26

2.3.3 Time-Varying Inductance 26

2.4 Actuation 27

3 METHODOLOGY 31

3.1 Electrostatically Tunable Multiferroic Inductor Dynamics 31

3.1.1 Linear Time-Invariant Inductor 32

3.1.2 Linear Time-Variant Inductor 34

3.1.3 Linear Time-Variant Inductor Simulation 47

3.1.4 Non-Linear Time-Variant Inductor 52

3.1.5 Non-Linear Time-Variant Inductor Simulation 61

3.1.6 Measurement Techniques Summary 62

iv

3.2 Actuator Electronics 63

3.2.1 Load Requirements 63

3.2.2 Custom Amplifier 65

3.2.3 COTS Amplifier 67

3.2.4 Amplifier Implementation 70

3.3 Laboratory Measurement System 73

4 RESULTS 78

4.1 Sinusoidal Method 80

4.2 Constant Current Method 83

5 DISCUSSION 87

5.1 Tuning Speed 87

5.2 Measurement Techniques 89

5.3 Future Work 92

6 CONCLUSION 94

7 REFERENCES 95

APPENDIX A 100

v

LIST OF FIGURES

Figure 1.1: Voltage-Controlled Oscillator 3

Figure 1.2: Tunable Voltage Controlled Oscillator 4

Figure 1.3: Second Order Band-Pass Filter 5

Figure 1.4: Tunable Filter Implemented with Only Tunable Capacitors 9

Figure 1.5: Voltage-Controlled Oscillator (RF MEMS Implementation) 10

Figure 2.1: Electrostatically Tunable Multiferroic Inductor 13

Figure 2.2: Multiferroic Inductor Inductance 17

Figure 2.3: Multiferroic Inductor Tunability 17

Figure 2.4: Multiferroic Inductor B-H Curve 18

Figure 2.5: Multiferroic Inductor Quality Factor 19

Figure 2.6: Butterworth Van Dyke Piezoelectric Circuit Model 22

Figure 2.7: Simulated PZT Impedance vs. Frequency (BVD Model) 23

Figure 2.8: Simulated Impedance Magnitude vs. Frequency 24

Figure 2.9: Simulated Impedance Phase vs. Frequency 24

Figure 2.10: Guan Piezoelectric Circuit Model with Mechanical Loading 25

Figure 3.1: Two-Port Network 33

Figure 3.2: Modified Two-Port Network 35

Figure 3.3: P(s) Frequency Response 38

Figure 3.4: Simulated PZT Charge Step Response 39

Figure 3.5: M(s) Frequency Response 43

Figure 3.6: SPICE Circuit Model of a Linear Time-Variant Electrostatically

47

Tunable Multiferroic Inductor

Figure 3.7: Inductor Voltage Step Response 50

Figure 3.8: Inductor Voltage Step Response (Zoomed) 51

Figure 3.9: Inductance Step Response 52

Figure 3.10: Magnetic Flux vs. Current vs. Permeability 55

Figure 3.11: Magnetic Flux vs. Current vs. Permeability (2D) 56

vi

Figure 3.12: Static Inductance vs. Current vs. Permeability 57

Figure 3.13: Dynamic Inductance (Current) vs. Current vs. Permeability 58

Figure 3.14: Dynamic Inductance (Permeability) vs. Current vs. Permeability 59

Figure 3.15: SPICE Schematic with Constant Current Source 61

Figure 3.16: Magnetic Flux Response to Determine Tuning Speed 62

Figure 3.17: PZT Impedance Magnitude and Phase 64

Figure 3.18: Custom Amplifier Voltage Output 67

Figure 3.19: Apex Microtechnology MP111 Equivalent Schematic 69

Figure 3.20: Unassembled MP111 Amplifier and EV57 Evaluation Kit 71

Figure 3.21: Assembled MP111 Amplifier and EV57 Evaluation Kit 71

Figure 3.22: MP111 Implementation Schematic 72

Figure 3.23: MP111 SPICE Model Simulated Step Response 73

Figure 3.24: Laboratory Measurement System 1 74

Figure 3.25: Laboratory Measurement System 2 75

Figure 3.26: Measurement Circuit 76

Figure 4.1: PZT Sample 1 78

Figure 4.2: PZT Sample 2 79

Figure 4.3: Measured Inductor Voltage Step Response 80

Figure 4.4: Measured PZT Current Response 81

Figure 4.5: Measured Inductor Voltage Step Response (Reverse Polarity) 82

Figure 4.6: Measured PZT Current Response 83

Figure 4.7: Measured Inductor Voltage Step Response 84

Figure 4.8: Measured Inductor Voltage Step Response (Reverse Polarity) 85

Figure 4.9: Measured Inductor Voltage Step Response 86

Figure 4.10: Measured PZT Current Response 86

Figure 5.1: High-Level Control Block Diagram 93

Figure A.1: Custom Amplifier Design Schematic 101

vii

LIST OF TABLES

Table 2.1: Piezoelectric Driver Circuit Comparison 29

Table 3.1: Apex Microtechnology MP111 Amplifier Specifications 68

1

1 INTRODUCTION

Multiferroic materials and devices are a popular research area due to their

electrostatically tunable magnetic properties and potential application as voltage-

controlled filters, phase shifters, and inductors in radio frequency (RF) circuits [1].

Comprised of magnetoelastic and piezoelectric components, the multiferroic

heterostructure features a magnetoelectric (ME) coupling effect. The ME effect induces a

change in the magnetoelastic permeability through excitation of the piezoelectric

element. This change in permeability can lead to the alteration of external magnetic fields

in exchange for very little static energy dissipation, as the piezoelectric material is a high-

impedance device that requires only a constant electric field (voltage).

Development of the ME effect for tunable inductors has been driven by the

inefficient, bulky electromagnet tuning found in traditional tunable inductors [2], and

continued innovation has led to multiferroic materials exceeding the tunable range of

competing novel devices [3]. Although multiferroics have shown wider tuning frequency,

other RF MEMS devices possess desired tuning response time in the microsecond range

[4]. To ensure successful application in the RF domain, similar tuning speed must be

achieved.

In this study, the performance of the novel multiferroic tunable inductor is

researched to ascertain its performance in high-speed RF applications. Particularly, the

response time of the inductance change with respect to the control input is characterized

and a methodology for achieving repeatable high-speed operation is developed. By

establishing the inductors suitability in high-speed applications, scientists and engineers

can confidently integrate these low-power components into a range of devices.

2

Ultimately, the employment of efficient, high-performance tunable components will

provide benefits across the electronics spectrum.

1.1 Motivation

The inspiration for researching the transient response of a tunable inductor

originates from the general need for fast-performing tunable inductors in high-frequency

applications. Since present applications for tunable devices derive mainly from the RF

domain, the following discussion cites two specific RF applications to support the study

of tunable inductors in general and, more specifically, multiferroic tunable inductors and

their transient characteristics.

1.1.1 Tunable Inductors

Many wireless (RF) communications technologies require a passive tunable

component in the front-end to facilitate various operational states. One example is the

voltage-controlled oscillator (VCO), a device employed to generate the internal operating

frequencies essential to wireless transceivers. The VCO outputs a periodic signal whose

frequency is a function of the circuit topology. Figure 1.1 on the following page provides

an example schematic representation of the VCO [5].

In the example, the passive inductor (L) and passive capacitor (C) form a resonant

circuit alongside four active cross-coupled complementary metal oxide semiconductor

field-effect transistors (MOSFETs). Together, the components generate a particular

output frequency. Given the above topology in Figure 1, the output frequency is a direct

function of L and C, or:

3

𝑓0 = 𝑓(𝐿,𝐶) (1.1)

To achieve a fixed operating frequency, the VCO must employ static components

with values that do not drift over time. If either the inductor or capacitor is variable,

however, the output frequency will change as the component values change. Hence,

tunable components can directly tune the VCO frequency of operation to a desired value.

Figure 1.1: Voltage-Controlled Oscillator

M M

M M

C

L

VD

D

4

Figure 1.2: Tunable Voltage Controlled Oscillator

Given the option, designers may choose to implement a tunable inductor and a fixed

capacitor, a tunable capacitor and a fixed inductor, or a tunable inductor and tunable

capacitor as shown in Figure 1.2. The choice is typically dictated by performance,

availability, and/or component maturity. Since tunable capacitor research has yielded

many advancements in the tuning range of capacitors, these devices are easily

implemented as a VCO frequency-modifying component [5][6]. There are drawbacks to

solely using a capacitor, however, including increased device current consumption and

M M

M M

C

LV

DD

5

energy usage [5]. Thus, the inclusion of the tunable inductor as a complementary device

is often necessary and preferred to achieve the highest device performance. Examination

of another RF application, the tunable band-pass filter, further verifies the inductors

importance in tunable applications by illustrating an improved tunable range.

The tunable filter is required to selectively attenuate out-of-band signals

especially as reconfigurable RF front-end circuits are becoming increasingly popular for

multi-standard receivers [7]. Thus, a band-pass filter is useful in RF applications where

various carrier frequencies are received by an antenna, and a specific frequency is

required to be isolated for signal processing. Figure 1.3 demonstrates a second-order

resistor-inductor-capacitor (RLC) filter.

Figure 1.3: Second Order Band-Pass Filter

To illustrate the tunable inductors appeal, a nodal analysis is performed on the

circuit in Figure 1.3 to yield the following transfer function:

R

LCIN OUT

6

𝐻(𝑠) =𝑉𝑂𝑈𝑇(𝑠)𝑉𝐼𝑁(𝑠)

=𝑠 ∙ 𝑅𝐿

𝑠2 + 𝑠 ∙ 𝑅𝐿 + 1𝐿 ∙ 𝐶

(1.2)

For comparison, the general form of a second-order band-pass filter is as follows

[8]:

𝐻(𝑠) =𝑉𝑂𝑈𝑇(𝑠)𝑉𝐼𝑁(𝑠) =

𝑠 ∙ 𝜔0𝑄

𝑠2 + 𝑠 ∙ 𝜔0𝑄 + 𝜔0

2 (1.3)

Contrasting equations 1.2 and 1.3 demonstrates that the resonant frequency squared, or

𝜔02, is equivalent to the inverse of the product of inductance and capacitance in the filter,

or 1𝐿∙𝐶

. Also, the ratio of resonant (or center) frequency to quality factor, 𝜔0𝑄

, is equivalent

to the ratio of resistance to inductance, 𝑅𝐿. Both of these ratios help to define the filter

characteristics and both ratios include an inductive term, a clue to recognizing the value

of a tunable inductor.

The quality factor is an often used indicator of reactive efficiency. Specifically, 𝑄

relays the ratio of stored energy to dissipated energy per signal cycle. In a second-order

filter, the quality factor also signifies the bandwidth range, where low quality represents a

wide band of passable frequencies and high quality signifies a narrow band. In RF

applications, filters with a high quality factor are preferred as they support more

frequency selective processing.

Given 𝜔0 = � 1𝐿∙𝐶

and 𝜔0𝑄

= 𝑅𝐿, solving for 𝑄 yields

𝑄 =𝐿 ∙ 𝜔0

𝑅=

1𝑅∙ �

𝐿𝐶

(1.4)

7

Equation 1.4 suggests that a high quality factor can be achieved through manipulation of

the component values in three different ways:

1. Decrease resistance

2. Decrease capacitance

3. Increase inductance

Since resistance is inversely proportional to quality factor, the value of 𝑅 should

be as low as possible. Regarding the reactive components, inductance is proportional to

quality factor while capacitance is inversely proportional, or

𝐿 ∝ 𝑄 (1.5)

𝐶 ∝1𝑄

(1.6)

In other words, if a tunable inductor has its value increased while the capacitance

remains fixed, the quality factor will increase until the inductor reaches its maximum

tunable value. Similarly, if a tunable capacitor has its value reduced in equation 1.4 while

the inductor remains fixed, the quality factor will increase until the capacitor reaches its

maximum tunable value. Thus, if a designer can utilize a tunable inductor in addition to a

tunable capacitor, the quality factor may be extended beyond the range capable with only

a single tunable component.

The analysis presented above can also be applied to moving the resonant (center)

frequency for the selection of different bands. Since inductance and capacitance are

inversely proportional to resonant frequency, modification of both terms in the

inductance/capacitance product results in a greater tunable range than modification of a

single term.

8

𝐿 ∙ 𝐶 ∝1𝜔𝑜

(1.7)

Hence, designing a tunable filter with both a tunable inductor and a tunable

capacitor offers increased range versus a tunable filter without the complementary pair.

The concepts presented here also extend to the VCO and other tunable applications

requiring pairs of reactive components, ultimately suggesting that achieving high

performance in RF applications requires the inclusion of tunable inductors.

1.1.2 High-Performance Tunable Inductors

In the above examples, the tunable inductor was shown to achieve improved

circuit performance when paired with a complementary device. An arbitrary inductor,

however, is not suitable for all designs. In RF applications, designers must ensure that the

tunable inductor meets certain performance criteria related to high-frequency operation

before implementation in a particular circuit. These criteria include quality factor, tuning

range, and tuning speed. If the high-speed criteria are not met, alternative options that

sacrifice performance may be required. An example is a device comprised of only tunable

capacitors when no sufficient tunable inductor is available [9]. Therefore, researching the

performance of these inductor parameters and finding a balance between high tuning

range, quality factor, and tuning speed is critical to developing devices that include

tunable inductors and achieving maximum RF performance.

9

Figure 1.4: Tunable Filter Implemented with Only Tunable Capacitors [9]

As mentioned, one key performance parameter is the tuning response time, or

tuning speed. This parameter defines the amount of time it takes for the inductance to

settle to a new value (within some percentage) given an instantaneous change at the

control input. In an example RF application, a VCO may be required to change output

frequencies within a specified time range. Since the typical operating frequencies

required by RF applications are above the megahertz band, this may require the VCO to

change frequencies within an oscillation period, or less than one microsecond.

