4-2Sinusoids in Circuitsece2040.ece.gatech.edu/videos/Module4Handouts.pdf · 1/5/2014 1 School of Electrical ... Identify how past techniques apply to impedances in AC circuit analysis

9/22/2013

1

Linear Circuits

An introduction to linear electric components and a study of circuits containing such devices.

Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering

School of Electrical and Computer Engineering

Concept Map

2

Background Resistive Circuits

Reactive Circuits

Frequency Analysis

Power

1 2

3 4

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Resistive vs Reactive Circuits

3

Time

Volta

ge

Concept Map

4


Reactive Circuits

Frequency Analysis

Power

Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)

RC, RLC circuits

Frequency Domain

Impedance AC Circuit

Analysis

Transfer Function

Frequency Response

Filters

Frequency Analysis

1/5/2014

1



Sinusoids in Circuits

Review sinusoidal properties and introduce their representation in circuits

Identify sinusoid properties Examine sinusoids in circuits

(Alternating Current)

Lesson Objectives

4

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Sinsoids

5

Amplitude: VmPeriod: T secFrequency (Hz): Frequency (rad/sec):Phase Angle:

v(t) = Vmcos(t + )v(t)

Vm

-Vm

Circuit Responses

6

T Tvin

vout

If the output is from the input, output phase input phase

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3

Cosines and Sines

7

cos(100t)sin(100t)

sin(t) = cos(t 90o)-sin(t) = cos(t +90o)

Sinusoids and Capacitors

8

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Reviewed sinusoid properties Frequency (Hz, rad/sec), amplitude, phase

Identified sinusoid behavior in linear circuits AC Phase lag/lead

Summary

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Phasors

Use phasors to represent sinusoids

Introduced sinusoids in circuits Alternating Current (AC)

Previous Lesson

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Introduce phasors to represent sinusoids

Lesson Objectives

5

Why? Easier than solving differential equations!

Phasorsv(t)=Vmcos(t + )

Polar: V = Vm Rectangular: V = a+bj

6

Im

Re

i(t)=Imcos(t + )Polar: I = Im Rectangular: I = a+bj

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Signal Phasor in Polar Form Phasor in Rectangular Formv(t) = 10cos(100t 45o) V = 10-45o V = 102 j102

v(t) = 10cos(1000t + 90o) V = 1090o V = 0 + 10j = 10j

i(t) = 10cos(500t) I = 100o I = 10 + 0j = 10

i(t) = 10sin(1000t + 20o) = 10cos(1000t + 20o 90o)

I = 10-70o I = 3.42 9.40j

Examples

7

Adding Sinusoids with Phasors

8

Phasorv1 (t) = 7cos(1t+30o) 730o 6.1 + 3.5jv2 (t) = 3cos(1t-60o) 3-60o 1.5 - 2.6jv1(t) + v2(t)

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Multiplying:VI = V1 I2 = V I 1+2

Dividing:V/I = V1 I2 = V /I 1-2

Multiplying/Dividing Phasors

9

V = 530oI = 2-60o

Sinusoids must have same frequencies Adding/subtracting phasors rectangular Multiplying/dividing phasors polar

Summary

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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering


Impedance

Identify impedances a mathematical tool to analyze reactive circuits with sinusoidal inputs.

Be able to describe impedance Calculate impedances of resistors,

capacitors, and inductors Identify the relationship between voltage and

current based on and impedance value

Lesson Objectives

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Definition of Impedance

6

Impedance of an Inductor

7

Inductor impedance purely imaginaryScales based on frequencyPositive imaginary, so current lags voltage

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Impedances

8

In-phase Current leads voltage Current lags voltageFrequency invariant Voltage attenuates for

high frequencyCurrent attenuates for high frequency

Defined impedance and calculated impedance of linear devices Described the relationship between the

current and the voltage given impedance

Summary

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1

Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering


AC Circuit Analysis

Identify how past techniques apply to impedances in AC circuit analysis.

Apply techniques from DC analysis to sinusoidal systems Find equivalent impedances for devices in

series/parallel Use superposition for analysis: particularly for

systems with multiple frequencies Be able to analyze a system using these techniques

Lesson Objectives

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Impedance is Linear

6

Impedances in Series

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Impedances in Parallel

8

Kirchhoffs Laws

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Source Transformations

10

Superposition

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Kirchhoffs Laws Superposition Node-voltage Mesh-current Thvenin and Norton Equivalent Circuits Source Transformations

Valid Impedance Techniques

12

Example

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Showed how DC analysis techniques are applied in sinusoidal systems Used superposition to analyze a system with

multiple frequencies Solved an example system using these

techniques

Summary

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Transfer Functions

Transfer functions characterize the input to output relationship of a system.

Introduce transfer functions to characterize a circuit to find sinusoidal output

Lesson Objectives

5

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Behavior of Sinusoids in Linear Systems

6

y(t) = Aoutcos(t + out)x(t) = Aincos(t + in)

xin youtLinear Circuit

Transfer Function

7

x(t) = Ain(t + in) H() y(t) = Aoutcos(t + out)

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Series RC

8

+

Vo-

Vi

Series RC

9

+Vo-

Vi

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RLC Example

10

+

Vo-

Vi

Series RLC

11

Vi+Vo-

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Using the Transfer Function

12

Vi+Vo

-

R = 20k, L = 3.3mH, C = 0.12F, f = 50Hz

Introduced the concept of a transfer function (output phasor)/(input phasor)

