5. Linear and Nonlinear (Time-Invariant)Electrical Elements

5.1 Introduction to the chapter.

In this chapter, we discuss time-invariant (TI) linear and nonlinear electricalelements that are the building blocks for electrical circuits. In section 5.2, weprovide the definitions for ideal energy sources and discuss the implications ofsuch assumptions from a real-world perspective. Next, in section 5.3, we discussvarious electrical elements. The two key quantities, power and energy, are usedto define the various elements. The discussion leads to passive elements orelements that consume power (lossy elements) or store power as energy andreturn back later with no net loss of energy (lossless elements), and activeelements that generate energy. The energy storage property leads to what isknown as devices with memory. We first consider one-port and multi-port linearand nonlinear memoryless devices and then move on to devices with memory.These elements or building blocks can be used to form complex circuits anddynamics as we discussed before and as we will see in later chapters.

Though the discussions in this chapter are on mostly TI passive devices, wealso discuss specific interconnection of some of these passive devices withindependent and or dependent sources that lead to complex one port circuits thatexhibit the passivity property. Such one port circuits and their models can beused to design nonlinear controllers for plants that are unstable.

5.2 Basic Concepts: Electrical Source, Power, andEnergy

Energy, and energy generation, consumption, and or storage capability ofphysical devices are very fundamental concepts that play very important role notonly in electrical engineering but in all other areas as well. In this section, welook at the basic definition of an electrical source capable generating electricalenergy and the related terms such as electrical potential, power etc. and the inter-relationship among them. These basic quantities will then be used to definevarious electrical elements.

An electrical source is a physical device with two terminals for connection toother electrical elements and sources. An ideal source provides a voltage acrossits terminals (or fixed current through its terminals) that is independent of thecurrent drawn from it (voltage that exists across its terminals). Constant-valued-and periodic- (sinusoidal) waveforms are the common waveforms for idealelectrical sources (Fig. 5.1) The current (voltage) supplied by an ideal voltagesource (current source) depends only upon the electrical elements or loadconnected to the source. The instantaneous power ps (t) supplied by a source

(with the voltage v s(t) polarity and current i s (t) direction as shown in the

figure) is given by:

ps (t) = v s(t)i s (t) (5.1)

and the energy E s (t0 ) delivered by the source up to time to is given by:

E s (t0 ) = ps (t)dt−∞

t 0

∫ (5.2)

As the current supplied (voltage supplied) by an ideal voltage source (currentsource) is dependent on the load connected to the source and not the source, the

(b2)(a1) (a2) (b1)

t

vs(t) = sin[ωt]; ω =1rad / sec

is (t)

+

-

v s (t)

v s (t)

v0

v s (t)

i s (t)+

-

0 4 8-1

-0.5

0

0.5

1

t

Figure 5-1. a1) & a2) Ideal voltage & current source; b1) & b2) Commonwaveforms for the ideal electrical sources. We assume the power capacity of thesedevices as infinite implying that the power delivered (for a given value of thesource) is solely determined by the elements connected to the sources and not by thesources.

instantaneous power supplied, and the energy supplied up to certain time can bepositive, zero, or negative. By placing certain constraints on the power drawnfrom the source by the load, and the energy delivered by the source to the load,we will be able to arrive at useful electrical network elements as we will see inthe next section. Further, the assumption that the supplied current in the case ofa voltage. source (or voltage, in the case of a current source) is solelydetermined by the load, implies that an ideal source is capable of supplying hugeamounts of power and infinite amount of energy. This, of course, is not feasiblein the real world. We will look into physically meaningful models for electricalsources after we define the various electrical elements.

5.3 Linear and Nonlinear (Time-Invariant) ElectricalElements

5.3.1 Passive (Lossless and Lossy) and Active Elements

The simplest form of an electrical element is one with two terminals (one-portelement) as shown in figure 5.2a as a black box. In the figure, we have assumeda certain polarity for the voltage v(t) and for the direction of flow of the currenti(t) for the elements that is different from the ones for independent sources. Wecan also have multi-terminal (multi-port) elements with associated voltages andcurrents as defined in Figures 5.2b and c. Such one-port and M-port elementscan be interconnected to form a M-port (M > 1) circuit or network. Theinstantaneous power p(t) delivered to an M-port element or network is given

by:

p(t) = v k (t)ik (t)k=1

M

∑

= p k (t)k=1

M

∑(5.3)

where pk (t) is the power entering through the k-th port. The energy delivered to

the M-port element at time t 0 is given by:

E(t0 ) = p k (t)dt−∞

t0

∫k=1

M

∑ (5.4)

If we consider the expression for power and energy without linking them toany physical device or circuit, we may argue that the instantaneous power and/or

the energy can be positive, zero of negative. However, we end up with arestricted set for instantaneous power and energy as we introduce the notion ofphysical realizability. Three important cases are:

1) p(t) ≥ 0 (and hence E(t) ≥ 0) for all t and all possible

v(t), i(t) waveforms;

2) p(t) = 0 (and hence E(t) = 0) for all t and all possible

v(t), i(t) waveforms;

3) E(∞) = p(τ)dττ=−∞

∞

∫ = 0 for all square integrable waveforms

v(t), i(t)

(5.5)

The first condition implies that we have a device or circuit that consumes

An ElectricalElement

(a two-terminaldevice)

+

v(t)

-

i(t)

(b)(a)

A Multi-portElectricalElement

(with common ground)

+

-

+

+

v1(t)

v 2 (t)

vM(t)

1

M

M+1

A GeneralMulti-portElectricalElement

+

-

+

-

+

-

1

M

2

vM

(t)

v2 (t)

v1(t)

i1(t)

i2 (t)

iM (t)

(c)M'

2'

1'

Figure 5-2. Block diagrammatic representation of electrical devices. a) One-portdevice; b) Multi-port device with one common negative terminal; c) Multi-portdevice with no common terminals.

power all the time leading to continuous energy consumption. From the law ofenergy preservation, we can infer that the electrical energy that has beenconsumed by the device has been transformed into another form of energy (suchas heat). The second possibility implies that the device consumes no power atany time. In the case of a one-port element this leads to the trivial cases of open-circuit and short-circuit elements. However, a number of multi-port devicesobeying this condition are possible. Here, the condition that power consumed iszero all the time implies that finite, non-zero valued power may be entering anyone (or more) port(s) of the multi-port device, and the same amount of powerflows out through the other ports of the device. The third category correspondsto the case where the power flowing into the device is stored in the device asenergy and returned back with net zero energy absorption. The elementsbelonging to these three categories are known as passive elements. The firstcategory represents lossy memoryless elements, the second category representslossless memoryless devices, and the third category leads to lossless elementswith memory as we will see in the following pages. Elements which are notpassive are known as active elements.

5.3.2 One-Port Memoryless Devices

5.3.2.1 Passive, Active, linear and nonlinear resistors

The condition that a two-terminal element be memoryless can be satisfied ifa functional relationship between its voltage and current v(t), i(t) :

f[v,i] = 0 for all t (5.6)

exists. That is, depending upon the properties of f[.], we should be able to findone response parameter (v(t) or i(t) ) at any given time t by a knowledge of the

other (source) parameter value at that time instant. It is a memoryless devicesince its response at any given time depends on the source parameter value atthat time instance only and not at the source parameter values that might haveexisted before. An element with a voltage-current relationship as given byequation (5.6) where the condition that p(t) ≥ 0 for all t may or may not be

satisfied is known as a resistor.We can note that p(t) ≥ 0 for all time t can be satisfied if the function f[v, i]

leads to a v-i characteristics that is confined to the first- and third-quadrants inthe v-i plane and pass through the origin. Since such a resistor consumes powerall the time, it can be called a lossy or passive resistor. We can have a number ofv-i characteristics that satisfy these requirements. Let us look at some of thepossibilities.

The simplest category is a functional relationship of the form:

f[v R , i R ] = vR − Ri R ; R> 0

= R[iR − GvR ] ; G = R −1

= 0

or vR(t) = G i R (t)

or iR(t) = Rv R (t)

= pk (t)k =1

M

∑

(5.7a)

This corresponds to a straight line in the v-i (i-v) plane with a slope of R(G)(Fig. 5.3). A device having such a property is known as a linear resistor whereR is the resistance (G the conductance). The two different equations in (5.7a)simply indicate that neither one of the two variables, vR(t), iR (t) can be the

independent variable with the other the dependent variable. The resistorequation expressed in the discrete form when the current is the independentvariable and the voltage the dependent variable is:

vR(nT) = R i R (nT) (5.7b)

where T represents the sampling interval. The equation when the voltage is theindependent variable and the current the dependent variable is:

i R (nT) = G vR(nT) (5.7c)

The symbol for a linear resistor is also shown in Fig. 5.3. It can be noted thatfor a linear resistor, f[-v, -i] = -f[v, i] (anti-metric characteristic). This propertysimply implies that the two terminals of a linear resistor can be interchangedwithout affecting the characteristics of a circuit in which it is embedded. Anelement having such a characteristic is known as a bilateral element.

The power consumed by a linear resistor is given by:

pR(t) = vR (t)i

R(t)

= Gv R2 (t)

= R i R2 (t)

(5.7d)

That is, the power consumed by the linear resistor increases as the second powerof the applied source amplitude. In practice, we can expect the device to startdrawing more current (voltage) if the voltage (current) applied exceeds somemaximum value (linear relationship no longer holds) and eventually burn out.Hence the concept of a linear resistor is an idealization of the real characteristicsof a physical device to make modeling and or analysis simpler.

When the value of R is negative, the element starts supplying power orbecomes an active element. Such a device is called a negative resistor.