Researching tunable inductors with fast tuning response times has yielded a few

solutions that achieve microsecond response time. Each developed technology, however,

has its own set of constraints. One frequently published tunable inductor implementation

is comprised of RF MEMS switches. The basic design consists of a number of switches

that produce series and/or parallel combinations of fixed-value inductors. These switches

are desirable as they achieve contact closure in the microsecond range [4]. Additionally,

with their low on-resistance, the RF MEMS switch aids in the pursuit of high quality

factor as lower resistance reduces energy dissipation (refer to the discussion on tunable

filters and equation 4).

10

Figure 1.5: Voltage-Controlled Oscillator (RF MEMS Implementation)

In Figure 1.5, an example VCO implementation is illustrated with an array of RF

MEMS switches for tuning inductance and capacitance. A control signal (not shown)

actuates each switch to select a combination of inductance and capacitance. This process

produces the required oscillation frequency.

Although tunable RF MEMS implementations are fast and feature high quality

factor, they do have drawbacks. Fabricating an array of inductors and capacitors will

require some area for each component. The required area may exceed that available to a

designer and the tuning range will be limited. Further, due to the individual component

instances, the discrete tuning steps may be unsuitable for some applications. When

operating a device in such an “open-loop” fashion, component drift and aging will affect

M M

M M

L L

LL

C C

CC

VD

D

IN+ IN-

IN+ IN-RF

MEMSSwitches

RF MEMSSwitchImplementation

11

performance over time. Thus, other tunable inductor implementations may strike a better

balance between the discussed RF criteria.

The electrostatically tunable multiferroic inductor is one such device that has the

potential to offer high quality factor, tuning range, and tuning speed. As will be

discussed, researchers at Northeastern University have proven the multiferroic tunable

inductor to have increased tuning range and quality factor in the RF region as compared

to RF MEMS and other tunable inductor implementations [3]. Therefore, by

demonstrating a high tuning speed, the multiferroic inductor may be positioned as a

leading component in the design of RF devices.

1.2 Objective

This thesis aims at revealing the multiferroic inductor as a viable solution for

high-speed tunable applications. To accomplish the stated goal, the multiferroic tunable

inductor should achieve a tunable transient response in the low microsecond range. The

microsecond response time is required to compete with other tunable inductors, such as

the RF MEMS switch implementation discussed above and other current inductor

technologies [4][10][11]. By achieving the defined tuning response time, multiferroic

tunable inductors may be implemented in various RF applications that require fast tuning

of critical RF circuits while offering the improved range and quality factor proven by

multiferroic devices.

12

1.3 Outline

The research in this thesis builds upon cited work in the fields of multiferroic

device applications, piezoelectric modeling and physical behavior, and high-power

amplifier design. The subsequent chapters document the steps required to prove the

multiferroic inductors suitability as high-speed electronic component.

In chapter two, past research related to the multiferroic tunable inductor is

discussed. This includes:

• Tuning response and settling time of piezoelectric devices

• A survey of suitable models for mathematical analysis

• A review of high-power amplifiers required to drive piezoelectric devices

In chapter three, the methodology for establishing the high-speed performance of the

inductor is developed. Covered topics span:

• The derivation of mathematical models

• Simulation techniques

• Electronic driver circuit development

• Measurement techniques

Results are presented in chapter four, along with an improvements discussion in chapter

five. The chapter six conclusion is followed by an appendix providing additional

technical information.

13

2 LITERATURE REVIEW

This chapter provides an introduction to the electrostatically tunable multiferroic

inductor through the evaluation of peer-reviewed literature. Included in the examination

are published tuning response data, device models, and state-of-the-art actuating

electronics required to tune the device. Through this summary a foundation for furthering

study of the tunable inductor’s dynamic response is provided.

2.1 Background

Electrostatically tunable multiferroic inductors are composite devices.

Specifically, they are constructed from multiple materials and their design requires

knowledge of ferromagnetics and piezoelectrics. With experience in both areas,

researchers at Northeastern University have spent the last few years developing these

novel devices. This has resulted in several published papers detailing their design and

performance [3][12].

Figure 2.1 depicts an electrostatically tunable multiferroic inductor. As illustrated,

the inductor includes a coil and two materials labeled in the figure as Metglas 2605CO

and lead zirconate titanate (PZT). Additionally, two sets of terminals can be identified:

The inductor coil terminals and control voltage terminals attached to the PZT slab.

Figure 2.1: Electrostatically Tunable Multiferroic Inductor [3]

14

At the core of the inductor are the ferromagnetic and piezoelectric materials, or

Metglas and PZT. Metglas is a magnetoelastic ribbon that possesses a variable

permeability when strained by a mechanical force [3][13]. Conversely, lead zirconate

titanate, or PZT, provides a mechanical strain when subject to an electrical charge [14].

As a composite, the interaction of the two materials forms the magnetoelectric (ME)

effect where the injection of charge into the PZT creates a coupling strain to the Metglas

to vary the permeability. Thus, by wrapping the composite material in a coil of wire, the

PZT/Metglas combination acts as a permeable inductor core with a permeability that is

controlled by the PZT charge.

When changing the permeability of the multiferroic inductor’s core, a change in

inductance is realized [3]. This is illustrated by reviewing the constituent equations that

define permeability and its relationship to inductance. The dynamic permeability of a

ferromagnetic material is defined incrementally as

𝜇𝑑 =𝜕𝐵𝜕𝐻

(2.1)

where 𝐵 is the magnetic flux density of the core and 𝐻 is the magnetic field [15]. If the

ferromagnetic material is a linear device with no hysteresis, equation 2.1 can be

simplified to

𝜇𝑠 =𝐵𝐻

(2.2)

which describes static permeability. Similarly, the incremental, or dynamic, inductance

can be defined as

𝐿𝑑 =𝜕Φ𝜕𝑖

(2.3)

and the inductance of a linear, or static, device can be expressed as

15

𝐿𝑠 =Φ𝑖

(2.4)

where Φ is the magnet flux through the inductor core and 𝑖 is the current through the wire

[15][16].

Assuming a linear time-invariant (LTI) inductor with 𝑁 ideal coil windings and a

multiferroic core material, the permeability in equation 2.2 and the inductance in equation

2.4 are related by the equation for inductance of a multiferroic inductor [3], or

𝐿 =𝜇 ∙ 𝑁2 ∙ 𝐴

𝑙 (2.5)

where 𝐴 is the cross-sectional area of the inductor core and 𝑙 is the length of the solenoid

formed by the stacking of 𝑁 coils. Rearranging equation 2.5 for the permeability results

in

𝜇 =𝐿 ∙ 𝑙𝑁2 ∙ 𝐴

(2.6)

Based on the LTI assumption and citing the definition of static inductance, substituting

equation 2.4 into equation 2.6 produces

𝜇 =Φ𝑖∙

𝑙𝑁2 ∙ 𝐴

(2.7)

Finally, the second product in equation 2.7 can be condensed into a constant term,

resulting in

𝜇 =Φ𝑖∙ 𝐶 = 𝐿𝑠 ∙ 𝐶 (2.8)

where 𝐶 = 𝑙𝑁2∙𝐴

. From equation 2.8 it can be observed that permeability is linearly

proportional to inductance by some constant 𝐶 representing the geometric design of the

inductor. Thus, a change in permeability will result in a proportional inductance change.

16

If non-linear inductance is analyzed, the terms in equation 2.8 can be expressed with their

dynamic counterparts, or

𝜇𝑑 = 𝐿𝑑 ∙ 𝐶 =𝜕𝐵𝜕𝐻

=𝜕Φ𝜕𝑖

∙ 𝐶 (2.9)

thereby suggesting that the slope of the B-H curve is proportional to the slope of the

Φ− 𝑖 curve.

2.2 Performance

While the device has a unique design that requires a fundamental understanding

of device physics, materials science, and electromagnetics, its importance lies in the

performance. The electrostatically tunable multiferroic inductors fabricated at

Northeastern University achieve documented improvement in two of the performance

categories referenced in chapter 1. Both the tuning range and quality factor are improved

over other tunable inductor technologies. Since the tuning speed is currently unreported, a

similar device with published tuning speed is referenced for completion.

2.2.1 Tuning Range

Figure 2.2 on the next page dictates the inductance versus frequency for various

PZT input voltages applied to a laboratory fabricated multiferroic tunable inductor. Each

line color represents a different electric field applied to the PZT with units of 𝑘𝑉𝑐𝑚

, which

can also be represented as volts when the thickness of the PZT is known.

17

Figure 2.2: Multiferroic Inductor Inductance [3]

Given the properties of this particular PZT sample, the electric field in Figure 2.2

represents voltages from 0 to 600 V while producing inductances that range from 0.03 to

0.2 mH. As presented, the inductance plots suggest a tunable range of 450%, 250%, and

50% when operating at 100 Hz, 100 kHz, and 5 MHz respectively. Figure 2.3 reveals

additional tenability data measured by Lou, et al [3].

Figure 2.3: Multiferroic Inductor Tunability [3]

18

Portrayed another way, the B-H curve is presented in Figure 2.4. By applying an

input voltage to the PZT, the slope of each B-H curve changes to reflect the changing

inductance. Recall from previous analysis that permeability and inductance are

proportionally related by constant, C.

Figure 2.4: Multiferroic Inductor B-H Curve [3]

2.2.2 Quality Factor

The quality factor for the multiferroic tunable inductor is shown in Figure 2.5.

Recall that the quality factor describes the ratio of stored to dissipated energy. Although

the example in chapter 1 was for a tunable filter, the same definition holds true for an

inductor with a coil of non-ideal (resistive) wire.

19

Figure 2.5: Multiferroic Inductor Quality Factor [3]

Based on the graphic, the quality factor ranges from roughly 3 at zero electric field to 8.5

with a voltage applied to the PZT terminals, which is a marked improvement over

competing inductor technologies [3]. Similar to the tunability, the quality factor contains

peaks that vary with frequency.

2.2.3 Tuning Speed

The last performance characteristic to review is the tuning speed. As discussed,

little information exists regarding the transient response of multiferroic devices, and, in

particular, the multiferroic tunable inductor. However, researchers at Northeastern

University have studied the short-term and long-term dynamic performance of a

multiferroic heterostructure and published results describing the change in the

magnetoelectric effect over time [17][18].

In the first document [17], a Metglas/PZT heterostructure was exposed to a high-

voltage square wave switching at a frequency of 0.4 Hz. The input signal was applied for

20

1000 cycles and the magnetoelectric effect was observed via a vibrating sample

magnetometer. The result indicates the magnetoelectric effect features a nonlinear aging

process, thereby causing the permeability associated with a given control voltage to drift

over time as well. As reported by Chen, et al. [17], the aging effect can be attributed to

the physical construction of the heterostructure (bonding), relaxation of the Metglas

ribbon, or ion relaxation of the PZT crystal. While not explicitly investigated, these

findings suggest that the interaction of the physical structure of the multiferroic

heterostructure may have other impacts on temporal dynamics, including tuning speed.

In the second paper, Chen, et al. [18] generally defined the magnetoelectric effect

and studied a PZT/Metglas heterostructure’s transient characteristics in response to an

800 V peak-to-peak step input voltage applied to the PZT. The magnetoelectric effect

was found to fully settle within 0.6 seconds of the applied input voltage, thereby

indicated a 0.6 second tuning response time.

A research group at Oakland University has documented the switching speed of a

multiferroic heterostructure formed by ferromagnetic YIG and ferroelectric barium

strontium titanate (BST) [19]. In this research, a 500 V step voltage was applied to the

BST terminals with rise and fall times of 15 ns to generate a change in the resonate

frequency produced by the YIG/BST composite. The fabricated microwave resonator

experienced a full variation in resonate frequency 100 – 200 µs after application of the

step voltage. The settling time was mainly dictated by the resistor-capacitor (RC) circuit

formed by the input electrode and the BST plate capacitance.

In another study by the same group, a microwave resonator was constructed with

ferromagnetic YIG and piezoelectric PZT [20]. The results indicate a switching time in

21

the low microsecond range. A notable improvement over the YIG/BST resonator, the

decreased settling time can be attributed to the greater than order of magnitude electrode

thickness (more conductivity, or less resistance) and reduced voltage applied to the

YIG/PZT sample (20 V).

2.3 Modeling

To properly realize the stated hypothesis, the multiferroic tunable inductor’s

physical behavior must be modeled mathematically to facilitate the design of actuating

electronics. Without a model, one may have difficulty specifying the type of load a

tunable inductor’s control terminals present to an external amplifier, thereby reducing

optimal tuning speed. Since the direct interface to an amplifier output is the PZT control

terminals, published piezoelectric circuit models are discussed with arbitrary mechanical

loads. Research regarding Metglas and time-varying inductance modeling is also

discussed.

2.3.1 Piezoelectric

The classic piezoelectric circuit model is the Butterworth Van Dyke (BVD) model

presented schematically in Figure 2.6 [21].

22

Figure 2.6: Butterworth Van Dyke Piezoelectric Circuit Model

The BVD model is comprised of a capacitor (C0) in parallel with a series resistor-

inductor-capacitor (RLC) circuit. The dominant parallel capacitance (C0) approximates

the crystal dielectric separating the PZT terminals. The additional RLC components are

necessary to model the mechanical resonance of the PZT. The electrical equivalents are

described as resistor RM (damping coefficient), inductor LM (equivalent mass), and

capacitor CM (spring constant).

The model suggests primarily capacitive behavior over frequency due to C0. The

resonant peak, however, must be accounted for if control input frequencies will sweep

through the resonance band. Figure 2.7 illustrates the impedance magnitude to enlighten

these points. The data was collected by Guan, et al. [21] for a 500 µm thick PZT sample

from Piezo Systems, Inc. and incorporated into a Simulation Program with Integrated

Circuit Emphasis (SPICE) simulation to produce the plot.

C0

RM

LM

CM

INO

UT

23

Figure 2.7: Simulated PZT Impedance vs. Frequency (BVD Model)

Referring to Figure 2.7, the capacitive nature of PZT is evident. Specifically, impedance

tends toward infinity and 0 as frequency approaches 0 and infinity, respectively. Around

30 kHz, a notch in the impedance is also visible, corresponding to the mechanical

resonance of the PZT.

To further refine the accuracy of the BVD model, two additional resistances are

added:

• A parallel leakage resistance, Rp

• A series current limiting resistance, Rs

These resistances more closely mimic the actual behavior of PZT, as Rp approximates the

small charge leakage that occurs in PZT and Rs is correlated with the energy dissipation

in the hysteretic relationship between dielectric displacement and electric field [21].

24

Accounting for the additional resistances results in simulated responses plotted in Figure

2.8 and 2.9.

Figure 2.8: Simulated Impedance Magnitude vs. Frequency

Figure 2.9: Simulated Impedance Phase vs. Frequency

25

The dashed line is the Guan impedance that features magnitude limits and phase shifting

as frequency tends towards 0 and infinity, substantiating the Guan model as a more

realistic model.

The above data was compiled for an unloaded sample that features a single

resonant mode. If the PZT is bonded to another material, however, the mechanical

boundary conditions are changed [21]. This alteration produces additional modes which

must be accounted for in the model. Thus, Guan, et al. [21], proposed additional parallel

RLC circuits to model the increased resonance frequencies. Figure 2.10 provides a

schematic representation of fully loaded Guan piezoelectric model with three resonant

modes.

Figure 2.10: Guan Piezoelectric Circuit Model with Mechanical Loading

The dominant capacitive nature of PZT results in some designers choosing to

model the piezoelectric device as a single capacitor [22]. Care must be taken to ensure

C0

R1

L1

C1

Rs

Rp

R2

L2

C2

R3

L3

C3

INOU

T

26

that input frequencies do not excite the resonant modes; else the behavior will take on

second-order effects. This effect can be readily seen when dealing with step input, square

wave, and switching voltages that feature fast rise and fall times [23].

2.3.2 Ferromagnetic

As reported by Chen, et al. [18], the response of the Metglas to a mechanical

strain is on the order of a microsecond. Therefore, its dynamics may be assumed

negligible until the response of the PZT approaches within an order of magnitude or less

(i.e., 10 µs).

2.3.3 Time-Varying Inductance

Another important aspect of the tunable inductor that requires modeling is the

time-varying inductance formed by the interaction of a wire coil and the Metglas/PZT

heterostructure. Assuming an LTI core, researchers at the University of Defence Brno

discussed the proper modeling of a time-varying inductance for use in SPICE circuit

simulations [24]. While geared toward computer simulation, the derived mathematics

hold true.

The literature indicates standard textbook equations that define the voltage across

the terminals of an inductor do not persist when dealing with an inductor that changes

values over time. Equation 2.10 presents the voltage across an LTI inductor.

𝑣 = 𝐿 ∙𝑑𝑖𝑑𝑡

(2.10)

If the inductor is time-varying, however, the expression changes to [24]

27

𝑣(𝑡) = 𝐿(𝑡) ∙𝑑𝑖𝑑𝑡

+ 𝑖(𝑡) ∙𝑑𝐿𝑑𝑡

(2.11)

After integrating, the expression for computing inductance is

𝐿(𝑡) =𝐿(0) ∙ 𝑖(0) + ∫ 𝑣(𝑡) ∙ 𝑑𝑡𝜏

0𝑖(𝑡)

(2.12)

Biolek, et al. [24] assert that equation 2.12 should be applied to situations where an

inductance is time-varying.

2.4 Actuation

To deliver the proper charge to the PZT plates, high-performance driver circuits

are necessary. As discussed, PZT devices require hundreds of volts to achieve a

displacement that appropriately strains the Metglas, thus affecting the permeability. If the

PZT/Metglas heterostructure is modeled as a mechanically loaded PZT sample, research

published in the piezoelectric driver domain may be useful in assessing state-of-the-art

driver circuit technology.

A number of amplifiers were surveyed, and each device was selected to compare

performance in four key categories:

• Voltage

o Sufficient to displace the PZT and bonded Metglas to modify Metglas

permeability, thus producing a tunable inductance. Based on previous

research [3], greater than 100 V is preferred.

• Current

28

o Sufficient to supply charge to the PZT plates consistent with the desired

tuning speed (frequency). Based on PZT modeling, greater than 2 A is

preferred.

• Slew Rate

o If several hundred volts are required to tune the inductor to the desired

frequency, and this tuning value is requested on the order of a few

microseconds, a slew-rate approaching 100 V/µs is desirable.

• Bandwidth

o In high-frequency applications, varying inductance may necessary at

differing periods. To switch frequencies in this fashion, higher bandwidth

amplifiers are desirable.

In Table 2.1 on the subsequent page, the best of the surveyed driver circuit

specifications are compared to determine if a published amplifier suits the above

requirements.

29

Table 2.1: Piezoelectric Driver Circuit Comparison

Based on these criteria, the first amplifier by Lewis [25] is impressive in its power

handling capability and slew rate. However, the amplifier is designed for switching

applications and cannot produce a continuous tuning voltage to a PZT. The amplifier

could, however, be useful in characterization and testing.

The second amplifier by Borg [26] features an impressive slew-rate with very

high bandwidth. The voltage and current range, however, are low in comparison to other

models. Out of all the amplifiers surveyed, this is the only one to feature integrated

circuit fabrication.

The next driver [27] did not specify a number of critical performance factors and

its operation in comparison to other amplifiers is unknown.

Fleming [28] produced a very capable amplifier that could potentially be used in

the proposed application, as it has very high voltage, large current output, high

Author G. K. Lewis [25] J. Borg [26] C. Wallenhauer [27] A. Fleming [28] M. S. Colclough [29]Peak-to-Peak Voltage (V)

100 40 100 200 400

Current (A) 10 0.4 - 1.9 0.68

Unloaded Bandwidth (MHz) 8 40 - 8 0.1

Slew Rate (V/µs) 1250.00 13333.33 - 1000.00 300.00Output Impedance (Ω)

0.05 High 5 Low Low

Linear Amplifier No Yes Yes Yes Yes

Feedback No No Yes Yes YesEfficient Efficient Efficient High Voltage High Voltage

Fast High Voltage Fast Slew-RateResistive Loads Only

(Feedback Required for Capacitive)

Poor Linearity Requires Op-Amp

0.35 µm, 50 V CMOS Requires Op-Amp Requires Op-Amp Inefficient

Low Voltage Low Bandwidth Inefficient Low Bandwidth

Notes

Switching Applications

Only

High Bandwidth

30

bandwidth, and fast slew-rate. The amplifier is based on an Apex Microtechnology power

amplifier, and a discrete transistor output stage.

Finally, Colclough [29] advertised an amplifier that achieved impressive voltage,

but was lacking in other categories.

Given the above data, the amplifier by Fleming is most suited for driving the

tunable inductor. It’s possible, however, that the current rating may not be sufficient for

providing the necessary PZT charge. This fact will be considered in later chapters.

31

3 METHODOLOGY

The literature in the previous chapter provides a foundation from which the tuning

response time of a tunable inductor may be theoretically analyzed and experimentally

measured. In this chapter, the study of electrostatically tunable multiferroic inductors is

advanced through

• the development of mathematics that describe and simulate the interaction

between the piezoelectric material and the time-varying inductance.

• the development of appropriate actuating electronics.

• novel laboratory measurement techniques to capture the changing inductance over

time.

The research builds upon cited literature in an attempt to further the development of

electrostatically tunable multiferroic inductors and multiferroic heterostructures in

general.

3.1 Electrostatically Tunable Multiferroic Inductor Dynamics

The goal of this section is to develop a method for accurately measuring a

changing inductance over time. In doing so, the tuning speed of the multiferroic inductor

may be established. To determine the tuning response of a tunable inductor, however, a

measureable parameter must first be identified that properly conveys the variation in

tunability. A seemingly obvious parameter that lends itself to measurement is the

inductance itself, as inductors are well defined in circuit analysis textbooks [8][15]. To

properly measure the inductance of an electrostatically tunable mutliferroic inductor,

32

however, the physical behavior of the PZT and Metglas must be understood, as well as

the mathematical equations that represent this behavior.

In chapter two, the physical behavior of the PZT/Metglas permeable core was

discussed to gain insight into how the PZT and Metglas react to step voltages applied to

the control inputs. The following paragraphs further this work by analyzing the dynamic

properties of PZT and Metglas and how they relate to measuring inductance through

mathematical analysis and simulation.

3.1.1 Linear Time-Invariant Inductor

To determine a measurable inductance, a circuit model that incorporates all

elements of the multiferroic inductor is a useful in supporting the analysis. Figure 3.1

depicts a typical two-port network used in the modeling of small-signal amplifiers [30] to

represent the PZT/Metglas behavior and the core’s interaction with a coil of wire. As

discussed, a dependent source can be used in SPICE modeling of time-varying elements

such as capacitors and inductors [24].

33

Figure 3.1: Two-Port Network

In Figure 3.1, element Z on the left side of the schematic represents the PZT electrical

impedance seen by a driving source. On the right is a dependent current source that

represents the inductor current as a function of the terminal voltage, Vo. In other words,

any load connected across the inductor that applies a voltage, Vo, will experience a

current through it dictated by f(Vo) and vice versa.

The function f(Vo) can be expanded more explicitly to describe the inductor

current dependence on voltage. The terminal voltage of an inductor is related to current

through Faraday’s law, which states that the time derivative of magnetic flux, Φ, is

equivalent to the voltage across an inductors terminals, or

𝑣 =𝑑Φ𝑑𝑡

(3.1)

Since the magnetic flux is equivalent to the product of inductance and current through the

inductor as described by equation 2.4, equation 3.1 expands to

i=f(Vo)Z VoVi

34

𝑣 =𝑑(𝐿 ∙ 𝑖)𝑑𝑡

= 𝐿 ∙𝑑𝑖𝑑𝑡

(3.2)

The expression in equation 3.2 implies that the inductance can be determined if the

voltage and current are measurable. With the proper laboratory setup, voltage and current

are measurable and 𝐿 can be extracted by dividing the voltage time-series by the current

derivative. Or, equation 3.2 can be integrated to obtain

𝑖(𝑡) = 𝑖(𝑡0) +1𝐿� 𝑣(𝑡) ∙ 𝑑𝑡 = 𝑓(𝑉𝑜)𝑡

𝑡0 (3.3)

which satisfies the dependent source relationship in Figure 3.1. In equation 3.3, the

inductance is found by integrating the voltage-time series and dividing by the current

time-series minus the initial current value.

There are fundamental flaws with the presented scenario, however. Noticeably

absent is any relationship between the input Vi and Z to the output dependent source.

Modeling the inductance in this way fails to capture the inductor’s true dependence on

permeability, µ, as described in equation 2.5. Further, the assumption in this section is the

inductor behavior is linear and time-invariant (LTI). As will be shown in the next section,

time invariance is not preserved when modeling the relationship from input to output in

Figure 3.1, and additional steps must be taken to properly extract the inductance value.

3.1.2 Linear Time-Variant Inductor

Referring back to Figure 3.1, the permeability of the Metglas – and, ultimately,

the inductance – can be shown to depend on the PZT impedance. More importantly, the

permeability will be shown to change over time and cause a violation in the time-

invariant assumption of the LTI inductor presented above.

35

To begin, Figure 3.1 can be modified to illustrate a direct depiction of inductance.

Figure 3.2: Modified Two-Port Network

In Figure 3.2, the dependent source is replaced directly by an inductor that is a function

of permeability. The inductance of a multiferroic tunable inductor was previously

investigated by Lou, et al. [3] as

𝐿 =𝜇 ∙ 𝑁2 ∙ 𝐴

𝑙 (3.4)

with permeability plainly influencing the inductance value. Additionally, the impedance

of the PZT material was researched by Guan, et al. [21] and found to be

𝑍(𝑠) =𝑠2 ∙ 𝐿𝑀 ∙ 𝐶𝑀 + 𝑠 ∙ 𝑅𝑀 ∙ 𝐶𝑀 + 1

𝑠3 ∙ 𝐿𝑀 ∙ 𝐶0 ∙ 𝐶𝑀 + 𝑠2 ∙ 𝑅𝑀 ∙ 𝐶0 ∙ 𝐶𝑀 + 𝑠 ∙ (𝐶𝑀 + 𝐶0) (3.5)

for the BVD model.

To relate the impedance in equation 3.5 to the current-voltage characteristics of

the inductor in Figure 3.2, the charge of the PZT is analyzed. As mentioned, the strain

ZVi VoLL=f(u)

36

provided by PZT is related to the amount of charge on its plates. The impedance seen by

a source driving the PZT, however, dictates the amount of charge transferred to the PZT

plates. This is explained by the following Laplace-domain expression for impedance,

which states that the ratio of voltage to current is equal to impedance, or

𝑍(𝑠) =𝑉(𝑠)𝐼(𝑠) ⇒ 𝐼(𝑠) =

𝑉(𝑠)𝑍(𝑠)

(3.6)

Also suggested in equation 3.6 is that the ratio of voltage to impedance yields the current

flow. Given that the impedance of the PZT for the BVD model was provided, substituting

into equation 3.6 yields

𝐼(𝑠) =𝑉(𝑠) ∙ 𝑠 ∙ 𝐶𝑀

𝑠2 ∙ 𝐿𝑀 ∙ 𝐶𝑀 + 𝑠 ∙ 𝑅𝑀 ∙ 𝐶𝑀 + 1+ 𝑉(𝑠) ∙ 𝑠 ∙ 𝐶0 (3.7)

Since current is equivalent to the quantity of charge transferred through a point in an

electric circuit per second, current and charge are related by

𝑞𝑃𝑍𝑇(𝑡) − 𝐶𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 = �𝑖(𝑡) ∙ 𝑑𝑡 (3.8)

Equation 3.8 demonstrates that the time integral of current will sum the amount of charge

transferred to the PZT plates, 𝑞𝑃𝑍𝑇(𝑡). Since impedance was provided in the Laplace

domain, using the Laplace property of integration will return equation 3.8 to the proper

domain. Assuming the no charge is present on the plates as an initial condition

(𝐶𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 = 0), the following relationships are defined.