When the v-i characteristics of a resistor is not linear, we end up with anonlinear resistor. The general form of the v-i characteristics of a nonlinearresistor is as shown in Fig. 5.4. In the figure, we have indicated the possibilitiesthat:

1) the characteristics and hence the device may not be bilateral;2) the mapping from the input variable to the output variable may

be one-to-many or many-to-one. The former implies awaveform which has infinite slope (or is discontinuous). Thelater implies that the slope might be zero over a range of valuesof the input variables and changes sign often.

Since the device may not be bilateral, the symbol for the nonlinear resistorshould indicate this property as shown in the fig. 5.4b. From a practical

perspective, lack of bilateral property implies that we cannot interchange theterminals of a nonlinear resistor without affecting the behavior of the circuit inwhich the nonlinear resistor is embedded. Of course, in situations where wehave the choice of selection, we may ask if the non-bilateral property is reallynecessary and worth the trouble.

Similar to the definition of resistance and conductance for a linear resistor,we may call R[iR ] = ∂vR ∂i R G[v R] = ∂iR ∂vR{ } as the small-scale resistance

{conductance} of a nonlinear resistor which may be positive, zero, or negative(Fig. 5.4c) with out making the resistor non-passive.. The one-to-many mappingis not desirable in practical devices (and perhaps not possible to realize) since itwill only lead to confusion. The many to one mapping implies that thecharacteristic is non-invertible or from a practical point of view, implies that the

iR

vR

iRmax

vRmax

iR

vR

slope = R

vR

(t)

+

-

iR (t)

iR

vR

slope = G

(a) (b)

(c) (d)

Figure 5-3. a) Symbol of a linear resistor; b) I/O functional relationship of alinear resistor driven by a voltage source; c) Functional relationship whendriven by a current source; d) The characteristic of a non-ideal resistor. vR (or iR )

∂vR

∂iR

(or∂iR

∂vR

)

(c)

(a)

(b)

+

-

iR (t)

vR

(t)

vR (or iR )

iR (or vR )

Figure 5-4. a) Symbol of a nonlinear resistor; b) Typical v-i characteristic ofa nonlinear resistor; c) Small-scale resistance (or conductance) of a nonlinearresistor.

use of the element should be restricted such that the independent variable anddependent variable designation is preserved. Some circuit examples consistingof nonlinear resistors along with other linear elements (yet to be defined) areshown in fig. 5.5. and the kind of nonlinear resistors to be used for proper circuitbehavior are indicated.

The slope becoming zero (over a region of the independent variable) orinfinite may be mathematically possible but not possible in practice, at least inanalog systems. Hence we can omit these two possibilities and assume that thewaveform is continuous with finite first-order and second-order derivatives. Theslope becoming negative (leading to many to one mapping) implies that thedependent variable decreases (increases) when the independent variableincreases (decreases) in value. Such a characteristic may lead to exotic circuitbehavior when used with similar complex elements in applications such asneural nets. However, we may omit such characteristics in engineering designapplications such as control where we invariably go for "conservative design".

In summary, a current controlled nonlinear resistor will be represented as:

vR(t)= vR i R(t)[ ] (5.8a)

and a voltage controlled nonlinear resistor by:

i R(t)= i R vR(t)[ ] (5.8b)

with the corresponding discrete models as:

vR(nT)= vR i R(nT)[ ] (5.8c)

and

i R(nT)= iR vR(nT)[ ] (5.8d)

where, for most practical purposes, we can restrict the nonlinear resistorcharacteristics to be anti-metric with respect to the origin of the v-i plane (orbilateral element), continuous and monotonously increasing. Further relaxationcan be made if and when necessary.

5.3.2.1.1 Non-passive and negative (active) nonlinear resistors:

In the case of linear resistors, we have only two choices: a positive resistor(R and G > 0 ) or a negative resistor ( R and G < 0 ). In the case of nonlinear

resistors, we have a few more possibilities. In Fig. 5.6a. we show thecharacteristic of a passive, nonlinear resistor. In fig. 5.6b we show a resistor forwhich p(t) ≤ 0 for all values of the independent variable leading to a active or

negative nonlinear resistor, similar to the case of a negative linear resistor.Figures 5.6c to 5.6e show the other cases where we have nonlinear resistorcharacteristics with p(t) ≥ 0 (a passive or lossy nonlinear resistor) for certain

values of the independent variable and p(t) ≤ 0 (an active nonlinear resistor) for

other values of the independent variable. The behavior of circuits composed ofsuch elements and other elements will obviously be more complex as comparedto the response from circuits with linear elements only. Such elements can beused to model systems which are not passive always, to explain concepts such aschaos, and to design complex neural network architectures as we will see in laterchapters.

5.3.3 Multi-port Memoryless Devices

The condition that the power, p(t) = 0 for all time t can be satisfied in a non-trivial manner only in the case of multi-port devices. Let us first consider two-port devices with a block diagrammatic representation as shown in Fig. 5.7a. Wehave 4 parameters for a two-port-device of which two can be the independentvariables and the other two the dependent variables leading to two equations.For example, if v1 (t) and v 2 (t) are the independent variables, we can represent

the dependent variables i1 (t) and i 2 (t) as:

(a)

Current controllednonlinear resistor

+

-

iR (t)

vR(t)

+ -iL (t)+

-

v s (t)

is (t)

(b)

Voltage controllednonlinear resistor

+ -

vL(t)

+

-

v s (t)

is(t)

+

-

i c (t)

vc(t)

iR(t)

vR (t)

+

-

Figure 5-5. Interconnection of nonlinear resistors with other electrical elements;a) A nonlinear resistor connected in series with an inductor or a current sourcemust be current controlled; b) Voltage controlled nonlinear resistor in parallelwith a capacitor or a voltage source.

i1 (t) = y 1 v1 (t), v 2 (t)[ ]i 2 (t) = y 2 v1 (t), v 2 (t)[ ]

(5.9)

In addition to the choice of the independent and dependent variables givenby this example, there exist five more choices (Table 5.1). The condition thatp(t) = 0 can be satisfied by constraining the functions f1 •[ ] and f 2 •[ ] such that:

p(t) = v1 (t)i1 (t) + v2 (t)i 2 (t)

= 0(5.10)

is satisfied. Let us consider useful devices arising from the six possibilitiessubject to the above constraint.

iR

vR

iR

(b)

vR

iR

(c)(a)

vR

vR

iR

vR

iR

(d) (e)

Figure 5-6. Nonlinear resistors v-i characteristics; a) Fullypassive with monotone characteristic; b) Fully passive with non-monotonic characteristic; c) Fully active or a negative; d) Non-passive (active near the origin & passive when the magnitude ofthe independent variable is large); e) Another non-passive(passive near the origin & active when the magnitude of theindependent variable is large).

(a)

(b)

+

-

i1(t)

+

-

v2 (t)

i2 (t)1 : N

•

v1(t)

i1(t)

v2 (t)

i2 (t)

+

-

+

-

1

1'

2

2'

A Two-portDevice

v1(t)

RL

+

-

iR (t)+

-

i1(t) i2 (t)1 : N

•

vR (t)

+-

+

-

R s

v s (t)

(c)

v1(t)

+

-

v2 (t)ˆ R L ⇒

= RL

N2

Figure 5-7. a) Block diagrammatic representation of a 2-portdevice; b) Symbol for an ideal transformer; c) Resistancescaling property of an ideal transformer.

5.3.3.1 Transformer with constant turns ratio

An ideal transformer is one two-terminal element characterized by the followingtwo equations:

v 2 (t) = N v1 (t)

i1 (t) = −N i2 (t)(5.11)

or in a matrix form:

v2 (t)

i1 (t)

=

N 0

0 −N

v1 (t)

i 2 (t)

(5.12)

where N is a constant called the turn ratio. The effect of having a negative valuefor N is equivalent to changing the polarity and direction for v 2 (t) and i1 (t) .

Hence, we can restrict N to be positive and represent the resulting voltagepolarity by a dot in the symbol for transformer as shown in Fig. 5.7b.

The power p(t) consumed by a transformer is exactly equal to zero. Thus, thedevice neither dissipates nor generates energy and the power entering one portgets delivered to the load at the other port. Thus a transformer is a memorylessdevice and functions as power-transfer unit or a power broker. If we terminatethe second-port of a transformer by a resistive load R L as shown in fig. 5.7c, we

can show that:

v1 (t) =RL

N2 i1 (t) (5.13)

That is, the load seen by the source v s(t) connected to the first port is given by

z in = R L N2 and can be varied by varying N without changing the power

supplied to the actual load R L .

The equations given so far are for an ideal memoryless transformer1 and inpractice we can expect some loss in the device. However, we can use the idealtransformer as one basic building-block and represent the loss by a two-portnetwork formed by other building blocks such as resistors. This statementapplies equally to other lossless, multi-port devices discussed.