ℒ{𝑞𝑃𝑍𝑇(𝑡)} = 𝑄𝑃𝑍𝑇(𝑠) (3.9)

ℒ �� 𝑖(𝑡) ∙ 𝑑𝑡� =1𝑠∙ 𝐼(𝑠) (3.10)

𝑄𝑃𝑍𝑇(𝑠) =1𝑠∙ 𝐼(𝑠) (3.11)

37

Equations 3.9, 3.10, and 3.11 illustrate the Laplace transformation of time domain charge

to its Laplace domain equivalent, 𝑄𝑃𝑍𝑇(𝑠) . Current flow to the PZT in the Laplace

domain is now divided by the Laplace variable, 𝑠, and equated to charge.

By further substitution, the PZT charge can be related to the voltage and

impedance ratio in equation 3.6 by

𝑄𝑃𝑍𝑇(𝑠) =1𝑠∙ 𝐼(𝑠) =

1𝑠∙𝑉(𝑠)𝑍(𝑠)

(3.12)

Multiplication of the 1𝑠 term into equation 3.12 allows for cancelling of common terms

after expanding 𝑍(𝑠), yielding

𝑄𝑃𝑍𝑇(𝑠) = 𝑉(𝑠) ∙ 𝐶𝑀

𝑠2 ∙ 𝐿𝑀 ∙ 𝐶𝑀 + 𝑠 ∙ 𝐶𝑀 ∙ 𝑅𝑀 + 1+ 𝑉(𝑠) ∙ 𝐶0 (3.13)

Not surprisingly, after converting from the Laplace domain to the frequency domain by

substituting 𝑗𝜔 for 𝑠, the expression confirms that charge across the PZT plates at 0 Hz

(DC) is equivalent to

𝑄𝑃𝑍𝑇(0) = 𝑉(0) ∙ 𝐶𝑀 + 𝑉(0) ∙ 𝐶0 = 𝑉(0) ∙ (𝐶𝑀 + 𝐶0) (3.14)

or, the steady-state voltage multiplied by the total capacitance of the PZT plates.

Additionally, by dividing the numerator and denominator of the first term in equation

3.13 by 𝐿𝑀 ∙ 𝐶𝑀 and dividing both sides by 𝑉(𝑠), the low-pass form is similar to equation

1.3 discussed in chapter 1, plus a capacitive offset.

𝑃(𝑠) =𝑄𝑃𝑍𝑇(𝑠)𝑉(𝑠) = 𝐶𝑀

1𝐿𝑀 ∙ 𝐶𝑀

𝑠2 + 𝑠 ∙ 𝑅𝑀𝐿𝑀+ 1𝐿𝑀 ∙ 𝐶𝑀

+ 𝐶0 (3.15)

The quality factor of frequency domain charge is similar to that of the band-pass filter

discussed in chapter one. Given that 𝑅𝑀 is usually several orders of magnitude larger than

38

𝐿𝑀 [21], the PZT material can have a high quality factor that leads to peaking in the low-

pass frequency response [8]. This can correspond to a resonance when excited by a

voltage step. Using the data obtained from Guan, et al. [21] for the BVD model, Figure

3.3 demonstrates the capacitive frequency response and Figure 3.4 reveals the under-

damped charge in response to a 100 V step voltage.

Figure 3.3: 𝑃(𝑠) Frequency Response

In Figure 3.3, the capacitive behavior is seen before and after the resonant peak at 30

kHz. Specifically, the PZT appears as an 18.9 nF capacitor below resonance, and a 17.3

nF capacitor beyond resonance.

39

Figure 3.4: Simulated PZT Charge Step Response

In Figure 3.4, as expected, the charge settles to a steady-state value relative to the total

capacitance of the PZT and the voltage applied to the plates, or

100 𝑉 ∙ (17.3 𝑛𝐹 + 1.6 𝑛𝐹) = 100 𝑉 ∙ (18.9 𝑛𝐹) = 1.89 𝜇𝐶

Since it was previously determined that PZT produces a strain in relation to an

applied charge, the following general association is defined:

𝐴𝑃𝑍𝑇(𝑠) =𝑆𝑃𝑍𝑇(𝑠)𝑄𝑃𝑍𝑇(𝑠) (3.16)

Equation 3.16 states mathematically what was described in the aforementioned

paragraph, where 𝑆𝑃𝑍𝑇 is the strain produced by the charge, 𝑄𝑃𝑍𝑇(𝑠) , and 𝐴𝑃𝑍𝑇(𝑠)

describes the relationship between strain and charge. Consistent with published analysis,

𝐴𝑃𝑍𝑇(𝑠) is equivalent to the g constant that describes the strain developed over an applied

40

charge density with units of meters squared per coulomb [32]. Therefore, the Laplace

variable 𝑠 can be dropped and 𝐴𝑃𝑍𝑇 treated as a coefficient for further analysis, or

𝐴𝑃𝑍𝑇(𝑠) = 𝐴𝑃𝑍𝑇 (3.17)

Similarly, a coupling factor can be introduced to describe the relationship between

the PZT and Metglas strain, or

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) =𝑆𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)𝑆𝑃𝑍𝑇(𝑠) (3.18)

where 𝑆𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) and 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) are the strain experienced by the Metglas and the

coupling factor, respectively. Assuming that equal strain is applied to the entire surface of

the Metglas and the bonding material does not introduce dynamic effects,

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) represents a coefficient that describes the amount of deflection

transferred from one material to the other. For example, a value of one would indicate

full transfer of strain. Hence, the Laplace variable 𝑠 in 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 is dropped for further

analysis, or

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) = 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 (3.19)

Finally, a relationship between the strain on the Metglas and its permeability can

be established as

𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)𝑆𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) (3.20)

where 𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) and 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) are the Laplace domain permeability of Metglas and

gain of the strain to permeability ratio. Given the research by Chen, Harris, et al. [18], it

was shown that the Metglas has a very fast response time on the order of one

microsecond. Assuming a first-order response, the transfer function in equation 3.20 will

41

have a time constant approximately 5 times less1 than one µs, or no greater than 200 ns.

Placing equation 3.20 into the form of a general first-order transfer function yields

𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) =1

𝑠 ∙ 𝜏 + 1 (3.21)

with 𝜏 signifying the time constant in seconds [8]. Given this first-order low-pass filter

transfer function, the 1/𝜏 pole location is no less than 5 MHz. By this assumption, the

pole due to the Metglas strain will not likely influence the lower frequency poles

introduced by the PZT impedance, and 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) can be assumed a constant, or

𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) = 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 (3.22)

Given equations 3.16, 3.18, and 3.20, the permeability of Metglas can be related

back to the charge across the PZT plates by making the following substitutions. First,

multiply both sides of equation 3.20 by the strain term, 𝑆𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠), and divide by the

constant 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 to create the following expression:

𝑆𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠

(3.23)

Substituting the above expression into equation 3.18 yields

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 ∙ 𝑆𝑃𝑍𝑇(𝑠) (3.24)

and rearranging for 𝑆𝑃𝑍𝑇(𝑠) produces

𝑆𝑃𝑍𝑇(𝑠) =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 ∙ 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 (3.25)

Substituting for 𝑆𝑃𝑍𝑇(𝑠) by the ratio in equation 3.16 yields

1 After five time constants, a first-order system will have achieved approximately 99% of its final value [8]

42

𝐴𝑃𝑍𝑇 =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 ∙ 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 ∙ 𝑄𝑃𝑍𝑇(𝑠) (3.26)

Rearranging the constant terms to the left-hand side of equation 3.26 gives the following

expression relating charge to strain:

𝐴𝑃𝑍𝑇 ∙ 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 ∙ 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 = 𝐴 =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)𝑄𝑃𝑍𝑇(𝑠) (3.27)

Multiplying both sides of equation 3.27 by 𝑄𝑃𝑍𝑇(𝑠) illustrates how the permeability

response is a scaled replica of the charge response.

𝐴 ∙ 𝑄𝑃𝑍𝑇(𝑠) = 𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) (3.28)

Since 𝐴𝑃𝑍𝑇, 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠, and 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 are all constant values, they are consolidated into

a single variable, 𝐴, to represent a coupling gain factor. The gain may be positive or

negative. According to Lou, et al. [3], the increased charge in PZT decreases permeability

due to negative coefficients, implying 𝐴 will be negative.

One final relationship can be established from the above analysis: The impedance

term, 𝑍(𝑠) , and the actuating electronics driving voltage, 𝑉(𝑠) , can be inserted into

equation 3.28 to form a complete expression from voltage input to permeability output.

Substituting equation 3.5 into equation 3.28 yields

𝐴 =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

1𝑠 ∙𝑉(𝑠)𝑍(𝑠)

⇒𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

𝑉(𝑠)=

𝐴𝑠 ∙ 𝑍(𝑠)

(3.29)

and expanding to include the unloaded BVD model results in transfer function 𝑀(𝑠):

𝑀(𝑠) =𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

𝑉(𝑠) (3.30)

43

𝑀(𝑠) = 𝐴 ∙ �𝐶𝑀

1𝐿𝑀 ∙ 𝐶𝑀

𝑠2 + 𝑠 ∙ 𝑅𝑀𝐿𝑀+ 1𝐿𝑀 ∙ 𝐶𝑀

+ 𝐶0� = 𝐴 ∙ 𝑃(𝑠) (3.31)

The expression in equation 3.31 is similar to that in equation 3.15, save for the additional

constant, 𝐴 . Figure 3.5 demonstrates the frequency response assuming the BVD

impedance with the Guan, et al. [21] data and assuming the gain factor, 𝐴, is unity.

Figure 3.5: 𝑀(𝑠) Frequency Response

In general, 𝑀(𝑠) will be a scaled replica of 𝑃(𝑠) described in equation 3.15. By

re-substituting equation 3.15 back into equation 3.31, the resulting expression simplifies

to equation 3.28:

𝑀(𝑠) = 𝐴 ∙ 𝑃(𝑠) = 𝐴 ∙𝑄𝑃𝑍𝑇(𝑠)𝑉(𝑠) =

𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)𝑉(𝑠) ⇒𝐴 ∙ 𝑄𝑃𝑍𝑇(𝑠) = 𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)

44

As a result, 𝜇𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑡) is a reflection of 𝑞𝑃𝑍𝑇(𝑡) in the time domain, including all

second-order effects such as overshoot, under-damped oscillations, and settling time [33].

In the time domain expression, the relationship is simply:

𝐴 ∙ 𝑞𝑃𝑍𝑇(𝑡) = 𝜇𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑡) (3.32)

It should be noted, however, that if the terms 𝐴𝑃𝑍𝑇 , 𝐴𝑃𝑍𝑇/𝑀𝑒𝑡𝑔𝑙𝑎𝑠 , and 𝐴𝑀𝑒𝑡𝑔𝑙𝑎𝑠 are a

function of Laplace variable 𝑠due to dynamic behavior, additional poles and zeros will be

introduced into the transfer function in equation 3.31. This may alter the dynamic

response and equation 3.31 will no longer hold true.

Finally, it should be noted that the capacitance of piezoelectric material can be

hysteretic [34] and the capacitive terms in equation 3.31 may not always be constant after

a strain has been applied. Thus, permeability may not always reflect a one-to-one

relationship with an applied voltage that strains the PZT in such a way that the hysteresis

curve comes into play.

The main focus of the above analysis is to review time domain response of the

permeability. Once a voltage step input is applied to the PZT, the permeability reacts as a

function of the voltage and may feature overshoot, ringing, and/or settling. Since

permeability of the multiferroic tunable inductor is a demonstrated function of time, the

inductance will also be a function of time, or time-variant.

𝐿(𝑡) =𝜇(𝑡) ∙ 𝑁2 ∙ 𝐴

𝑙 (3.33)

Referring back to equation 3.2, the derivative term no longer simplifies to a product of

the static inductance and the time derivative of the current through the inductor coils.

Rather, it expands to

45

𝑣(𝑡) =𝑑�𝐿(𝑡) ∙ 𝑖(𝑡)�

𝑑𝑡=𝑑𝐿𝑑𝑡

∙ 𝑖(𝑡) +𝑑𝑖𝑑𝑡∙ 𝐿(𝑡) (3.34)

under the assumption that the inductance is still linear.

To move forward and isolate the inductance term such that it may be solved for, a

slightly different approach must be used as 𝐿(𝑡) is no longer a unique variable. One

approach is to integrate both sides of equation 3.34. The result is

𝐿(𝑡0) ∙ 𝑖(𝑡0) + � 𝑣(𝑡) ∙ 𝑑𝑡 = 𝐿(𝑡) ∙ 𝑖(𝑡)𝑡

𝑡0 (3.35)

and rearranging yields

𝐿(𝑡) =𝐿(𝑡0) ∙ 𝑖(𝑡0) + ∫ 𝑣(𝑡) ∙ 𝑑𝑡𝑡

𝑡0𝑖(𝑡)

(3.36)

This expression is identical to that derived by Biolek, et al. [24] in their paper on the

SPICE modeling of time-varying components. In equation 3.36, several constant terms

are generated due to the integration. These terms are 𝐿(𝑡0) and 𝑖(𝑡0), or the inductance

and current values at zero time in the circuit.

Interestingly, if the inductance is time-invariant, or 𝐿(𝑡) = 𝐿(𝑡0) = 𝐿, multiplying

both sides by 𝑖(𝑡) and dividing by 𝐿 yields the familiar circuit expression for current

through an inductor [8] derived in equation 3.3.

𝑖(𝑡) =𝐿(𝑡0) ∙ 𝑖(𝑡0) + ∫ 𝑣(𝑡) ∙ 𝑑𝑡𝑡

𝑡0𝐿(𝑡0)

= 𝑖(𝑡0) +1𝐿� 𝑣(𝑡) ∙ 𝑑𝑡𝑡

𝑡0 (3.37)

Consequently, equation 3.36 provides an expression that allows time-varying

inductance to be computed by measuring the voltage and current at the inductor terminals

and by taking into account any initial conditions that exist. Therefore, the initial

inductance must be known at the time the measurement is started. With this knowledge,

46

any change in permeability via an applied voltage to the PZT inputs will be accurately

represented by equation 3.36 as a measurable inductance change.

Although the above method is suitable for determining inductance, there is

another measurement method worth discussing. In the previous analysis, a sinusoidal

voltage source (or similar) is required to drive the inductor such that a change in

inductance would present a change in the inductor voltage and, thus, a changing

inductance. If, however, the voltage source was modified to a constant current source, the

mathematics simplifies. By equation 3.34, a constant current will zero the derivative term

and leave only the following expression:

𝑣(𝑡) =𝑑𝐿𝑑𝑡

∙ 𝑖(𝑡0) + 0 ∙ 𝐿(𝑡) =𝑑𝐿𝑑𝑡

∙ 𝑖(𝑡0) (3.38)

Whether integrating or deriving from equation 3.38, the resulting measurement equation

is

𝐿(𝑡) =𝐿(𝑡0) ∙ 𝑖(𝑡0) + ∫ 𝑣(𝑡) ∙ 𝑑𝑡𝑡

𝑡0𝑖(𝑡0)

(3.39)

𝐿(𝑡) = 𝐿(𝑡0) +1

𝑖(𝑡0)∙ � 𝑣(𝑡) ∙ 𝑑𝑡

𝑡

𝑡0 (3.40)

Now, when the permeability changes due to some input voltage at the PZT terminals, the

inductance change is reflected solely in the voltage across the inductor terminals. In the

next few sections, the constant current method will reveal its importance to tuning speed

characterization.

47

3.1.3 Linear Time-Varying Multiferroic Inductor Simulation

To further demonstrate the above points, Figure 3.6 shows a circuit schematic

designed in SPICE to simulate the multiferroic inductor with the aforementioned Guan

PZT parameters. The model is based on the two-port network schematic shown in Figure

3.2.

Figure 3.6: SPICE Circuit Model of a Linear Time-Variant Electrostatically Tunable

Multiferroic Inductor

Inside the rectangle in Figure 3.6 are the primary components that model the multiferroic

tunable inductor. Components C0, RM, LM, and CM make up the BVD model of the

L

f lu x= ta n h ( ' u 0 '* ( ' A '* V (s t ra in ) / ' u 0 '+ 1 )* x)1 7 . 3 n

C 0

6 3R M

1 6 . 5 m

L M

1 .6 nC M

A C 1

P U L S E (0 1 0 0 1 0 0 e -6 1 e -6 1 e -6 )

V _ IN

B 1

I= id t ( I (V _ IN ))

R 1 0

1

R _ M E A S U R E

R 1

1

B 2

I= id t (V (V L ))

R 3

1

B 3

I= ( ' u 0 '* ( 'D C _ O F F S E T ' ) + I (B 2 ) ) / I (L )

V _ O U T

S T R A IN

VL

48

PZT, and L represent the inductor and its terminal behavior. Voltage sources V_IN and

V_OUT are step and sinusoidal signals, respectively. The additional current sources

make up mathematical modeling elements.

• B1 is used to integrate the electrical current into charge (represented by net

STRAIN).

• B2 integrates the voltage across the inductor terminals to compute magnetic flux.

• B3 receives the integrated voltage (or magnetic flux) plus initial conditions and

divides it by the current through the inductor to compute static inductance.

Note that if the 𝑖(𝑡0) initial condition is zero, the product of initial conditions in equation

3.36 is also zero and B3 simply divides magnetic flux by current.

To model the inductance, SPICE defines a keyword flux. As seen in Figure 3.6,

the flux term is equated to a product of initial permeability (u0), permeability divided by

initial permeability plus an offset (A*V(strain)/u0+1), and current (x). This is determined

by referring back to equation 2.4 where linear inductance is defined as the ratio of

magnetic flux to current. To derive the flux equation used in the inductor SPICE model,

rearrange equation 2.4 and substitute equation 2.5 to form

Φ(t) = 𝐿(𝑡) ∙ 𝑖(𝑡) =𝜇(𝑡) ∙ 𝑁2 ∙ 𝐴

𝑙∙ 𝑖(𝑡) (3.41)

For simplicity, the 𝑁2∙𝐴𝑙

term equates to unity. Therefore,

Φ(t) = 𝜇(𝑡) ∙ 𝑖(𝑡) (3.42)

Referring to the data collected by Lou, et al. [3] in Figure 2.4, there is baseline

permeability that exists before a change in charge is applied to the PZT material. This is

49

evident by the slope of the B-H curve, or the incremental permeability discussed in

chapter 2. If the initial slope is represented as 𝜇0, then 𝜇(𝑡) can be expanded to

𝜇(𝑡) = 𝜇𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑡) + 𝜇0 = 𝐴 ∙ 𝑞(𝑡) + 𝜇0 (3.43)

where 𝐴 is the previously described gain factor due to PZT/Metglas coupling and 𝜇0 is a

permeability offset. 𝐴 ∙ 𝑞(𝑡) is equivalent to A*V(strain) in the flux expression of Figure

3.6. Substituting back into equation 3.42 yields

Φ(t) = (𝐴 ∙ 𝑞(𝑡) + 𝜇0) ∙ 𝑖(𝑡) (3.44)

Extracting the initial permeability term yields

Φ(t) = 𝜇0 ∙ �𝐴 ∙ 𝑞(𝑡)𝜇0

+ 1� ∙ 𝑖(𝑡) (3.45)

Equation 3.45 now represents the flux expression from the SPICE simulation in Figure

3.6. Of note, equation 3.45 states that if zero charge exists on the PZT plates, the flux will

be equal to the initial permeability multiplied by the current through the coil. Dividing by

𝑖(𝑡) produces linear static inductance, or the ratio of flux to current as shown in equation

3.46.

Φ(t)𝑖(𝑡)

= 𝐿 = 𝜇0 = 1 µ𝐻 (3.46)

The result of equation 3.46 is valid when no charge is applied to the PZT and (still)

assuming 𝑁2∙𝐴𝑙

to be unity.

The above analysis will now be demonstrated and applied to the formula derived

for measuring the time-varying inductance. Figure 3.7 and 3.8 represent the simulated

response of the multiferroic inductor to a 100 V step input voltage to the PZT with a one

microsecond rise time (100 V/µs slew rate) occurring at 100 µs into the simulation. The

plot depicts the voltage change across the inductor terminals when driven by sinusoidal

50

source with amplitude of 1 V and frequency of 1 MHz through a 47 Ω resistor. Figure 3.7

illustrates the response over 1 ms and Figure 3.8 illustrates a close-up between 75 and

150 µs.

Figure 3.7: Inductor Voltage Step Response

51

Figure 3.8: Inductor Voltage Step Response (Zoomed)

As expected, the voltage response varies in a similar fashion to the charge response in

Figure 3.4. Specifically, the envelope of the sinusoidal signal reflects the under-damped

response of the PZT charge reacting to the voltage step. Per equation 3.36, as the

inductance changes, the voltage and current through the inductor change accordingly and

produce the signal in Figure 3.7 and 3.8.

Plotting the output of source B3 yields the calculated time-varying inductance in

Figure 3.9.

52

Figure 3.9: Inductance Step Response

Based on the mathematics derived above and input into SPICE, the static inductance is

numerically computed and plotted over time in response to the PZT input step voltage.

This plot should be identical to that of the PZT charge response since the coupling

between PZT and Metglas was assumed to be unity. With this result, the inductance may

now indicate the tuning speed of a particular device.

3.1.4 Non-Linear Time-Variant Multiferroic Inductor

The linear time-variant assumption is valid for many inductors. However,

referring back to the B-H curves collected by Lou, et al. [3], a clear non-linear behavior is

present as magnetic field (current) is increased through the coil. At a certain field value,

the flux no longer has a linear relationship with field and the linear assumption is invalid.

53

This fact affects the previously derived measurement equations and requires further

investigation.

If magnetic flux features a non-linear relationship with current, it can be

represented generally as

Φ(t) = Φ�𝑖(𝑡), 𝜇(𝑡)� (3.47)

where 𝑖(𝑡) and 𝜇(𝑡) are functions of time. Since Faraday’s law is related to the flux rate-

of-change with respect to time, revising equation 3.1 with the new flux expression

produces

𝑣(𝑡) =𝑑Φ𝑑𝑡

=𝜕Φ𝜕𝜇

∙𝑑𝜇𝑑𝑡

+𝜕Φ𝜕𝑖

∙𝑑𝑖𝑑𝑡

(3.48)

Equation 3.48 now represents a more general case of equation 3.34 with two product

terms containing flux rate-of-change with respect to current and permeability. In equation

2.3, the dynamic inductance was introduced as the partial derivative of flux with respect

to current. Substituting into equation 3.48 yields

𝑣(𝑡) =𝜕Φ𝜕𝜇

∙𝑑𝜇𝑑𝑡

+ 𝐿𝑑(𝑡) ∙𝑑𝑖𝑑𝑡

(3.49)

Another dynamic inductance exists as a partial derivative of flux with respect to

permeability, or

𝐿𝑑𝜇(𝑡) =𝜕Φ𝜕𝜇

(3.50)

Renaming equation 2.3 results in a unique variable for dynamic inductance due to

current, or

𝐿𝑑𝑖(𝑡) =𝜕Φ𝜕𝑖

(3.51)

Substituting both into equation 3.48 produces

54

𝑣(𝑡) = 𝐿𝑑𝜇(𝑡) ∙𝑑𝜇𝑑𝑡

+ 𝐿𝑑𝑖(𝑡) ∙𝑑𝑖𝑑𝑡

(3.52)

Equation 3.52 suggests that there are at least two non-unique dynamic inductance

definitions that will require alternate methods for calculation based on measured data.

The steps taken in the previous section to determine inductance as a function of time are

useful for determining static inductance, but not dynamic.

Referring again to Figure 2.4, the trend of each B-H curve follows a hyperbolic

tangent function (the small hysteresis effect notwithstanding). This is typical behavior of

an inductor. In fact, the inductor SPICE model documentation suggests using a

hyperbolic tangent in the flux equation discussed in the previous equation [31].

Therefore,

Φ(𝑡) = tanh (𝑖(𝑡) ∙ 𝜇(𝑡)) (3.53)

is defined. By assuming a non-linear hyperbolic tangent flux, three inductance definitions

are now possible: a static inductance due to the ratio of flux to current, a dynamic

inductance due to changing current, and a dynamic inductance due to changing

permeability. These are presented using the hyperbolic tangent flux expression in

equations 3.54, 3.55, and 3.56.

𝐿𝑠(𝑡) =Φ𝑖

=tanh (𝑖 ∙ 𝜇)

𝑖 (3.54)

𝐿𝑑𝑖(𝑡) =𝜕Φ𝜕𝑖

= 𝜇 ∙ sech(𝑖 ∙ 𝜇)2 (3.55)

𝐿𝑑𝜇(𝑡) =𝜕Φ𝜕𝜇

= 𝑖 ∙ sech(𝑖 ∙ 𝜇)2 (3.56)

The flux and inductances can be plotted as functions of current and permeability to

further illustrate their complex nature.

55

Figure 3.10: Magnetic Flux vs. Current vs. Permeability

For each permeability value, the magnetic flux follows a hyperbolic tangential curve as

current is swept through its axis. An alternate two dimensional view is plotted in Figure

3.11.

56

Figure 3.11: Magnetic Flux vs. Current vs. Permeability (2D)

Figure 3.11 is similar to Figure 2.4 in chapter 2, although with more plotted curves. The

minimum and maximum permeability values correspond to the minimum and maximum

slopes of each curve. The outlines of the data plotted in Figure 3.11 suggest the relative

values of each slope at the tangent points. Note that the permeability can be extracted

from this Φ− 𝑖 curve when constant C is known, as discussed in chapter 2.

The next three figures depict the three inductances, starting with the static

inductance in Figure 3.12.

57

Figure 3.12: Static Inductance vs. Current vs. Permeability

Figure 3.12 plots the ratio of magnetic flux to current. The shape is similar to the

subsequent curve, but the inductance diverges as the absolute value of current and

permeability grow in value. Note this is valid for a positive coupling gain between charge

and permeability. As Lou, et al. [3] researched, the gain factor can be negative. If so, the

plot would then be reversed on the permeability axis.

58

Figure 3.13: Dynamic Inductance (Current) vs. Current vs. Permeability

Here, the inductance tends to roll off more sharply as current is increased than in

Figure 3.12 since the inductance is a derivative with respect to current. As permeability

increases (decreases), the roll-off becomes more drastic.

59

Figure 3.14: Dynamic Inductance (Permeability) vs. Current vs. Permeability

Finally, the dynamic inductance due to permeability is plotted in Figure 3.14. Dissimilar

to Figures 3.12 and 3.13, the shape of the dynamic inductance is closer to the flux-current

curve in Figure 3.10. The minimum and maximum values are different than the other

inductances, and the inductance may actually produce negative values when the current

becomes negative. This is evident by the expression for dynamic inductance due to

permeability in equation 3.56.

Given the non-linear complexities of the inductances, it is difficult to produce a

measurement technique that yields accurate results. The static inductance may be found

by first integrating the voltage across the inductor terminals to compute flux. Then, the

resulting expression is divided by current to produce the linear inductance. The dynamic

inductances, however, are more difficult and require additional measurement techniques

to compute the partial derivatives of flux with respect to permeability and current and the

derivatives of permeability and current with respect to time.