5.3.3.2 Time-Varying/Nonlinear Transformer

By allowing the turns-ratio N in the equations to vary as a function of time t, wearrive at a time-varying transformer that is still lossless with the I/O relationshipas:

v2 (t)

i1 (t)

=

N(t) 0

0 −N(t)

v1 (t)

i2 (t)

(5.14)

The reader may wonder how the device may be constructed, and what kindof characteristics to choose for N(t). Some analog realizations have beenproposed where the main objective is to control the amount of power deliveredto the load. We can also construct such a device in the digital domain where wesimply implement the relationship:

v2 (nT)

i1 (nT)

=

N(nT) 0

0 −N(nT)

v1 (nT)

i2 (nT)

(5.15a)

1 Later, we will see the definition of an ideal transformer with memory.

Independentvariables

DependentVariables

I/O Relationship

v2(t), i1(t)

v2(t), i1(t)

v1(t), i1(t)

v1(t), i1(t)

v1(t), i2(t)

v1(t), v2 (t) i1(t), i2(t)

i1(t), i2(t)

v2(t), i2(t)

v1(t), v2 (t)

v2(t), i2(t)

v1(t), i2(t)

i1(t) = y

1v

1(t), v

2(t)[ ]

i2(t) = y2 v1(t), v2 (t)[ ]

v2(t) = h11 v1(t), i2(t)[ ]

v2 (t) = t11 v1(t), i1(t)[ ]

v1(t) = h21 v2(t), i2 (t)[ ]i

1(t) = h

22v

2(t), i

2(t)[ ]

v1(t) = z1 i1(t), i2 (t)[ ]v2 (t) = z2 i1(t), i2(t)[ ]

i1(t) = h12 v1(t), i2 (t)[ ]

i2(t) = t12 v1 (t), i1(t)[ ]

Type

Voltage controlled

Hybrid # 1

Transmission # 1

Hybrid # 2

Transmission # 2

Current Controlled

i2(t) = t22 v2(t), i1(t)[ ]v1(t) = t 21 v2 (t), i1(t)[ ]

1)

2)

3)

4)

5)

6)

Table 5-1. Six representations for a two-port network.

using software or hard-wired digital logic. A pseudo-code for implementing theI/O relationship of a circuit corresponding to a transformer loaded by a linearresistor at the second-port is:

Given v1 (nT), N(nT)

v 2 (nT) = N(nT)v1 (nT) ; Transformer first equation

i2 (nT) = −GLv 2 (nT) ; Due to load

i1 (nT) = −N(nT)i2 (nT) ; Transformer second equation

Return i1(nT)

(5.15b)

In both analog and digital realizations, instead of simply making N(t) anindependent function of t, we can (and mostly we will) make N(t) a function ofthe state-variables of the network in which it is embedded. For example, supposethere exists an error function e(t) that is a well defined, continuous function ofthe state-variables and expected to go to zero. We can connect the time-varyingtransformer (with the choice of N(t) = e(t) and terminated at its second port witha fixed load R L ) to the rest of the circuit as shown in Fig. 5.8. The load seen at

the terminals 1-1' can be shown to be varying inversely to the square of the errore(t). Thus a load is present at the terminals 1-1' as long as e(t) ≠ 0 and gets

removed as and when e(t) becomes zero. Thus, we are able to include a damperin the circuit whose effect varies as a function of suitably chosen error function.We will see the use of such configurations in adaptive systems in the laterchapters.

A number of points are worth noting here. As suggested above, in mostapplications, the turns ratio will be made a function of the state variables and notan independent time-varying function. Hence we can call such a device Non-linear transformer. Secondly, though an analog realization is possible, thedevice is highly suited for use in digital realization of complex nonlinearsystems where we can preserve the lossless property even under finite precisionrepresentation for N •[ ] and other variables. Further, in practice, N •[ ] will be a

function of e(t − τ) ( τ > 0) so that a physical realization is possible.. Thus, it

may be argued that it is a device with memory. However, from a functional viewpoint, a nonlinear transformer is a memoryless device and we will continue touse this interpretation.

5.3.3.3 Two port, linear Gyrators:

Another useful device called a two-port gyrator (Fig. 5.9a) results by lettingv1 (t), v2 (t) as the independent variables and i1 (t), i 2 (t) as the dependent

variables with the following relationship:

i1 (t)

i2 (t)

=

0 G

−G 0

v1 (t)

v 2 (t)

(5.16)

where G, a constant, is called the gyrator conductance. It can be noted thatp(t) = v1 (t)i1 (t) + v 2 (t)i2 (t) = 0 regardless of the value of G. In the case of the

two-port gyrator, the I/O relationship can also be written in the form:

v1 (t)

v2 (t)

=

0 R

−R 0

i1 (t)

i2 (t)

(5.17)

where R = G−1 . That is, i1 (t), i 2 (t) can be considered as the independent or

source variables and v1 (t), v2 (t) the dependent or response variables.

When a gyrator is terminated at the second-port with a linear resistor of R L

ohms (Fig. 5.9b), it can be shown that the load seen at the first-port is:

+

-

i1(t)

+

-

v2 (t)

•

1 : N(t)

An ElectricalCircuit

Nonlinear and ortime-varyingtransformer

1

1'

i2 (t)

vR (t)

+

-

iR

(t)

RL

ˆ R L (t) ⇒

= RL

N 2(t)

v1(t)

Figure 5-8. Use of a nonlinear/time-varying transformer to implement a time-varying load.

R in ( t )=v1 (t)

v1 (t)=

1

G2RL

(5.18)

That is, a gyrator 'inverts' the behavior of the load. This inversion propertyallows the realization (or simulation of the property) of an inductor using acapacitor2 and hence eliminates the need for the use of an inductor as a basicbuilding block of electronic circuits. Also, when terminated with a current(voltage) controlled nonlinear resistor at the end of the second port, theinversion property leads to voltage (current) controlled nonlinear resistor at thefirst port. Thus, we can form one type of nonlinear resistor from the other typeof nonlinear resistor, a useful property when only one such nonlinear resistorcan be implemented.

5.3.3.4 Multi-port, linear Gyrators:

Multi-port gyrators result from a straight forward extension of the definition ofthe two-port gyrator to a M-port device. For example, the I/O relationship of avoltage controlled multi-port gyrator is given by:

2 Formal definitions for inductors and capacitors are given in section 5.3.4.

i1 (t)

i2 (t)

MiM (t)

=

0 y12 K y1M

−y12 0 K y 2M

M M O M−y1M − y2M K 0

v1 (t)

v 2 (t)

MvM(t)

(5.19a)

or

i = Yv (5.19b)

where yij are real constants and Y is the admittance matrix3. Note that the

property:

Y + Y t = 0 (5.20)

still holds as well the lossless property. Thus, the multi-port gyrator also servesthe function of transferring power from certain ports to other portsinstantaneously with no loss.

The current controlled representation of a multi-port gyrator is:

v1(t)

v 2 (t)

MvM (t)

=

0 z12 K z1M

−z12 0 K z2M

M M O M−z1M −z2M K 0

i1(t)

i 2 (t)

MiM (t)

(5.21a)

or

v = Zi (5.21b)

where

Z + Zt = 0 (5.21c)

with Z the impedance matrix. It can be observed that the relationship:

Y = Z−1 (5.22)

holds when the inverse exists. We can show that the inverse doesn't existalways. This simply implies that under such circumstances, the gyrator will be

3 Definitions such as admittance matrix are normally made in the frequency domain (s-

plane). We use the definition/terminology in the time-domain itself as we aredealing with nonlinear and time-varying elements.

+

-

i1(t)

v1`(t)

+

-

i2 (t)

•

v2(t)

1 : G

(b)(a)

vR(t)

+

-

iR(t)

RL

v1(t)

+

-

i1(t)

v1`

(t)

+

-

i2 (t)

•

v2 (t)

1 : G

v1(t)

ˆ R L

⇒

= G2RL−1

Figure 5-9. a) The symbol for a linear, two-port gyrator; b) The impedanceinversion property of a linear gyrator.

either voltage controlled or current controlled and must be used as such. It canalso be noted that the admittance (and the impedance) matrix Y (Z) of a losslessgyrator is positive-definite or semi-definite and anti-metric. As we will see later,most mathematically oriented approaches to control or signal processing, omitsuch matrices (and hence such devices) from any consideration. In fact, thedefinition of positive definiteness and positive semi-positive definitenessinvolves only symmetric matrices and ignores anti-metric matrices completely.However, considered as a device, gyrators can play an important role in linearand nonlinear systems.

5.3.3.4.1 Circulator: A special three-port gyrator:

A special device called a three-port circulator can be obtained from the multi-port gyrator definition by letting M = 3 and constraining the impedance matrixelements to some specific values as shown below:

v1 (t)

v2 (t)

v3 (t)

=0 R − R

−R 0 R

R −R 0

i1 (t)

i2 (t)

i3 (t)

(5.23)

Note that the determinant of the impedance matrix is zero. Let us demonstratethe importance of such devices. Suppose we connect load resistors of value R(same as the value of the impedance matrix element of the circulator) to portstwo and three and a non-ideal voltage source represented by an ideal voltagesource in series with a resistance R to port one as shown in Fig. 5.10. It can beshown that:

i1 (t) = i2 (t)

i 3 (t) = 0

R in at port # 1= v1 (t)

i1 (t)= R

(5.24)

and that the power entering Port one is transferred completely to port two withno power going to port-three. A similar phenomena occurs if the non-idealsource is connected to port-2 (or port-3) and the loads to the port-3 (port-1) and1 (and port-2) whereby port-3 (port-1) receives all the power and the port-1(port-2) none, leading to the name circulator. Such devices find usefulapplications in communications and measurements.

Note that the determinant of this matrix is zero which implies, for thisdevice, i 3 = i1 + i2 Thus, we should not connect inductors in series to all three

ports. Issues of interconnecting various devices and the effects are discussed inchapter 6.

5.3.3.5 Nonlinear Gyrators:

By making the admittance matrix elements (or impedance matrix elements)functions of the state-variables, we obtain a nonlinear, lossless gyrator with a v-irelationship:

i1 (t)

i2 (t)

MiM (t)

=

0 y12 [.] K y 1M [.]

−y12 [.] 0 K y 2M[.]

M M O M−y1M [.] −y2M [.] K 0

v1 (t)

v 2 (t)

MvM(t)

or i = Y[.]v

with Y[.]+ Y t [.] = 0

(5.25a)

0r

1

1'

2

3' 3

Ri1(t) i2 (t)

v2 (t)

i3 (t)

+

-

+

-

+-

RL = R

R s = R

Rin

= R

+

-2'

RL = R

v1(t)v s (t)

Figure 5-10. A three-port circulator connected to a source at one port and loads atthe other two ports. The source and the load resistances have the same value.

v1(t)

v 2 (t)

MvM (t)

=

0 z12 [.] K z1M [.]

−z12 [.] 0 K z2M[.]

M M O M−z1M[.] −z2M [.] K 0

i1 (t)

i 2 (t)

Mi M(t)

or v = Z[.]i

with Z[.] + Zt [.] = 0

(5.25b)

where

Z−1[.] = Y[.] (5.25c)

when the inverse exists. However, in the case of nonlinear gyrators, it is better tospecify which are the input variables and which are the output variables andstick to that convention. It can be noted that though a nonlinear gyrator isobtained by a straight forward extension of the definition of a linear gyrator, thisnew device can play an important role in nonlinear system modeling and designas we will see later in this chapter and other chapters. It should be noted that thebasic properties such as losslessness and positive definiteness or semi-definiteness are still valid and can be preserved in a digital implementation.

5.3.4 One Port elements with memory

The fourth possibility:

E(∞) = p(t)dt−∞

∞

∫

= v(t)i(t)dt−∞

∞

∫= 0

(5.26)

is possible in a nontrivial two terminal (one - port) element only if:1) a)the power delivered to the device can be both positive and

negative, and2) the energy received by the device (the area enclosed by the

waveform p(t) ≥ 0 and the t-axis) is returned (the area

enclosed by the waveform p(t) ≤ 0 and the t-axis) completely

(see Fig. 5.11).

Such a behavior implies that the element is capable of storing energy and islossless. The energy storage property can also be considered as leading to"memory" in the element since the net energy returned by the device depends onthe energy previously delivered to the element.

The property that the element has memory rules out the possibility of ainstantaneous relationship of the form:

f[v(t),i(t)] = 0

or v(t) = f1 [i(t)]

or i(t) = f2 [v(t)]

between the voltage and the current of the two-terminal element. Rather, therelationship has to include dynamic operators such as integrators. Two suchbasic one-port devices are possible as explained below.

5.3.4.1 Capacitors:

5.3.4.1.1 Linear, Time-Invariant Capacitors

A physical device can be formed with two flat parallel metal plates separated bya distance d and the space between the plate filled with ferro-electric materialssuch as barium titanate or left free. If the distance between the plates d is keptconstant, and a voltage v c (t) is applied to the two plates, a charge q c (t) will be

Negativepowerflow

Positivepowerflow

t

pc(t) = v c(t)i c(t)

Figure 5-11. Power flow into a two-port lossless device. The power flowing intothe device is stored as energy and returned back completely at a later time. It is apassive device as it returns only the energy given to it before and nothing more, itis lossless as it consumes no energy, and is a device with memory in the sense thatthe energy returned is limited to what was stored before.

induced on the upper plate and a charge of value -q c (t) on the lower plate. The

magnitude of the charge will depend on:1) the distance d,2) the plate area A,3) the dielectric constant of the material between the plates and4) the magnitude of the voltage applied.

If the first three are held constant, then the charge will be directly proportionalto the voltage leading to a linear time-invariant capacitor (LTI) with a symbolas shown in Fig. 5.12a. We can denote the relationship between the charge andthe voltage of LTI capacitor as:

q c (t) = cv c (t) (5.27)

where c (c > 0 ) is a constant known as the capacitance. The current through aLTI capacitor is given by:

i c (t) =dq c (t)

dt

= cdvc (t)

dt= c˙ v c

(5.28)

We can write v c (t) , the voltage, as a function of i c (t) as:

v c (t) = 1

ci c (τ)dτ

τ=−∞

t

∫ for t> -∞

=1

ci c (τ)dτ

τ=−∞

t0

∫ +1

cic (τ)dτ

τ= t0

t

∫ for t≥ t0

= v c (t0 ) + 1

ci c (τ)dτ

τ=t 0

t

∫

(5.29)

That is, the capacitor voltage v c (t) at any instant t depends not just on the value

of current at that time instant but also on the past values of the current goingback to t = −∞ (entire past history), confirming the memory property of the LTIcapacitor. Equation (5.29) also indicates that in practice we do not have tospecify the entire past history. We can choose a convenient initial time t 0 and a

single value, the voltage v c (t 0 ) across the capacitor at that time, to denote the

effect of the current i c (τ) for −∞≤ τ ≤ t 0 on v c (t) ( t > t 0 )

Equation (5.29) also implies that a capacitor with past history represented bythe voltage v c (t 0 ) can be considered as equivalent to a series connection of a

constant voltage source of v c (t 0 ) volts and a capacitor with the same

capacitance value and no past history as shown in Fig. 5.12b. This equivalencecan help in determining the minimal but sufficient number of elements to buildcomplex circuits. Also this observation will become useful as we move onto theanalysis of complex circuits with elements with past history.

Finally, the net energy entering the LTI capacitor during any time interval(t1 , t2 ) is given by:

E c [t1 ,t 2 ] = pc (τ)dτ t2 ≥ t1

τ= t1

t2

∫

= vc (τ)i c (τ)dττ= t1

t2

∫ = vc (τ)c ˙ v c (τ)dττ=t1

t2

∫

= c v c (τ)dv c

v c(t1)

vc (t2 )

∫ = 12 c{vc

2 (t 2 ) − v c2 (t1 )}

(5.30)

That is, the net energy E c [t1 ,t 2 ] entering a LTI capacitor for any interval

(t1 , t2 ) depends only on the value of the voltage (or charge) at the two end

points t 1, t 2 and is independent of the voltage or charge waveform that existed

in that time interval. Further, if v c (t) is periodic (perhaps a very complex)

signal with a period Tp , we can show that:

E c [t1 ,t 1 + kTp ] = 0 k an integer (5.31)

That is, the energy entering a LTI capacitor in a complete cycle of a periodicexcitation is exactly equal to zero. This implies that the power entering a LTIcapacitor during certain parts of a cycle must be negative to result in a zero netenergy consumption in a full period. Thus, the energy is stored during that partof the cycle and released during the other part of the cycle. From (5.30), it canbe seen that maximum energy release takes place when v c (t2 ) = 0 and is given

by (letting v c (t1 ) = v c ):

E c [vc ] = 12 cvc

2 = 12c q c

2 (5.32)

Therefore the energy stored in a LTI capacitor is proportional to the square ofthe voltage across (or charge on) the capacitor. When q c = 0, E c = 0 implying

that the capacitor has no energy to release. Thus, we can say that the capacitor isin a relaxed state and q c = 0 can be called the relaxation point for the capacitor.

5.3.4.1.1.1 Interconnection of LTI Capacitors and Independent & orControlled Sources

We noted that a charged capacitor, a passive device, is equivalent to a seriesconnection of a capacitor with no charge and an ideal constant voltage source,an active device. This is perhaps little bit surprising as we know that an idealvoltage source is capable of supplying unlimited amount of energy whereas aninitially charged capacitor has only finite amount of energy to give out. We canshow that there is no discrepency by calculating the energy that can come out ofthis series connection. In fact, this calculation will help us to arrive at morecomplex interconnections involving a capacitor and ideal and controlled sourcesthat have similar properties.

The power coming out of the circuit of Fig. 5-12b is given by:

pout (t) = −v1 (t)i1 (t)

= − v c (t) − vDC{ }c˙ v c (t)(5.33)

and the energy that can be delivered from time t 1 to t 2 by this circuit as:

Eout [t1 , t2 ] = pout (τ)dτ t2 ≥ t1τ= t1

t2

∫

= − v c (τ) − vDC{ }c ˙ v c (τ)dττ=t1

t2

∫

= −c v c (τ) − vDC{ }dv cvc( t1 )

v c(t2 )

∫

= − 12 c vc (t 2 ) − vDC{ }2 − vc (t1 ) − vDC{ }2[ ]

(5.34)

That is, the energy that can be delivered by this circuit (which has an idealsource) is restricted by the voltage that exists across the capacitor at the two endpoints and not the voltage (at other times) or the current values. Using similararguments that we made for a charged capacitor, we can see that the maximumnet energy release takes place from this circuit when v c (t2 ) = vDC and just

depends on the initial voltage on the capacitor and the ideal voltage sourceamplitude. That is:

Eout_max (vc ) = 12 c v c (t) − vDC{ }2[ ] (5.35)

where we have dropped the subscript '1' associated with the initial time. Thisexpression is identical to the one for the stored energy of a capacitor except forthe shift by the constant factor vDC . Therefore, the maximum energy that can be

released is still limited by the initial charge in the capacitor even though we havean ideal voltage source. In other words, the capacitor still calls the dance, so tospeak.

We can extend further this concept of a circuit with only a limited storedenergy to give out to include controlled sources as well as time varying sources.Referring to equation (5.34), we find that a circuit as shown in Fig. 5.12c will dothe trick. In the circuit, x(t) can be a state variable of a complex circuit (to whichthis sub-circuit has been connected) or an independent source and k is aconstant. The corresponding energy expressions are:

+

-

+

c

-

-

+

v1(t)v c(t)

i1(t)

k xn (t)

ic (t)

cknx n −1(t) ˙ x (t)

(c)

(a) (b)

+

-

+

-

cv c(t)

vc(0) ≠ 0

+

-

+

c

v1(t)

= v c(t)

ˆ v c(t)

ˆ v c(0)= 0

-

+

-

≡

+

-

v c(t)

charge = qc(t)

ic(t) = ˙ q

c(t)

c

v c (0)

Figure 5-12. a) Symbol of a Linear time-invariant capacitor. b) A LTI capacitor withinitial charge and its equivalent representation in terms of a capacitor with no chargeand a constant voltage source. c) Interconnection of a capacitor with independent ordependent voltage sources and certain current sources that lead to a passive circuit.