60

In any case, one is left with three inductance time-series data sets to determine the

settling time, and each will give varying results. To reduce the ambiguity, another option

is presented. A common theme among the presented methods for linear time-invariant,

linear time-variant, and non-linear time-variant is the measurement of flux via the

inductor voltage integral.

Φ− 𝐶𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 = � 𝑣(𝑡) ∙ 𝑑𝑡𝜏

0= �

𝑑Φ𝑑𝑡

𝜏

0∙ 𝑑𝑡 (3.57)

All of the inductance dependent variables in equation 3.57, whether linear or non-

linear, are represented by the changing flux. If the permeability is changed due to a step

voltage input to the PZT, the flux will represent the inductance transient per equation 2.3

or 2.4. Further, if current is varying through the inductor and forcing the permeability

slope to change due to non-linearity, this will also be accounted for. Although a specific

value of inductance will not be present in the measured time-series2, the tuning response

will be clearly indicated by integrating the voltage across the inductor terminals.

Additionally, since the change in permeability creates a change in flux, the flux

derivative with respect to time will also indicate the tuning response transient.

Conveniently, the flux rate-of-change has been already defined in equation 3.1 as the

voltage across the inductor terminals. When dealing with small signals that create

numerical integration errors (in an effort to compute flux), the flux rate-of-change may be

used as an alternative for determining the tuning response transient. Therefore, the flux

rate-of-change provides a rapid analytical tool that will be illustrated in chapters four and

five.

2 Methods for determining the inductance vs. PZT input voltage have been studied by Lou, et al. [3].

61

3.1.5 Non-Linear Time-Varying Multiferroic Inductor Simulation

Figure 3.15 portrays an updated schematic with a 1 A constant current source

providing a DC bias to the inductor to establish a magnetic field instead of a sinusoidal

voltage source. The voltage step applied to the PZT will produce a change in

permeability that is reflected in source B2, which is integrating the voltage across the

inductor terminals. Source B3 continues to compute the static inductance. Additionally,

the flux equation has been updated to include a hyperbolic tangent function to simulate

non-linear behavior. In response to the step, the simulated flux is plotted in Figure 3.16.

Figure 3.15: SPICE Schematic with Constant Current Source

L

f lu x= ta n h ( ' u 0 '* ('A '* V (s tra in )/ ' u 0 '+ 1 )* x)1 7 .3 n

C 0

6 3R M

1 6 .5 m

L M

1 .6 nC M

A C 1

P U L S E (0 1 0 0 1 0 0 e-6 1 e -6 1 e -6 )

V _ IN

B 1

I= id t (I (V _ IN ))

R 1 0

1

R _ M E A S U R E

R 1

1

B 2

I= id t (V (V L ))

R 3

1

B 3

I= ( ' u 0 '* ( 'D C _ O F F S E T ' )+ I (B 2 ) ) /I (L )

I_ O U T

S T R A IN

VL

62

Figure 3.16: Magnetic Flux Response to Determine Tuning Speed

3.1.6 Measurement Techniques Summary

To summarize, two measurement methods are useful in generally determining the

tuning response time of a tunable inductor that features non-linear and time-varying

behavior. After providing a step voltage to the PZT, the inductor flux rate-of-change may

be measured by observing the inductor terminal voltage and

1. exciting the inductor with a sinusoidal voltage source.

2. exciting the inductor with a constant current source.

The voltage may then be integrated to compute magnetic flux and converted to static or

dynamic inductance as determined by the previous analysis with the understanding that

each inductance will result in differing values over time. If sufficient measurement

resolution is not available to perform the numerical integration required for computing

63

flux, the magnetic flux time rate-of-change is a valid substitute for determining the tuning

response time.

3.2 Actuator Electronics

To produce a sufficient input to the PZT terminals, a driving circuit is required.

Specifically, an electronic amplifier that produces hundreds of volts, provides amps of

output current, features a slew-rate of 100 V/µs, and maintains a high bandwidth is

studied to allow for microsecond switching of the PZT. The next sections discuss the

PZT load requirements, a custom amplifier design, and a commercial-off-the-shelf

(COTS) solution.

3.2.1 Load Requirements

To determine the proper specifications for the amplifier, the load must be known

so that the voltage and current requirements are explicitly defined. In support of this, an

impedance analyzer was connected to the PZT input of a multiferroic inductor and its

impedance curve generated. Figure 3.17 illustrates the magnitude and phase of tested

PZT impedance. The tested sample was obtained from a single-layer PZT sheet

manufactured by Piezo Systems that was bonded to a piece of Metglas ribbon.

64

Figure 3.17: PZT Impedance Magnitude and Phase

As predicted by the Guan circuit models, a loaded piezoelectric features multiple

resonant modes. The PZT, now loaded by Metglas and terminal contacts, features

multiple resonant modes. In practical applications, the amplifier will be switching high-

voltage step voltages, so a square-wave voltage can be approximated for frequency

analysis. A square wave will produce odd harmonic amplitudes according to [35]

𝑓(𝑥) =4𝜋

� �1𝑛

sin �𝑛𝜋𝑥𝐿��

∞

𝑛=1,3,5,…

(3.58)

where 𝑥 is either 𝐿2 or 3 ∙ 𝐿

2 and 𝑛 represents the odd harmonics.

Since the majority of literature suggests switching times on the order of a few

microseconds are high-performance, ten microsecond period will be selected. Assuming a

ten microsecond switching period between pulses, a 100 kHz square wave is required. At

100 kHz, a fundamental frequency will exist with harmonics at n·100 kHz. Given the

65

voltage ranges published in the literature and used by Lou, et al. [3] in previous testing, a

100 V square wave is likely suitable for driving the piezoelectric and generating a

suitable tuning response. Therefore, at the fundamental frequency of 100 kHz, the

sinusoidal voltage output is

(100 𝑉) ∙4

𝜋 ∙ 𝑛= 127.3 𝑉

and drawing an impedance value from the magnitude in Figure 3.17 at 100 kHz, the

current requirement is

𝑉𝑅

=127.3 𝑉100 Ω

= 𝐼 = 1.27 𝐴

Similarly, at 300 kHz, the current requirement is

𝑉𝑅

=42.4 𝑉20 Ω

= 𝐼 = 2.12 𝐴

and at 500 kHz the value is

𝑉𝑅

=25.5 𝑉46 Ω

= 𝐼 = 0.554 𝐴

Continuing to sum the currents dictated by the remaining of the square wave harmonics

totals a maximum current of 5 A. Therefore, the amplifier may be required to produce up

to 100 V at 5 A, or 500 W of power, to an external load3. This rules out the Fleming [28]

amplifier discussed in chapter 2.

3.2.2 Custom Amplifier

Upon reviewing the literature in chapter 2, it was determined that many published

amplifiers were not suitable to produce the charge required by the PZT due to either

3 The analysis assumes ideal square-wave signals (infinite bandwidth). The amplifier naturally band-limits the square- wave, thereby reducing the amplitude of each harmonic and reducing the required current.

66

frequency or power limitations. To remedy the situation, a custom discrete amplifier was

designed. Appendix A contains a complete amplifier schematic that will be referred to

going forward.

The amplifier design features a fully differential architecture to support high-

speed current source inputs and high-voltage outputs. Beginning with the input stage, a

differential common-base stage (Q7/Q3) converts the DAC current outputs to a voltage

input while level-shifting for the next amplifier stage, similar to the ubiquitous 741 op-

amp [30]. At the input to the subsequent differential cascode stage, the signal is amplified

by the common-emitter NPN pair (Q5/Q4) and buffered by the common-base pair

(Q1/Q2). Bias is supplied by the R6/R4 resistor divider, and the 4.8 kΩ resistor pair

provides voltage gain via the currents flowing from the common-base stage. The

differential voltages are buffered by a Darlington-type configuration [36] before power

gain is supplied by the class AB output stage with paralleling for current-sharing.

Notable features include a common-mode feedback configuration to zero the

output offset voltage4 and beta-helper current mirrors. Most prominent is the lack of any

feedback compensation via the open-loop design. Without a compensation capacitor

between the first and second gain stages, high slew-rates are achievable with this

amplifier, on the order of 100 V/µs according to simulation results.

Figure 3.18 depicts a square wave voltage applied across a Guan PZT model. In

the figure, the total peak-to-peak voltage swing is 100 V at 100 kHz.

4 The common-mode feedback amplifier is represented by an ideal DC servo amplifier in the Appendix A schematic [37].

67

Figure 3.18: Custom Amplifier Voltage Output

Although the custom amplifier performance is adequate, the power consumption

is quite high (40 W quiescent). Additionally, to operate correctly, the amplifier should be

part of a larger system that drives the DAC chip with an FPGA or microcontroller.

Similarly, an ADC would be required to sample the current and/or voltage drive into the

PZT, plus some sort of control and compensation. These efforts are beyond the scope of

this thesis.

3.2.3 COTS Amplifier

In lieu of a custom design, COTS amplifiers were also investigated. Although not

prevalent, a few high-power discrete drivers exist. Particularly, Apex Microtechnology

specializes in the design of piezoelectric drive amplifiers. The typical Apex amplifier is

68

capable of supplying hundreds of volts with many amps of current while supporting high

frequency outputs in the kHz region.

To support the 500 W power delivery and 100 V/µs slew-rate requirement, the

Apex Microtechnology MP111 was selected. The MP111 performance features are

highlighted in table 3.1.

Table 3.1: Apex Microtechnology MP111 Amplifier Specifications

Parameter Value

Peak-to-Peak Voltage ~90 V

Current Output (Pulse) 50 A

Current Output (Continuous) 15 A

Slew Rate 130 V/µs

Power Bandwidth 500 kHz

Gain Bandwidth Product 6 MHz

Power Supply ±50 V

Power (Quiescent) 16 W

Notable specifications are the large output currents and high power bandwidth, or the

highest supportable bandwidth while delivering a full voltage swing to the desired load.

Additionally, the MP111 requires less power than the custom design.

69

Figure 3.19: Apex Microtechnology MP111 Equivalent Schematic [39]

For completeness, the equivalent circuit schematic is shown in Figure 3.19. The

MP111 is a hybrid class AB design comprised of junction field effect transistors (JFET),

bipolar junction transistors (BJT), and metal oxide semiconductor field effect transistors

(MOSFET). Similar to the custom design, the input stage is a JFET/MOSFET cascode

configuration followed by a PMOS common-source amplifier. The class AB output stage

is biased by BJT transistors while power MOSFETs handle delivering current to the load.

The compensation capacitor is external to the circuit and parallels the second gain stage.

70

3.2.4 Amplifier Implementation

Given the two amplifier options, the COTS amplifier was selected as the PZT

driver as it featured acceptable specifications compared to the custom amplifier,

including high-voltage, fast slew-rate, high bandwidth, and large current output.

Although pursuing the custom option may have led to increased performance, the COTS

specifications are suitable to provide a step input voltage and supply the necessary charge

to the PZT. Additionally, the MP111 was readily available and was priced well for the

purposes of this research. The following paragraphs discuss the methods required to

prepare the MP111 for integration into the test system.

The assembly of the Apex Microtechnology MP111 was facilitated by the

purchase of the EV57 evaluation kit. The kit features a large and small heat sink, a

printed circuit board (PCB) for power and signal routing, high-power interface

connectors, power supply bypass capacitors, and a user area for soldering feedback

resistors and a compensation capacitor. Without the EV57, integration of the amplifier

into the measurement system would prove difficult.

Figure 3.20 on the following page shows the MP111 amplifier and EV57

evaluation kit side-by-side. Power, ground, and the amplifier output are connected to the

black terminal strip at the bottom of the board as indicated in Figure 3.20. The user-

modifiable area is also depicted.

When assembled, the amplifier sits beneath the tall black heat sink located in the

center of the green PCB in Figure 3.20. Further below the amplifier is a larger heat sink

that conducts heat away from the sensitive discrete components on the amplifier PCB.

Figure 3.21 illustrates this more clearly on the following page.

71

Figure 3.20: Unassembled MP111 Amplifier and EV57 Evaluation Kit

Figure 3.21: Assembled MP111 Amplifier and EV57 Evaluation Kit

MP111

EV57

MP111

User Area

Terminal Strip

Passive Components

72

Also visible in Figure 3.21 are three passive components: two resistors and a

capacitor. These are soldered to the user-modifiable area of the PCB to set the gain and

compensation capacitance of the amplifier. Based on available function generators, the

resistance values were selected to achieve a gain of 10. Additionally, based on the MP111

datasheet section regarding slew-rate, a compensation capacitor value of 33 pF was

selected to achieve a slew-rate up to 130 V/µs. Since the datasheet made no mention of a

maximum output capacitance value, the amplifier was simulated in SPICE driving a step

voltage into the Guan, et al. [21] model parameters to verify stability. The schematic

version of the MP111 implementation and ensuing simulation results are given in Figures

3.22 and 3.23.

Figure 3.22: MP111 Implementation Schematic

M P 1 1 1

C _ C

3 3 p

R _ L IM

0 . 0 5

R _ F

2 2 0 k

R _ I

2 2 k

V _ P O W _ P O S

'V C C '

A C 1

R se r= 5 0

P U L S E (-4 .5 4 .5 1 e -6 1 e -9 1 e-9 2 e -3 4 e -3 )

V _ F U N C _ G E N

V _ P O W _ N E G

'V C C '1 7 . 3 nC 0

6 3

R M

1 6 . 5 mL M

1 . 6 n

C M

.t ra n 0 1 0 e-3 0 1 e -7 s ta r tu p

.p ara m V C C = 5 0;a c o c t 1 e 3 1 1 e 8

73

Figure 3.23: MP111 SPICE Model Simulated Step Response

As seen in Figure 3.23, the amplifier is stable driving the PZT capacitance. According to

the Apex MP111 SPICE model, no overshoot or ringing is present, so the physical

implementation is expected to be similar. Additionally, the rise-time of the voltage output

is well within a microsecond.