Eout [t1 , t2 ] = pout (τ)dτ t2 ≥ t1

τ= t1

t2

∫

= − v c (τ) − k xn (τ){ } c ˙ v c (τ) − ck n x n−1 (τ) ˙ x (τ){ }dττ=t1

t2

∫

= − 12 c vc (t 2 ) − k x n (t 2 ){ }2

− vc (t1 ) − k x n (t1 ){ }2[ ](5.36)

and

Eout_max = 12 c v c (t) − k x n (t){ }2[ ] (5.37)

Again, the energy that can be released is limited and depends on the initialcondition only.

In summary, we find that the circuit combinations shown in Figs. 5.12b and5.12c can be considered as passive building blocks (that are capable of gettingcharged or capable of releasing only a limited amount of energy) even thoughthey have active sources as sub-components. We can use such passive buildingblocks to form complex stable nonlinear dynamics as well as control complexnonlinear plants (stable or unstable) as we will learn later.

5.3.4.1.2 Non-linear, Time invariant Capacitors:

When the dielectric constant of the material between the plates change with theapplied voltage (or electrical field), a nonlinear, time invariant (NLTI) capacitorresults. The q-v characteristics of a general NLTI capacitor will take the form:

f c [qc ,vc ] = 0 (5.38)

As in the case of nonlinear, time invariant resistors, we can obtain a chargecontrolled (voltage controlled) capacitor if the above equation can be solved toobtain a single valued function of charge (voltage) for the voltage (charge). Thatis,

v c (t) = v c[q c (t)] (5.39)

for a charge controlled capacitor and

q c (t) = q c [vc (t)] (5.40)

for voltage controlled capacitors. The current through the NLTI capacitorssimilar to LTI capacitor, is given by

i c (t) =dq c (t)

dt

= ˙ q c (t)(5.41)

If the nonlinear q-v characteristics is continuous (differentiable), areasonable condition for physical devices, the above equation becomes, forvoltage controlled capacitors:

i c (t) =dq c (t)

dt

=dq c (t)

dv c (t)

dvc (t)

dt

= c[v c ] ˙ v c

(5.42)

where c[v c ] can be called the small-scale capacitance of a NLTI capacitor. For

charged controlled NLTI capacitors, we can obtain similar relationship byconsidering the time-derivative of the voltage. That is,

˙ v c (t) =dv c (t)

dt

=dv c (t)

dq c (t)

dq c (t)

dt

= s[qc ] i c (t)

(5.43)

where the unit for s[qc ] will be the inverse of c[v c ] (if it exists) and hence can

be called the small scale inverse capacitance. It should be noted that for bothtypes of nonlinear capacitors:

q c (t) = ic (τ)dττ=−∞

t

∫ for t> -∞

= ic (τ)dττ=−∞

t 0

∫ + i c (τ)dττ= t0

t

∫ for t≥ t 0

= q c (t 0 ) + ic (τ)dττ= t0

t

∫

(5.44)

confirming that nonlinear time-invariant capacitors also exhibit memory.

5.3.4.1.2.1 Charge controlled or voltage controlled NLTI Capacitors ?

Given the choice, which version of the NLTI capacitor is preferable in practice?We can obtain an answer to this question from a consideration of the types of

state equations that would result when such an element is used in a circuit andtheir solvability. The application of Kirchhoff's current and voltage laws4 to thecircuit leads to the state equations that characterize the behavior of the circuit. Inpractice KVL would lead to an equation of the form:

ijk (t) = 0j=1

n

∑ (5.45)

where i jk (t) are the currents flowing through the n elements which are

connected together at the k-th node. If we assume that one such element is acapacitor (we can assume that j = 1 refers to that capacitor without any loss ofgenerality), then for a charge controlled capacitor we can obtain from (5.43) and(5.45):

i1k (t) = ick (t) = ˙ q ck (t) = − i jk (t)j=2

n

∑ (5.46a)

and

v ck (t) = vck [qck (t)] (5.46b)

On the other hand, for a voltage controlled capacitor we can write from(5.42) and (5.45):

i ck (t) = c[v c ]˙ v c = − i jk (t)j=2

n

∑ (5.47a)

and

q ck (t) = qck [v ck (t)] (5.47b)

Thus, in the case of a charge controlled capacitor we can choose q c as the state

variable and solve (5.46a) numerically to obtain q c (n + 1) and equation (5.46b)

for v c (n + 1). In this case the mapping v c [qc ] can be many to one

(s[qc ] = dvc dqc can become zero for some values of q c or its inverse can

become infinite).For a voltage controlled capacitor, we need to choose v c as the state

variable. This in turn will require that the inverse of c[v c ] exists or

c[v c ] = dq c dvc does not become zero for finite values of v c . Hence the

mapping q c [vc ] has to be restricted to a one to one (monotonically increasing or

decreasing).

4to be defined formally in chapter 6.

From the above discussion, we can conclude that charge controlledcapacitors can have characteristics that are not possible using voltage controlledcapacitors and hence their use in nonlinear networks will lead to highly complexbehavior than is possible with voltage controlled capacitors. We will thereforeassume the use of charge controlled capacitors when the use of a nonlinearcapacitor is indicated. The symbol for a nonlinear time-invariant capacitor isshown in Fig. 5.13a. The symbol for the nonlinear capacitor indicates thepossibility that the v-q characteristics may not be bilateral as in the case ofnonlinear resistors.

The net energy entering a charge controlled capacitor during any timeinterval [t1, t 2 ] is

E c [t1 ,t 2 ] = pc (τ)dτ = vc (τ)i c (τ)dττ=t1

t2

∫ t2 ≥ t1τ= t1

t2

∫

= vc [qc (τ)] ˙ q c (τ)dττ= t1

t2

∫ = vc [qc ]dqcqc1=qc (t1)

q c2=qc (t2 )

∫(5.48)

A graphical interpretation of the above equation is shown in Fig. 5.13b. Itcan be noted that E c [t1 ,t 2 ] represents the net shaded area that depends on the

values of q c (t1 ) and q c (t 2 ) and the capacitor voltage V s charge characteristics,

and not on the exact waveform of the charge q c (t) ( t 1 ≤ t ≤ t 2 ). That is, any

waveform q c (t) would have resulted in the same value of E c [t1 ,t 2 ] as long as

q c (t1 ) and q c (t 2 ) are the same. When q c (t1 ) equals q c (t 2 ) , E c [t1 ,t 2 ]

becomes equal to zero. Thus similar to a LTI capacitor, a NLTI capacitor is alsocapable of storing and releasing energy.

From the above discussion, we can conclude that a NLTI capacitor islossless for any v c − q c characteristics. However, its choice will determine how

much energy is stored or released as q c (t) changes from q c1 to q c2 and the

dynamic behavior of the network in which the NLTI capacitor is embedded. Letus study the expression for the energy further to determine the possiblewaveforms for v c [qc ] . Let us first rewrite expression (5.48) as:

E c[t1 , t2 ] = v c [qc ]dq c

q c1

q cr

∫ + v c [qc ]dq c

q cr

q c2

∫ (5.49)

E c [t1 ,t 2 ] = vc [q c ]dq c

q c1

q cr

∫ + v c [qc ]dq c

qcr

q c2

∫ (5.49)

Where we assume that there exists a value q c = qc r on the v c − q c

characteristics such that:

v c [qc ]dq c

qcr

q c2

∫ ≥ 0 for all real q c2 (5.50)

and hence,

v c[q c ]dq c

q c1

q cr

∫ ≤ 0 for all real q c1 (5.51)

The above condition implies the following:1) v c [qcr ] = 0 ;

2) dv c[q c ]

dq c qc =q cr

> 0

3) The net area covered by the waveform v c [qc ] and the q c axis

must be non-positive for values of q c in the range

−∞≤ q c ≤ q cr and,

4) The net area under the waveform v c [qc ] and the q c must be

non-negative for all values of q c in the range q cr ≤ qc ≤ ∞ .

Given such a value q c r , the net energy expression (5.44) can be written as

E c [t1 ,t 2 ] = −E cr + E cs (5.52)

where, E cr > 0 stands for the energy released by the capacitor and E cs > 0 for the

energy delivered to the capacitor during the time t 1, and t 2 . Thus if

q c (t 2 ) = q c2 = qcr , a NLTI capacitor with an initial charge of q c (t1 ) = q c1 = q

releases an energy equal to:

E c [q] = v c [qc ]dqc

qcr

q

∫ (5.53)

and represent the energy stored in an initially charged capacitor. If q is equal toq cr , the energy that can be released or supplied by the capacitor is equal to zero

and hence the capacitor can be considered to be in a relaxed state. Therefore,any point q cr with the property in (5.50) can be called a relaxation point for the

NLTI capacitor. A number of v c [qc ] waveforms having at least one relaxation

point q cr are shown in the Fig. 5.14. It should be clear from the figure that a

number of possibilities for the v c [qc ] waveform exist and lead to different

energy storage properties. If the v c [qc ] waveform is bilateral (anti-metric with

respect to the y axis), the energy curve is symmetric with respect to the y axis. Ifthe v c [qc ] waveform is confined to first- and the third-quadrants, the energy

curve is monotonically increasing with only one minima (global minima). If thewaveform strays into second- or fourth-quadrants, the energy curve will havemore than one minima (local minimas and maximas in addition to the globalones). Based on these stored energy curves, we can easily figure out theresponse when such a capacitor is connected to other passive elements as wewill find later.

5.3.4.1.2.2 Importance of the existence of one or multiple relaxation points:

Though the energy expression for losslessness alone does not indicate that thereshould be a relaxation point (An arbitrarily chosen v c [qc ] waveform for a

lossless NLTI capacitor with no relaxation point is shown in Fig. 5.15), theconstraint that the element be passive dictate that the v c [qc ] characteristics

include that at least one relaxation point.Multiple true relaxation points (where the stored energy is exactly equal to

zero) are possible only from trivial mathematical models5 such asv c [qc ] = sin[q c ] {in general , v c [qc ] = An sin[k nq c +θ n ]

n∑ where kn and θn

5 The waveform is identical (due to periodicity) and the relaxation points are equally

spaced.

(a)

+

-

charge = q c(t)

ic (t) = ˙ q c (t)

vc(t) = v c[q c(t)]

or

q c(t) = q c[v c(t)]

q c

q cr

(b)

v c[qc]

Net area between and the waveformfor any value ofmust be negativeq c ≤ qcr

q c to q cr

Net area between and thewaveformfor any value ofmust be positiveq c ≥ qcr

qcr to q

c

Figure 5-13. a) The symbol of a nonlinear time-invariant capacitor. b) Requirementson a waveform to be the valid v c − q c characteristic of a nonlinear capacitor.

are constants}. On the other hand, we can have non-trivial v c [qc ] characteristics

with just one relaxation point and a number of mimimas for stored energy. Thishappens when the v c [qc ] moves into second- and or fourth-quadrants.

An example of such a waveform along with the resulting stored energy curve isshown in Fig. 5.15b. When storage elements with such characteristics arecombined with other elements we can get nonlinear dynamics with exotic

response {for example, behavior that depends on initial condition etc.}. Inconservative design, we may assume that the v c [qc ] characteristic will have

only one relaxation point q cr . Further, without any loss of generality, we can

assume the waveform to be such that q cr is equal to zero.

In summary a charge controlled NLTI capacitor will be characterized by av c [qc ] waveform that

1) passes through the origin,2) may or may not be symmetric with respect to the origin.

(bilateral and non-bilateral elements),

(c) (d)

(a) (b)

qc

vc [qc ]

qcR ≠ 0

vc[q

c] = tanh[q

c]

q cR = 0

-4 -2 0 2 4-1

-0.5

0

0.5

1

qcR = 0, 2π, 4 π etc.

vc [q c] = cos[qc ]

qcR = π, 3π, 5π etc.

vc[q

c] = sin[q

c]

-8 -4 0 4 8-1

-0.5

0

0.5

1

-8 -4 0 4 8-1

-0.5

0

0.5

1

qc

q c

qc

Figure 5-14. Valid voltage-charge waveforms of nonlinear capacitors: a)Monotonically increasing waveform (only one relaxation point, q cr = 0 ); b) Non-

monotonic (only one q cr ≠ 0 ); c) Non-monotonic, with multiple relaxation points

(q cr = 0, ± 2π, ± 4π.... ); d) Another non-monotonic characteristic, with multiple

relaxation points (q cr = ±π , ± 3π.... ).

-5 -2 0 2 5

-200

0

200

300

v c [q] = q(q 2 − 9)(q 2 − 16) SSIC[q]

E c [q]

v c [q], SSIC[q], E c [q]

(b)

(a)

f c[q c] = q c +q c2

q c

-2 0 2

0

4

Figure 5-15. a) A waveform that fails to meet the conditions for the characteristic ofa nonlinear capacitor; b) A valid waveform with two energy minima.

3) is differentiable at least twice for all values of q c so that the

reciprocal small signal capacitance is finite and continuous),4) has none or many local minimas and maximas, and5) becomes zero at a finite number of non-zero values of q c .

Thus, even with the restrictions of only one relaxation point, we havetremendous flexibility in the choice of v c − q c characteristics.

5.3.4.1.2.3 Interconnection of Nonlinear capacitors with Independent & orControlled Sources

We noted that LTI capacitors when connected with independent and orcontrolled sources in a special way lead to two-terminal circuits that behaveslike a lossless circuit. Therefore, we may rightly ask if such a property extendsto nonlinear capacitors as well. As we will find now, it does in a limited way.

Considering figure 5.16, where we show the series interconnection of anonlinear capacitor with a constant voltage source, we can write the powercoming out of this circuit as:

pout (t) = −v1 (t)i1 (t)

= − v c (t) − vDC{ } ˙ q c (t) (5.54)

and the energy that can be delivered bythis circuit from time t 1 to t 2 as:

Eout [t1 , t2 ] = pout (τ)dτ t2 ≥ t1τ= t1

t2

∫

= − v c (τ) − vDC{ }˙ q c (τ)dττ=t1

t2

∫ (5.55)

For a charge controlled nonlinear capacitor, the energy expression reduces to:

Eout [t1 , t2 ] = − ˆ v c (q c )dq c

qc (t1)

qc( t 2)

∫ (5.56a)

where

ˆ v c (q c ) = vc (q c ) − vDC (5.56b)

is a shifted version of the original voltage-current characteristic. Therefore, thecombined circuit's behavior is identical to a nonlinear capacitor.

Considering a voltage controlled nonlinear capacitor, the expression for netenergy coming out of the circuit becomes:

Eout [t1 , t2 ] = − v c (τ) − vDC{ } dq

dv c

dv c

τ=t1

t2

∫ (5.57)

which is also dependent only on the end values of the voltage. Thus, thiscombination also behaves as a lossless circuit.

Upon some reflection, we will find that we need to allow cancellation ofterms (capacitor current canceled by controlled current source) to form losslesstwo terminal circuits using nonlinear capacitors and time-varying sources as wedid using LTI capacitors. We show the circuit along with others in Table 5.2.Again such circuits (or such techniques) will help us in building complex stablenonlinear dynamics as well as in controlling complex nonlinear plants (stable orunstable) as we will see later.

5.3.4.2 Inductors:

An inductor is another device that can be formed by winding conducting wiresaround a toroid made of non metallic materials such as wood or other nonlinearferromagnetic materials such as supermalloy. In such a device, the application ofa current i(t) will lead to flux φ(t) whose value will depend upon the

dimensions of the toroid, the properties of the materials used for the toroid, andthe current. The behavior of an inductor is similar to that of a capacitor in thesense that its properties can be derived by replacing voltage, current and chargeand capacitance in the capacitor expressions by current, voltage, flux andinductance respectively. Hence, each element can be considered as the “dual ”of the other element.

5.3.4.2.1 linear time invariant inductor

Applying the above mentioned duality concepts to the expressions for a lineartime invariant capacitor, we can obtain for a linear time invariant inductor as:

φL (t) = Li L (t) (5.58)

where, L is a positive constant known as the inductance. The voltage across theterminals of a LTI inductor is given by:

+

-

+

-

-

+

v1(t)

vDC

v c(t)

i1(t) = ic(t) = ˙ q c(t)

Figure 5-16. Nonlinear capacitor with aconstant voltage source. The combinedcircuit behaves as a lossless circuit.

Energy Equations

+

-

+

c

i1(t) = i

c(t)

-

-

+

v1(t)

vDC

vc(t)

+

-

+ i1(t) = ic(t)

-

-

+

v1(t)

vDC

v c(t)

Eout[t1 ,t 2]

= −c v c(τ) − vDC{ }dv c

v c (t 1 )

v c (t 2 )

∫

Eout[t1,t 2 ] =− c z dzz(t 1 )

z(t 2 )

∫where z(t) = vc (t) − k xn (t)

Eout[t1,t 2 ] =− z dzz(t 1 )

z(t 2 )

∫where z(t) = vc q(t)[ ]− k xn (t)

Circuit

+

-

+

-

-

+

v1(t)v c(t)

i1(t)

k xn (t)

ic (t)

is (t)

is (t) = ic(t)

+dvc q(t)[ ]

dt

+d kx n (t){ }

dt

Eout[t1, t2 ] = − v c(τ) − vDC{ }dq c

vc qc (t 1)[ ]

vc qc (t 2 )[ ]

∫

Eout[t1, t2 ] = − vc(τ) − vDC{ } dq

dvc

dvc

vc (t 1 )

vc (t 2 )

∫

Charge controlled capacitor

Voltage controlled capacitor

+

-

+

c

-

-

+

v1(t)v c(t)

i1(t)

k xn (t)

ic (t)

cknx n −1(t) ˙ x (t)

Table 5-2. Two terminal (one-port) circuits made of reactive elements and active sourcesthat behave as lossless circuits.

Flux controlled inductor

Energy EquationsCircuit

Eout

[t1, t

2]

= −L iL(τ) − iDC{ }diL

i L (t 1 )

iL (t 2 )

∫

Eout[t1,t 2 ] =− L z dzz(t 1 )

z(t 2 )

∫where z(t) = iL(t) − k x n (t)

Eout[t1, t2 ] = − iL (τ) − iDC{ }dφLi L φ L (t1 )[ ]

i L φL (t 2 )[ ]

∫

Eout[t1, t2 ] = − iL (τ) − iDC{ } dφL

diL

diL

i L φ L (t1 )[ ]

i L φL (t 2 )[ ]

∫

+

-

iL (t)v1(t) =v

L(t)

i1(t)

iDC

v1(t)

i1(t)

is(t) =

k xn (t)

+

vL (t)

iL (t)

L

-

+

+

-

-

v s (t) =

cknx n −1(t) ˙ x (t)

+

-

iL (t)v1(t) =vL (t)

i1(t)

iDC

Current controlled inductor

Eout

[t1,t

2] =− z dz

z(t 1 )

z(t 2 )

∫where z(t) = iL φ(t)[ ] − k xn (t)

v1(t)

i1(t)

is (t) =k xn (t)

+

vL (t)

iL (t)

L

-

+

+

-

-

vs(t) =

cknx n −1(t) ˙ x (t)

Table 5-2 (Contd.)

vL (t) = dφL (t)

dt

= ˙ φ L

= Ldi L (t)

dt

= L˙ i L

(5.59)

Or we can write i L (t) as a function of vL (t) and φL (t) as:

i L (t) = 1L vL (τ)dτ

τ=−∞

t

∫ for t> -∞

= 1L vL (τ)dτ

τ=−∞

t0

∫ + 1L vL (τ)dτ

τ= t0

t

∫ for t≥ t0

= i L (t0 ) + 1L vL (τ)dτ

τ= t0

t

∫

(5.60)

Thus, all the concepts such as memory, representation of the entire past historyby current i L (t0 ) etc. do apply here as well. Also the net energy entering the

LTI inductor can be shown to be

EL [t1 , t2 ] = pL (τ)dτ t2 ≥ t1

τ= t1

t2

∫

= vL (τ)iL (τ)dττ= t1

t2

∫

= iL (τ)L˙ i L (τ)dττ= t1

t2

∫

= L i L diLiL(t1)

iL( t 2)

∫= 1

2 L{iL2 (t2 ) − iL

2 (t1 )}

(5.61)

leading to the energy storage, release and lossless property.