3.3 Laboratory Measurement System

To perform the tuning response data collection, several pieces of hardware are

required. Each item is listed below in no particular order.

• Two 50 V power supplies

• One power amplifier

• Two function generators

• One DC current source

74

• One digital oscilloscope

• One current measure resistor

• One electrostatically tunable multiferroic inductor

The complete measurement system is depicted pictorially in Figures 3.24 and 3.25

to illustrate the hardware required.

Figure 3.24: Laboratory Measurement System 1

Figure 3.24 displays – from left to right – the constant current source, two function

generators, the oscilloscope, and one multiferroic inductor with one spare (labeled in

figure). Figure 3.25 shows the connection between the amplifier and two power supplies.

Inductor

75

Two isolated, dual-channel 25 V supplies were each stacked in series to form one +50

and one -50 V supply. Then, the supplies were connected across the power amplifier as

discussed in the COTS amplifier implementation section.

Figure 3.25: Laboratory Measurement System 2

76

A circuit schematic representing the entire measurement setup is shown in Figure

3.26. V_FUNC1 and V_FUNC2 reference the two function generators, V_POW_POS

and V_POW_NEG reference the two power supplies, the MP111 is the power amplifier,

the rectangular box is the inductor under test, and R_MEASURE is the measurement

resistor.

Figure 3.26: Measurement Circuit

To collect the tuning response speed data, the following steps are taken with

reference to Figure 3.26. Steps labeled a correspond to the sinusoidal method, and b

correspond to the constant current method.

1. The inductor is connected in series with the 47 Ω measurement resistor

(R_MEASURE) and a

MP 111

C_C

33p

R_LIM

0.05

R_F

220k

R_I

22k

V_P OW_POS

V_P OW_NEG

V_FUNC1

L17.3nC0

63RM

16.5mLM

1.6nCM

R_MEASURE

V_FUNC2

C_BYP _NEG

C_BYP _POS

VL

Electrostatically T unableMultiferroic Inductor

77

a. function generator.

b. DC constant current source.

2. An oscilloscope is connected across the VL node.

3. The power amplifier is connected to the PZT control input.

4. The two power supplies are connected to the +Vs and -Vs inputs of the power

amplifier.

5. A second function generator is connected to the input of the power amplifier.

6. Power is applied to each device.

7. The inductor driving source is set to

a. drive a 1 V sinusoidal signal at 5 MHz into the resistor-inductor

combination

b. drive a 100 mA constant current through the resistor-inductor combination

8. The second function generator is set to drive a 4.5 V square wave at 0.5 Hz into

the power amplifier (to achieve a 90 V peak-to-peak output voltage into the PZT).

9. The oscilloscope is set to trigger on the power amplifier voltage output and the

voltage at VL is recorded by the oscilloscope.

78

4 RESULTS

In chapter 3, two methods for determining the tuning response time were

discussed. One involved driving the inductor with a sinusoidal voltage and the other

involved supplying a constant DC current source. Ultimately, it was theorized that

observing the inductor voltage (and flux rate-of-change) will indicate the tunable inductor

tuning speed. Integrating to obtain the inductor flux is left up to the user.

Two multiferroic inductors are tested in this chapter to verify the analytical results

of the previous chapter. The first device, pictured in Figure 4.1, is fabricated with a two-

layer PZT and bonded Metglas core (bimorph [34]).

Figure 4.1: PZT Sample 1

79

The second is comprised of a single PZT sheet bonded to a Metglas core. The

device is pictured in Figure 4.2 (damaged due to overheating).

Figure 4.2: PZT Sample 2

Results obtained from both the sinusoidal and constant current methods are

illustrated, as well as additional measured data to attempt further insight into the tuning

speed dynamics.

80

4.1 Sinusoidal Method

Figure 4.3 is a plot of the inductor voltage in response to a 90 V peak-to-peak

input step collected through the measurement procedure described in the previous

section. A sinusoidal voltage is applied to the inductor at a frequency of 5 MHz and

amplitude of 1 V peak-to-peak.

Figure 4.3: Measured Inductor Voltage Step Response

The step voltage triggers the PZT/Metglas dynamic response at 0 seconds, which

influences the voltage across the inductor terminals to respond in a second-order fashion

as predicted by the chapter 3 analyses and Figure 4.3. The envelope of the sinusoidal

signal indicates that the core permeability takes several hundred microseconds to

completely settle.

81

To confirm the inductor tuning response, the current flow into the PZT is plotted

in Figure 4.4. Although noisy, the general waveform is clear.

Figure 4.4: Measured PZT Current Response

The plot shows a current peak of approximately one ampere that decays over a period of

roughly 300 µs before dropping into the noise floor.

Another interesting response is the sinusoidal voltage response to a step voltage

applied to the PZT with opposite polarity. This data is presented in Figure 4.5.

82

Figure 4.5: Measured Inductor Voltage Step Response (Reverse Polarity)

Although the inductor has clearly changed its inductance as indicated by the voltage

envelope, the response is not quite the same as in Figure 4.3. Reversing the polarity of the

step input has some effect on the inductor’s tuning response. In this case, the tuning

speed has increased by a few hundred microseconds. Figure 4.6 shows the current flow

into the PZT for the reverse polarity step voltage.

83

Figure 4.6: Measured PZT Current Response

4.2 Constant Current Method

To measure data with the constant current method, the circuit was connected as

described in Figure 3.15. The primary difference is the connection of a constant current

source instead of a function generator (labeled V_FUNC2 in the figure). Once the change

is made, the steps described in chapter 3.3 can be followed.

The first multiferroic inductor sample tested with the sinusoidal method is tested

again using the constant current method. Figure 4.7 plots the measured data for a 90 V

step input applied to the PZT control input.

84

Figure 4.7: Measured Inductor Voltage Step Response

Before time zero, the inductor voltage sits at 0 V in an inert state. A magnetic flux bias

has been applied by the 100 mA constant current, but the flux does not change over time

due to current. Once the step is applied to the PZT voltage at time 0, the voltage across

the inductor responds to the changing permeability of the inductor. Similar to the

envelope response of Figure 4.3, the inductor takes several hundred microseconds to

settle to the newly tuned value.

Figure 4.8 plots the response for the same inductor but with the input step applied

with opposite polarity.

85

Figure 4.8: Measured Inductor Voltage Step Response (Reverse Polarity)

Consistent with Figure 4.5, the tuning speed is faster with the PZT input voltage polarity

reversed.

The second sample multiferroic inductor was tested with the constant current

method. The device is slightly smaller and features half the PZT capacitance, as well as a

single sheet of PZT. Figures 4.9 and 4.10 plot the collected data for both the inductor

voltage response and the current flow into the PZT. Featuring different physical

properties, the inductor pictured in Figure 4.2 changes its inductance in less time than the

inductor in Figure 4.1. In addition to the faster tuning speed, the voltage across the

inductor terminals peaks at a higher value; in this case, around 70 mV.

86

Figure 4.9: Measured Inductor Voltage Step Response

Figure 4.10: Measured PZT Current Response

87

5 DISCUSSION

The following section discusses the measured results obtained from experimental

methods conceived in the previous chapter. Specifically, the tuning response data for two

sample electrostatically tunable multiferroic inductors comprised of a PZT/Metglas core

and 26 AWG coil are discussed. Improvements to the work presented in this thesis are

also recommended.

5.1 Tuning Speed

The first tested device achieved a tuning speed on the order of several hundred

microseconds as shown in Figures 4.3 and 4.7. The results are improved over the data

collected by Chen, et al. [18], as the slow reported tuning speed can be attributed to the

reduced slew-rate and current capacity of the power amplifier referenced. In any case, the

tuning speed is still slower than desired; however, as another research group [20] has

seen multiferroic switching speed on the order of a few microseconds. The discrepancy

can be attributed to the physical properties of the devices tested as a second tunable

inductor achieved tuning speed on the order of several microseconds (Figure 4.9).

As discussed, the first tested device was constructed of with two-layers of PZT

sheets, forming a bimorph bending-type device. These devices are noted for their reduced

resonances over single-layer PZT sheets [34]. Generally speaking, open-loop PZT can

typically achieve nominal displacement in approximately 1/3 of the lowest resonant mode

period, or

𝑡0 ≈1

3 ∙ 𝑓0 (5.1)

88

albeit with strong overshoot [34]. With resonances in the low kHz region, this implies

settling times of tens to hundreds of microseconds (or more) for PZT bimorph devices.

These estimates are consistent with the measured results in Figures 4.3 and 4.7 for tested

two-layer device.

According to the PZT impedance plot in 3.17, the single-layer PZT has a low

resonant mode near 60 kHz, implying a nominal tuning speed of

13∙

160000

≈ 5.6 𝜇𝑠

According to Figure 4.9, the tuning response is beginning to decay rapidly after

approximately 5 µs which corresponds nicely to the nominal speed predicated by

equation 5.1.

Regarding the response of Figure 4.5, one possible explanation for the increased

tuning speed is the saturation point of the Metglas ribbons [3]. Once the negative step

voltage is applied, the permeability may be saturating and causing a decrease in the

change in inductance with respect to time. Referring to Figure 2.4, the permeability is

noted to decrease as applied voltage is increased. Further, the range of available

permeability values is large before saturation occurs. However, as negative (reverse

polarity) voltage is applied, the data is not depicted. It is possible that the permeability

range available in the negative voltage direction is not as large. In other words, given the

permeability of the device with 0 V applied, the range of permeability values may not be

symmetric for positive and negative applied voltages. This would suggest that the

permeability offset in equation 3.43 has a negative skew and is close to the saturation

region. Therefore, the behavior could be a possible non-linear effect.

89

Generally, as discussed by Chen, et al. [18], the Metglas response time is on the

order of one microsecond, so the majority of the lower frequency dynamics are

contributed by the PZT. If a PZT is proven to achieve switching speed in the low

microsecond range, a multiferroic inductor using that same PZT will likely achieve

similar tuning speed based on analysis presented in this document. The experimental data

contributes to this idea as well, as both current response in Figures 4.4 and 4.10 mimic

their inductor tuning response settling times.

5.2 Measurement Techniques

In chapter 4, two measurement techniques were used on the sample multiferroic

tunable inductors. The first involved driving a sinusoidal voltage into a series a resistor-

inductor combination and examining the envelope of the sinusoid across the tunable

inductor terminals as it is affected by changing inductance. The second method required a

constant current source that sets up a magnetic bias in the inductor. Then, when a step

voltage is applied to the PZT, the voltage at the inductor terminals reflects the changing

inductance via changing magnetic flux.

As a result of the experiments, the constant current method is preferred. The

principal reason is the resolution at which the dynamics of the PZT/Meglas core can be

visualized. For example, Figure 4.7 not only displays the lower frequency resonant mode

described by the voltage envelope in Figure 4.3, but a second resonance is seen at a

higher frequency. This second resonance was expected as a result of the load

characterization in Figure 3.17, but the sinusoidal measurement technique obscures the

resonance as seen in Figure 4.3. Further, the constant current method requires a simpler

90

setup. The measurement resistor in Figure 3.15 is not required for the test, and only a DC

source capable of providing a constant current is necessary.

One drawback with the constant current method is the required resolution of the

measurement system. Collecting data with a standard digital oscilloscope limits the

precision to the selected voltage range which, in the case of Figures 4.7 and 4.8, is

approximately 0.4 mV 5. Evident in the aforementioned figures is the lower resonant

mode (which is clearly visible in Figure 4.3) falling below the noise floor. Performing

any sort of numerical integration to compute magnetic flux or static/dynamic inductances

over time proves difficult due to the noisy data.

Theoretically, however, the constant current method is expected to produce lower

measurement voltages. This can be illustrated by a Laplace domain representation of

equation 3.52. When a constant current is applied to the multiferroic inductor, equation

3.52 for dynamic inductance simplifies to

𝑣𝐿(𝑡) = 𝐿𝑑𝜇(𝑡) ∙𝑑𝜇𝑑𝑡

+ 𝐿𝑑𝑖(𝑡) ∙ 0 = 𝐿𝑑𝜇(𝑡) ∙𝑑𝜇𝑑𝑡

(5.2)

Performing a Laplace transform yields

𝑉𝐿(𝑠) = 𝐿𝑑𝜇(𝑠) ∙ 𝑠 ∙ 𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠) (5.3)

assuming no initial conditions. Dividing by the Laplace-domain permeability expression

produces

𝑉𝐿(𝑠)

𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝑠)= 𝑠 ∙ 𝐿𝑑𝜇(𝑠) (5.4)

and in the frequency domain

5 80 mV/ 8 bits of resolution = 0.3125 mV

91

𝑉𝐿(𝜔)

𝑀𝑀𝑒𝑡𝑔𝑙𝑎𝑠(𝜔)= 𝑗 ∙ 𝜔 ∙ 𝐿𝑑𝜇(𝜔) (5.5)

The expression in equation 5.5 is analogous to the frequency domain representation of

LTI inductor impedance. For this application, equation 5.5 states that the output voltage,

VL, will have its amplitude dictated by the frequency of the driving source, or the Metglas

permeability. Larger frequencies will produce larger changes in the dynamic inductance,

and this will be represented by larger voltages. Since the Metglas frequency domain

response is dictated by the PZT, the frequencies in which equation 5.5 is subjected to are

typically less than 100 kHz as evidenced by Figure 2.7.

In the sinusoidal case, changes in inductance can be amplified by an order of

magnitude over the constant current method. This is evident by the frequency-domain

impedance expression for inductor impedance [8], or

𝑉𝐿(𝜔)𝐼𝐿(𝜔)

= 𝑗 ∙ 𝜔 ∙ 𝐿 (5.6)

If 𝐼𝐿 is a sinusoidal current source operating at 𝜔, any changes in 𝐿 will be gained by a

factor dependent on the frequency and be reproduced in the voltage across the inductor

terminals. In summary, the constant current method will typically produce smaller

amplitude voltages across the inductor terminals due the frequency of the driving source,

or the PZT/Metglas dynamic response.

To remedy the quality of the data collected and make the gain factor of equation

5.6 a moot point, a high-impedance instrumentation amplifier can be used across the

inductor rather than an oscilloscope. The amplifier offers the added benefit of reduced

input capacitance over the oscilloscope probes (which are upward of 130 pF). The

resulting data will feature a much lower noise floor and be more conducive to numerical

92

processing. In other words, the tuning speed may converted to units of magnetic flux,

static inductance, or dynamic inductance.

5.3 Future Work

To continue the research in this thesis, the methods for characterizing the tuning

speed of a multiferroic inductor should be applied to a large quantity of devices and

verified against traditional piezoelectric and Metglas characterization methods. The

quality of measurement should be improved through the use of instrumentation amplifiers

and sensitive measurement devices. Further, the fabrication of the multiferroic tunable

inductors will hopefully improve over time, thus leading to consistent measured data for

comparison.

To integrate this work into applied systems research, investigation must be done

into the control of the PZT/Metglas dynamic response. Driving step voltages to achieve

fast tuning speed causes the PZT to respond with overshoot, ringing, and prolonged

settling time. Many methods of control exist and are beyond the scope of this thesis.

However, a proposed block level diagram is presented in Figure 5.1 as a starting point.

93

Figure 5.1: High-Level Control Block Diagram

If the tunable inductor were to be used in a VCO-type application, a third method of

control could be implemented based on measurement of the oscillator frequency. A

known relationship between applied tuning voltages and expected output frequency

would be prerequisite. After the applied tuning voltage, as the oscillator frequency varies

due the PZT/Metglas dynamics, a control circuit may attempt to correct the erroneous

output based on the ideal voltage and frequency transfer function.

RSenseP

RSenseN

U1

U2

POS

NEG

INP

INN

OUTP

OUTN

DACProcessor (FPGA/uC)

IN+

IN-

Input Piezo

IN+

IN-

VoltageSampling

CurrentSampling

ADC

ADC

Digital Domain

PowerAmplifier

CurrentController(FPGA/uC)

VoltageController(FPGA/uC)

INP

INN

OUTP

OUTN

U3

Analog Domain

U4

94

6 CONCLUSION

A study on the transient analysis of an electrostatically tunable multiferroic

inductor has been presented. Throughout the discussion, the inductor’s importance to

current high-speed applications was established through a review of the state of current

research, a developed methodology for determining the tuning response speed,

experimental data collected in the laboratory, and a discussion highlighting the concerns

with the presented work. In doing so, the electrostatically tunable inductor was shown to

be suitable for integration into RF applications. Although the tuning speed is device

dependent, multiferroic inductors can be fabricated that approach tuning speeds on the

order of just a few microseconds, rivaling competing technologies such as RF MEMS

tunable inductors [4][6][9][11].

As the fabrication of electrostatically tunable inductors continues to improve, the

tuning speeds should continue to increase, further establishing the inductor as a leading

technology for RF applications. The work presented here can potentially contribute to the

continued research, design, and development of novel tunable devices.

95

7 REFERENCES

[1] C.-W. Nan, M. Bichurin, D. Shuxiang, D. Viehland, G. Srinivasan,

“Multiferroic magnetoelectric composites: Historical perspective, status, and

future directions,” J. of Appl. Physics, vol. 103, no. 3, pp. 031101-35, Feb. 2008.

[2] A. Tatarenko, V. Gheevarughese, G. Srinivasan, “Magnetoelectric microwave

bandpass filter,” IEEE Electron. Lett., vol. 42, no. 9, pp. 540-541, Apr. 2006.

[3] J. Lou, D. Reed, M. Liu, N. Sun, “Electrostatically tunable inductors with

multiferroic composites,” Appl. Physics Lett., vol. 94, no. 11, pp. 112508, Mar.

2009.

[4] G. Rebeiz, K. Entesari, I. Reines, S.-J. Park, M. El-Tanani, A. Grichener, A.

Brown, “Tuning in to RF MEMS,” IEEE Microwave, vol. 10, no. 6, pp. 55-

72, Oct. 2009.

[5] J. Kim, J. Choi, K.-H. Lee, F. Bien, K. Lim, C.-H. Lee, “Wideband CMOS

voltage-controlled oscillator using tunable inductors,” IEEE Electron. Lett., vol.

46, no. 20, pp. 1391-1393, Sep. 2010.

[6] C. Pavageau, O. Millet, A. Cao, “An approach of rf-mems technology platform

for multi-band multi-mode handsets,” DelfMEMS, Villeneuve-d'Ascq, France,

2011.

[7] A. Hussaini, R. Abd-Alhameed, J. Rodriguez, “Tunable RF filters: Survey and

beyond,” in 2011 18th IEEE Int. Conf. on Electron., Circuits and Systems

(ICECS), pp. 512-515, Dec. 2011.

96

[8] J. Nilsson, S. Riedel, Electric Circuits, 5th ed. Reading, MA: Addison-

Wesley, 2004.

[9] S. Yonghyun, W. Zhengzheng, M. Rais-Zadeh, “A high-performance

continuously tunable MEMS bandpass filter at 1 GHz,” IEEE Trans. Microw.

Theory Tech., vol. 60, no. 8, pp. 2439-2447, Aug. 2012.

[10] C. Leifso, J. Haslett, “A fully integrated active inductor with independent voltage

tunable inductance and series-loss resistance,” IEEE Trans. Microw. Theory

Tech., vol. 49, no. 4, pp. 671-676, Apr. 2001.

[11] G. Rebeiz, RF MEMS: Theory, Design, and Technology. Hoboken, NJ: J. Wiley,

2003.

[12] N. Sun and G. Srinivasan, “Voltage control of magnetism in multiferroic

heterostructures and devices,” SPIN, vol. 02, pp. 1240004, 2012.

[13] T. Carroll, M. Todd, F. Rachford, L. Pecora, “Spatiotemporal nonlinear dynamics

of a magnetoelastic ribbon”, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., vol. 63,

no. 5 pt. 2, pp. 056205, 2001.

[14] IEEE standard on piezoelectricity, ANSI/IEEE Standard 176-1987, 1988.

[15] J. Kraus, and D. Fleisch, “Dielectric and magnetic materials,” in

Electromagnetics with Applications. 5th ed. Boston, MA: WCB/McGraw-Hill,

1999, ch. 7, 441-442.

[16] F. Hyde, D. Tucker, “Parametric amplifiers: Static and dynamic inductance

and capacitance and their significance in the non-linear and time-varying

approaches,” IEEE Radio and Electronic Engineer, vol. 25, no. 4, pp. 353-355,

Apr. 1963.

97

[17] Y. Chen, A. Geiler, T. Fitchorov, C. Vittoria, V. Harris, “Time domain

analyses of the converse magnetoelectric effect in a multiferroic metallic glass-

relaxor ferroelectric heterostructure,” Appl. Physics Lett., vol. 95, no. 18, pp.

182501-3, Nov. 2009.

[18] Y. Chen, T. Fitchorov, A. Geiler, J. Gao, C. Vittoria, V. Harris, ”Dynamic

response of converse magnetoelectric effect in a PMN-PT-based multiferroic

heterostructure,” Appl. Physics A, vol. 100, no. 4, pp. 1149-1155, 2010.

[19] A. Ustinov and G. Srinivasan, “Speed of a ferrite-ferroelectric microwave planar

resonator,” Tech. Physics, vol. 55, no. 6, pp. 900-903, 2010.

[20] A. Ustinov, Y. Fetisov, S. Lebedev, G. Srinivasan, “Electric switching in

bistable ferrite-piezoelectric microwave resonator,” Tech. Physics Lett., vol. 36,

no. 2, pp. 166-169, 2010.

[21] M. Guan, W.-H. Liao, “Studies on the circuit models of piezoelectric ceramics,”

Int. Conf. on Information Acquisition, 2004. Proc., pp. 26- 31, Jun. 2004.

[22] S. Robinson, “Driving piezoelectric actuators,” Power Electron. Tech., pp. 40-44,

Apr. 2006.

[23] L. Jin, X. Yao, X. Wei, Z. Xi, “Observation of piezoelectric resonance in time

domain transient current of ferroelectric ceramics and crystals,” Appl. Physics

Lett., vol. 87, no. 7, pp. 072910-3, Aug. 2005.

[24] D. Biolek, Z. Kolka, V. Biolkova, “Modeling time-varying storage components in

PSpice,” Dept. of EE, FMT, University of Defence Brno, Dept. of

Microelectronics/Radioelectronics, FEEC, Brno University of Technology, Brno,

Czech Republic, 2007.

98

[25] G. Lewis, W. Olbricht, “Design and characterization of a high-power

ultrasound driver with ultralow-output impedance,” Review of Scientific

Instruments, vol. 80, no. 11, pp. 114704-8, Nov. 2009.

[26] J. Borg, J. Johansson, “An ultrasonic transducer interface IC with integrated push-

pull 40 Vpp, 400 mA current output, 8-bit DAC and integrated HV multiplexer,”

IEEE J. Solid-State Circuits, vol. 46, no. 2, pp. 475-484, Feb. 2011.

[27] C. Wallenhauer, B. Gottlieb, R. Zeichfusl, A. Kappel, “Efficiency-improved

high-voltage analog power amplifier for driving piezoelectric actuators,” IEEE

Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 1, pp. 291-298, Jan. 2010.

[28] A. Fleming, “A megahertz bandwidth dual amplifier for driving piezoelectric

actuators and other highly capacitive loads,” Review of Scientific Instruments, vol.

80, no. 10, pp. 104701-7, Oct. 2009.

[29] M. Colclough, “A fast high-voltage amplifier for driving piezoelectric

positioners Review of Scientific Instruments, vol. 71, no. 11, pp. 4323-4324, Nov.

2000.

[30] P. Gray, P. Hurst, S. Lewis, R. Meyer, Analysis and design of analog

integrated circuits. 4th ed. New York: Wiley, 2001.

[31] LTspice software manual, Linear Technology Corporation, Milpitas, CA, 2013.

Available: http://ltspice.linear.com/software/LTspiceIV.exe

[32] J. Phillips, “Piezoelectric technology primer,” CTS Wireless Components,

Albuquerque, NM, 2000.

[33] G. Franklin, J. Powell, A. Emami-Naeini, Feedback control of dynamic

systems, 5th ed. Upper Saddle River, NJ: Pearson, 2005.

99

[34] Piezoelectric actuators: Components, technologies, operation. Physik

Instrumente Ceramic, Lederhose, Germany, 2012.

[35] E. Weisstein. Square wave [Online].

Available: http://mathworld.wolfram.com/SquareWave.html

[36] A. Sedra and K. Smith, Microelectronic circuits, 5th ed. New York: Oxford

University Press, 2003.

[37] B. Cordell, Designing audio power amplifiers, 1st ed. New York, NY: McGraw-

Hill, 2011.

[38] A. Safari and E. Akdogan, Piezoelectric and acoustic materials for transducer

applications. Springer, 2008.

[39] MP111 power operational amplifier, MP111U REVF, Apex Microtechnology,

Tucson, AZ, 2012.

100

APPENDIX A (Custom amplifier schematic on following page)

101

Figure A.1: Custom Amplifier Design Schematic

V1

50

V2

-50

V3

1

I1

PULSE(1e-3 7e-3 1e-6 1e-6 1e-6 4e-6 10e-6)

R5

100

V4

-1

R3

200k

R8

50

R9

4.8kI2

PULSE(7e-3 1e-3 1e-6 1e-6 1e-6 4e-6 10e-6)

R1

200k

R2

100

R7

50

R10

4.8k Cm1

1.6e-9

Lm1

16.5e-3

Rm1

63

Rp1

7.5e6

Co1

17.3e-9

Rs1

40R4

14k

R6

100k

R11

200

C1

12p

C2

12p

I84e-3

AC 3e-3

I94e-3

AC -3e-3

R12

2.35k

R13

1.25k

Q3

PNP

Q7

PNP

Q33

PNP

Q34

PNP

Q10

PNP

Q14

PNP

Q16

PNP

Q24

NPN

Q25

NPN

Q1

NPN

Q2

NPN

Q4

NPN

Q5

NPN

Q23

NPN

Q26

NPN

Q27

NPN

Q28

NPN

Q29

NPN

Q6

PNP

Q9

PNP

Q30

NPN

Q32

NPN

Q11

NPN

Q12

NPN

Q13

NPN

Q15

NPN

Q17

NPN

Q18

PNP

Q20

PNP

Q22

PNP

Q31

PNP

Q35

PNP

Q8

NPN

Q19

NPN

Q21

NPN

R14

10k

R15

10k

E11e6

C3

1µ

R19

1e6

Q36

NPN

Q37

NPN

VCC

VEE

VD

ACL

IM-

VD

ACL

IM+

VD

ACL

IM+

VCC

VD

ACL

IM-

VEE

VCC

OU

T-

VEE

OU

T+

VEE

VCC

VCC

VCC

VEE

VCC

VEE

VCC

VEE

VEE

VEE

npnMirrorBase

VEE

VCC

VCC

inputOUT-inputOUT+

inputOUT-inputOUT+

VEE

VCC

VCC

VCC

VEE

VCC

VEE

VCC

VEE

VEE

VEE

VEE

npnMirrorBasenpnMirrorBase

npnMirrorBase

npnMirrorBase

npnMirrorBase

pnpMirrorBase

pnpMirrorBase

pnpMirrorBase

VCC

OUTCM

VCC

VCC

OU

TCM

.tran 0 1000e-6 0 1e-6

;ac oct 1e3 1 1e9

TI DAC900 producing 100 kHz differential square wave

Current Mirrors

Sources