5.3.4.2.1.1 Interconnection of LTI Inductors and or Independent & orControlled Sources

Similar to what we have seen for the case of LTI capacitor, we can connect anLTI inductor with independent and or controlled sources to form an one-port

circuit or building block that behaves as a lossless building block. It is very easyto arrive at the circuit and the equations using the duality principle and hencewill not be repeated here. We just show the circuits in table 5.2.

5.3.4.2.2 Nonlinear, Time-invariant Inductors:

Similar to a charge controlled NLTI capacitor, a flux controlled inductor will bepreferred to a current controlled inductor and will be characterized by:

iL (t) =i L [φL (t)]

vL (t) =dφL (t)

dt

˙ i L (t) =di L (t)

dt

= di L (t)

dφL

dφL (t)

dt

= τ[φL ]vL (t)

(5.62)

where τ[φL ] is called the small-scale inverse inductance. The use of a flux

controlled inductor will result in φL being the state variable and will allow

τ[φL ] to become zero ( i L[φL ] non-monotonic). Further, we can show that a

NLTI inductor is a lossless element capable of storing energy and releasing it,and can have i L −φ L characteristic that can have none, one or many relaxation

points. The symbol for linear and nonlinear inductors are shown in Fig. 5.17.The various equations and the properties for linear and nonlinear resistors,capacitors, inductors, transformers and gyrators are summarized in table 5.3.

(b)

+

-

iL (t)

flux =φ L (t)

vL (t) =˙ φ L (t)

i L = i L[φL ]

or

φL =φ L[iL ]

+

-

iL (t)

flux =φ L (t)

vL (t) =˙ φ L (t)

(a)

Figure 5-17. The symbol of a Linear (a),and a nonlinear (b) time-invariant inductor.

Elements Equations & propertiesSymbols

Linear resistor

Linearcapacitor

Linearinductor

Lineartransformer

Lineargyrator

vR (t)= Ri R (t)

or

iR (t)= Gv R (t)

ic (t) = c˙ v c(t)

E(∞) = p(t)dt−∞

∞

∫ = 0

Ec[v c ] = 1

2cv c

2

vL(t) = L ˙ i L

E(∞) = p(t)dt−∞

∞

∫ = 0

Ec[i L] =1

2Li L

2

+

-

i1(t)

+

-

v2 (t)

i2(t)

•

v1(t)

1:N

+

-

vL (t)

iL (t)

p(t) = v1(t)i 1(t) + v2 (t)i 2 (t)

= 0 for all t

v2 (t)

i1(t)

=

N 0

0 −N

v1(t)

i2(t)

p(t) = v1(t)i 1(t) + v2 (t)i 2 (t)

= 0 for all t

i1(t)

i2 (t)

=

G 0

0 −G

v1 (t)

v2 (t)

v 2 (t)v1(t)

i1(t) i2 (t)

+ +

- -

G

+

-

vR (t)

iR (t)

ic (t)+

-

v c(t)

Table 5-3. Linear and nonlinear time-invariant elements, the symbols, and thedefining equations.

Nonlineartransformer:turns ratio N[x]is a functionof the state ofthe system.

Elements Equations & properties

ic (t)+

-

vc(t)

+

-

vL (t)

iL (t)

+

-

i1(t)

+

-

v 2 (t)

i2 (t)1 : N[x]

•

G[x]

v 2 (t)v1(t)

i1(t) i2 (t)

+ +

- -

Symbols

+

-

vR (t)

iR (t)

Nonlinear resistor

Nonlinearcapacitor

Nonlinearinductor

Nonlineartransformer

Nonlineargyrator

vR (t) = vR iR (t)[ ]= 0 for iR = 0

≠ 0 for iR ≠ 0

iR (t)= iR vR (t)[ ]= 0 for vR = 0

≠ 0 for vR ≠ 0

Current controllednonlinear resistor

Voltage controllednonlinear resistor

v c(t) = v c[q c(t)]

E(∞) = p(t)dt−∞

∞

∫ = 0

Ec[q] = v c[q c]dq c

q cr

q

∫

Charge controllednonlinear capacitor

iL (t) = iL[φL (t)]

E(∞) = p(t)dt−∞

∞

∫ = 0

Ec[φ] = iL[φL ]di L

φcr

φ

∫

Flux controllednonlinear inductor

v2 (t)

i1(t)

=

N[x] 0

0 −N[x]

v1(t)

i2 (t)

p(t) = v1(t)i 1(t) + v2 (t)i 2 (t)

= 0 for all t

p(t) = v1(t)i 1(t) + v2 (t)i 2 (t)

= 0 for all t

i1(t)

i2 (t)

=

G[x] 0

0 −G[x]

v1(t)

v2 (t)

A two-portvoltagecontrollednonlinear gyrator.The gyratorcoefficient G[x] isa function of thesystem state.

v1(t)

Table 5-3 (Contd)

5.3.5 Multi-port devices with memory:

5.3.5.1 Two-port LTI coupled inductors

In section 5.2.3, we introduced the ideal transformer, a two-port memorylessdevice formed from magnetically coupled coils (inductors) and having certainideal characteristic. In this section, we introduce more general multi-port devicesmade up of magnetically coupled coils (multi-port. coupled inductors), anddiscuss their properties. We will also derive a special class of two-port deviceswith memory well known in the network theory as Brune Transformers (BT).Finally, we will also show the conditions under which the two port devices withmemory can become ideal transformers with no memory.

Consider a magnetic core with two coils wound on the core leading to a two-port device with voltage polarities and current directions as shown in Fig. 5.18{It should be obvious from the figure that a M-port device, where M > 2, ispossible by increasing the number of coils wound on the core}. If the first port isdriven by a time varying current source i1 (t) while the second port is kept open,

a time-varying magnetic flux will be produced on the core leading to a time-varying flux through the second coil. This time varying flux will lead to avoltage across the open terminals of the second port. The polarity of the voltagewill depend upon the construction of the device and will normally be indicatedby a dot as shown in the figure.

Assuming a linear model, the current produces the flux linkage φ1 given by:

φ1 = L11i1 (5.63)

where, L11 is the self-inductance of the coil 1. Similarly, the flux linkage in the

coil 2 by the current i1 (t) will be give by:

φ2 = M12 i1 (5.64)

where, M12 is known as the mutual inductance.

In general, when both currents i1 (t) and i2 (t) are present, the linear model

assumption leads to

φ1 = L11i1 + M21i 2 (5.65)

and similarly for the second coil

φ2 = M12 i1 + L22i 2 (5.66)

where in practice

M 21 = M12 = M (5.67)

The φ(t) −i(t) characteristics of the two-port linear coupled inductors can be

written in a matrix form as:

φ1

φ2

=

L11 M

M L22

i1i2

(5.68a)

or

φ = Li (5.68b)

leading to a current controlled representation. If det[L] = L11L22 − M 2 is not

equal to zero, the above equation can be rewritten in a flux controlledrepresentation as

i = L−1φ

= 1Det[L]

L22 −M

−M L11

φ1

φ2

= τφ

(5.69)

where τ = L−1 is the reciprocal inductance matrix. Since the voltage induced ineach coil is given by Faraday’s law as the time rate of change of flux, we have,

(a) (b)

i1(t)

v1(t) v2 (t)

i2 (t)+

-

i1(t)

+

-

v2 (t)

i2 (t)•

v1(t)

Figure 5-18. a) A magnetic core with two wound coils; b) Symbol for magneticallycoupled coils.

v1

v2

=

˙ φ 1˙ φ 2

=

L11 M

M L22

˙ i 1˙ i 2

(5.70a)

or

v = ˙ φ = L ˙ i (5.70b)

or˙ i = L−1φ= τv (5.71)

5.3.5.1.1 Stored Energy and the Inductance Matrix Parameters:

We can study the energy supplied to the coupled inductors and obtain conditionson the inductance matrix parameters for physical realizability. Let us assumethat i1 (t) and i2 (t) are the two sources connected to the two ports at time t = 0

with i1 (0) = i 2 (0) = 0 and φ1 (0 − ) = φ2 (0− ) = 0 .The energy delivered by the

sources to the coupled inductor over the interval [0, T] is given by:

E[0,T] = {v1(τ)i1 (τ) + v 2 (τ)i2 (τ)}dττ=0

T

∫

= {(L11

di1

dτ+ M

di2

dτ)i1 (τ) + (M

di1

dτ+ L22

di2

dτ)i2 (τ)}dτ

τ=0

T

∫

= L11i1di1 + M(i 1di 2 + i 2di1 ) + L22 i2di2

i(0)

i(T)

∫= 1

2 L11i12 (T) + 2Mi1 (T)i 2 (T) + L22 i2

2[ ]= 1

2 L11{i1 (T) +M

L11

i2 (T)}2 + {L22 −M 2

L11

}i22 (T)

(5.72)

A number of points can be observed from the above equations. If for some T> 0, i1 (T) = i 2 (T) = 0 and i1 (t) , i 2 ( t ) (0 < t < T) in general is not equal to

zero, E becomes non-zero for 0 < t < T and returns to zero value at t = T. That is,the net energy consumed by (or delivered to) the coupled coil is exactly equal tozero indicating that the device is lossless and capable of storing energy. Further,if we assume that the second coil were completely absent, the coupled coilbecomes a simple inductor and the energy delivered will be positive iffi1 (T) ≠ 0 and L11 > 0 . Thus, we can assume that L11 is positive. A similar

reasoning will show that L22 is also positive.

The value of the mutual inductance can in general be positive or negative.

Further, from the expression for the delivered energy, we can note that for alli(t) ≠ 0

E[0,T] ≥ 0 iff L 11L22 ≥ M2 (5.73)

That is, positive energy is delivered to the coupled inductors. Thusi1 = i2 = φ1 = φ2 = 0 can be considered as the relaxation points for the two-port

coupled inductors and E[0, T] as the energy stored in them.The conditions,

L11 > 0

L22 > 0

L11L22 ≥ M or Det[L] = L11L22 − M 2 ≥ 0

(5.74)

are the necessary and sufficient conditions for the real symmetric matrix L to bepositive semi-definite. Thus, we find that there is a direct connection betweenpassivity (a physical condition) and the positive definiteness (a mathematicalcondition).

We can define a new parameter k as

k = M L11L22 (5.75)

where k = 0 (M = 0) implies the one extreme case where there is no effect due tothe current in one coil on the other or no magnetic coupling between the twocoils. Thus, k can be called the coefficient of coupling. When k = 1( M 2 = L11L22 ) we can note that the second term in the energy expression (5.72)

becomes zero regardless of whether i 2 (T) equal zero or not. Further, we can

note that the first term in the energy expression also becomes zero wheni1 = − M L11( )i 2 ≠ 0 for any T. That is, the device becomes a memoryless

device for this particular combination of { i1 , i2 }. Physically this implies that all

magnetic field due to i1 is completely canceled by the magnetic field due to i 2 ,

a situation not possible in practice.Hence k = 1 represents the other extreme case. The coefficient of coupling in

general satisfies the inequality 0 < k < 1. A magnetically coupled inductors withunity coefficients of coupling ( k = 1) is known as a Brune Transformer innetwork theory and plays an important role in network synthesis as we will seelater.

5.3.5.1.2 Equivalent Circuit Representation of magnetically coupled multi-port inductors based on Ideal Transformers:

We can represent the magnetically coupled inductors by a circuit consisting oftwo terminal (one-port) inductors or inductors and ideal transformer

combination as shown in Fig. 5.19 a and b . The equivalence can be shown bycomparing the φ − i or v − i relationship for the two circuits shown in the

Figures with that of the equation of the magnetically coupled inductors given in(5.65). It can be noted from the T - equivalent circuit and the constraint on thecoefficient of coupling (0 < k < 1), that the value of at most one inductor in theequivalent circuit can be negative. In Fig. 5.19b, the equivalent circuit consistsof an ideal transformer and two inductors LL and Lm . It can be observed that

LL = 0 implies K = 1. Thus, L can be considered to be the inductance seen at

the first port due to the leakage flux or the lines of the magnetic field that do notlink both coils and hence the name leakage inductance. The inductor Lm which

appears at both ports represents the magnetic flux common to both the coils andcan be called the magnetizing inductance . Since both LL = 0,L L and Lm > 0,

from the figure, we can infer that a linear coupled inductor is not really a newdevice in the sense that it can be constructed from previously defined elements,the inductor and the ideal transformer.

5.3.5.2 M-Port (M > 2) LTI Coupled Inductors:

By using M ( > 2) magnetically coupled coils we can arrive at a M-Port coupledinductor with expressions similar to the expressions for two-port magneticallycoupled inductors. Since the concept is the same , we will not repeat them here.However, it should be noted that the L matrix of such an M-Port device will besymmetric, positive definite and the off-diagonal elements (mutual inductances)can be positive or negative.

5.3.5.3 Nonlinear Time Invariant Coupled Inductors:

Similar to the case of nonlinear inductors, we can obtain nonlinear, time-invariant coupled inductors by forcing the φ − i characteristics to be nonlinear

and independent of time. Thus for two port current controlled nonlinear coupledinductors we will have:

φ1 = φ1 [i1 , i2 ]

φ2 = φ2 [i1 , i2 ](5.76)

where are the scalar-valued functions of the two variables i1 and i 2 . Using

the relationship v k = ˙ φ k (k = 1,2) , we can obtain a relationship between

v and i as:

v =v1 (t)

v2 (t)

= ˙ φ =

˙ φ 1 (t)˙ φ 2 (t)

=

∂φ1∂i1 i

∂φ1∂i 2 i

∂φ2∂i1 i

∂φ2∂i2 i

˙ i 1 (t)˙ i 2 (t)

= L[i]˙ i

(5.77)

Ideal transformer N = L 22 M

General, magnetically coupled inductorsLa = L11 − M 2 L22 ≥ 0

La

Lm

+

-

i1(t)

v1` (t)

+

-

v2 (t)

i2 (t)1 : N

•

L =Lm NLm

NLm

N2Lm

Brune transformer

(k = 1); Lm = M2

L 22

(a) (b)

+

-

L1 = L11 − M L2 = L22 − M

L3 = M

i1(t) i2 (t)

v1(t)

+

-

v2 (t)

Figure 5-19. Representation of magnetically coupled inductors; a) T-equivalent circuit; b)Constraints that lead to a Brune transformer and an ideal transformer.

where ∂φ j ∂i k1 i(j, k = 1, 2) are the partial derivatives evaluated at i . This

matrix, denoted as L[i ] here is in general called the Jacobian matrix J of φ .

For a two-port flux controlled nonlinear inductor, the correspondingexpressions are:

i1 = i1 [φ1 , φ2 ]

i 2 = i2 [φ1 , φ2 ](5.78)

and

d[i]dt

=di1 (t) dt

di2 (t) dt

=

∂i1∂φ1 φ

∂i1∂φ2 φ

∂i 2∂φ1 φ

∂i2∂φ2 φ

˙ φ 1(t)˙ φ 1(t)

(5.79)

5.3.5.3.1 Energy Stored in a NLTI coupled Inductor:

We can study the energy supplied to the NLTI coupled inductors and obtainconditions on the nonlinear functions for physical realizability. Let us firstconsider a flux controlled device. The energy going into the coupled inductorover the interval [0,T] is given by:

E[0,T] = {v1(τ)i1 (τ) + v 2 (τ)i2 (τ)}dττ=0

T

∫

= {˙ φ 1 (τ)i1 (φ1, φ2 ) + ˙ φ 2 (τ)i2 (φ1, φ2 )}dττ=0

T

∫

= i1 (φ1, φ2 )dφ1 + i2 (φ1 , φ2 )dφ2

φ(0)

φ (T)

∫

(5.80)

which depends on the end conditions only and can become zero when the initialvalue of the flux vector equals the final value and regardless of what value theflux takes in between. Thus, energy will be going in at some times and returnedat other times. Therefore, similar to the LTI mutual inductor, its nonlinearcounterpart is also lossless and capable of storing energy. Again, to make thedevice a passive one (returned energy is less than or equal to the energy wentin), we need to restrict the current mappings such that E[0, T] is non-negative.An example of a flux controlled NLTI mutual inductor is given by:

i1 (φ1 , φ2 ) = τ11φ1 + τ12φ2 + aφ12m−1φ2

2n ,

i 2 (φ1 , φ2 ) = τ21φ1 + τ22φ2 + aφ12mφ2

2n −1 ; a > 0; m, n > 0 & integer (5.81)

where τ11 , τ22 > 0, τ11τ22 ≥ τ122 . The resulting stored energy expression is given

by:

E[φ1 , φ2 ] = 0.5(τ11φ12 +τ 22φ2

2 + τ12φ1φ2 ) + aφ12mφ2

2n (5.82)

which is always positive. The origin is the relaxation point for this NLTI mutualinductor.

When the NLTI mutual inductor is current controlled, the expression for netenergy entering the device will be given by:

E[0,T] = ∂φ1

∂i1

˙ i 1 (t) + ∂φ1

∂i 2

˙ i 2 (t)

i1 (τ) + ∂φ2

∂i1

˙ i 1 (t) + ∂φ2

∂i2

˙ i 2 (t)

i 2 (τ)

dτ

τ=0

T

∫

=∂φ1

∂i1

i1 +∂φ2

∂i1

i 2

di1 +∂φ1

∂i2

i1 +∂φ2

∂i2

i2

di 2

i(0)

i(T)

∫(5.83)

Again, the device exhibits the lossless property. The expressions for the fluxhave to be selected such that the net energy entering the device is alwayspositive.

5.4 Summary

In this chapter, we discussed time-invariant linear and nonlinear electricalelements that are the building blocks for complex electrical circuits. We firstprovided a formal definition for ideal energy sources and discussed theimplications of such assumptions from a real-world perspective. Next, using thetwo key quantities, power and energy, we defined passive and active elements.Various possibilities for passive elements were explored based on the number ofterminals and the characteristics of the devices (linear or nonlinear, memory orno memory). We found that a number of complex nonlinear devices are indeedpossible using the power and energy consumption property. Further, we notedthat when a LTI capacitor is connected to independent or dependent sources in acertain fashion, it can lead to an one-port circuit which is basically lossless.From the duality principles, similar circuits can be obtained using LTI inductors.These elements or building blocks (including the one-port circuit discussed) canbe used to form complex circuits and dynamics as we discussed before, and aswe will see in later chapters. We will also use the same approach to define time-varying elements for forming time-varying networks and dynamics.