Showed how to calculate a transfer function for a particular system Impedance method (voltage divider law) Showed how to use a transfer

function to compute the output phasor

Summary

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FrequencySpectrum

Understanding and displaying the frequency content of signals

Introduce the frequency spectrum as a way of showing the frequency content of signals Introduce both linear and log scales for displaying

frequency content

Lesson Objectives

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Summation of Sines

6

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plitu

de

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plitu

de

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plitu

de

0 0.5 1 1.5 2 2.5 3-1

0

1

Time (sec)

x 1(t

)

0 0.5 1 1.5 2 2.5 3-1

0

1

Time (sec)

x s(t

)

0 0.5 1 1.5 2 2.5 3-1

0

1

Time (sec)

x 2(t

)

x1 = sin(22t)

x2 = 0.2sin(26t)

xs = x1+x2

Summation of Sines

7

0 0.5 1 1.5 2 2.5 3-1

0

1

Time (sec)

x 1(t

)

0 0.5 1 1.5 2 2.5 3-1

0

1

Time (sec)

x 2(t

)

0 0.5 1 1.5 2 2.5 3-2

0

2

Time (sec)

x s(t

)

x1 = 0.2sin(22t)

x2 = sin(26t)

xs = x1+x2

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plit

ude

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plitu

de

0 1 2 3 4 5 6 7 80

0.5

1

Frequency (Hz)

Am

plitu

de

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3

Harmonics

8

)cos()( 01

0 k

N

kk tkAAtx ++=

=

Frequency (rad/sec)0 0 20 30

Frequency Spectrum (Log Scale)

9

Frequency (rad/sec) or f (Hz)1 10 100 1000 Some frequency components are better viewed in

log scale Larger dynamic range while maintaining resolution

at the low amplitude range Historical usage, going back to time when graphs

drawn by hand

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Example Spectra

10

0 50 100 150 2000

0.5

1

1.5

Frequency (rad/sec)

Mag

nitu

de

0 50 100 150 200-100

-80

-60

-40

-20

0

20

Frequency (rad/sec)

Mag

nitu

de (

deci

bels

)

0 5 10 15 200

0.5

1

1.5

Time (sec)

x(t)

A is a plot of the frequency content of signals

include a fundamental frequency and multiples of it Log scale is often preferred Units are or dB

Summary

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Lab Demo: Guitar String FrequencySpectrum

Understanding and displaying the frequency content of signals

Demonstrate the use of a , a common measurement instrument for computing and displaying the frequency spectrum

Lesson Objectives

5

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2

is an instrument to measure and compute the frequency spectrum

Guitar string produces a tone and

Summary

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FrequencyResponse: Linear Plots

Understanding and displaying the frequency response of systems

Introduce the frequency response as a way of showing how a system processes signals of different frequencies

Lesson Objectives

5

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2

Frequency Response

6

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Mag

nitu

de0 200 400 600 800 1000

-100

-80

-60

-40

-20

0

A

ngle

(de

g)

vs

R

-

+vcC+-

Transfer Function

)tan()(

)(1

1)(

1

1)(

2

RCaHRC

HRCjH

=+

=

+=

Circuit Response

7

Vin Vout0 0.05 0.1 0.15 0.2 0.25

-2

-1

0

1

2

Time (sec)

v(t)

0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

v(t)

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Frequency Domain

50 800

1

50 800

1

Time Domain

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3

A circuit has the frequency response plot shown. What is steady-state response, vo(t), to an input of vin(t) = 2 + cos(200t)?

Example

8

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Mag

nitu

de0 200 400 600 800 1000

-100

-80

-60

-40

-20

0

A

ngle

(de

g)

A is a plot of the transfer function versus frequency

The frequency response can be used to determine the steady-state sinusoidal response of a circuit at different frequencies

Summary

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FrequencyResponse: Bode Plots

Understanding and displaying the frequency response of systems

Introduce the Bode plot as a way of showing the frequency response

Lesson Objectives

5

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2

Bode Plots

6

Frequency (rad/sec) or f (Hz)1 10 100 1000

Frequency (rad/sec) or f (Hz)1 10 100 1000

Linear Plot and Bode Plot

7

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Bode Plot First-Order Characteristics

8

Bode Plot of RLC Circuit, Overdamped

9

vs

+

-vcC

L

vs+

--

R

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Bode Plot of RLC Circuit, Underdamped

10

Example

11

A circuit has the Bode plot shown. What is the steady-state response of an input of vs(t)=1+cos(100t-45o)+cos(3000t)?

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A is a plot of the transfer function versus frequency A is the frequency response on a log scale Units are or dB RC Circuit magnitude goes down by 20dB/decade phase goes from 0o to -90o

RLC Circuit magnitude goes down by 40dB/decade phase goes from 0o to -180o

RLC with low damping has resonant peak

Summary

12

9/22/2013

1

Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering


Lab Demo: RLC Circuit Frequency Response

Transient response of an RLC circuit

RLC Circuit Schematic

4

vs

20k +

-vc0.01f

3.3mH

+15v

-15v

+

-

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2


5

Low R means low damping and high resonant peak The Bode plot is generated by a sine

sweep Input sinusoids of different frequencies and

calculate the gain (Ao/Ai) and phase for each response

Compute and plot 20*log10(Ao/Ai) vs f Plot phase vs f

Summary

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Lowpass and Highpass Filters

Introduce lowpass and highpass filters

Introduce filtering concepts Show the properties of lowpass and highpass filters

Lesson Objectives

5

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2

An is a circuit that has a specific shaped frequency response to attenuate (or filter) signals with specific frequency content

Analog Filters

6

Lowpass Filter

Highpass Filter

Lowpass Filter Example

7

Vin Vout0 0.05 0.1 0.15 0.2 0.25

-2

-1

0

1

2

Time (sec)

v(t)

0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

v(t)

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Frequency Domain

50 800

1

50 800

1

Time Domain

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Pass low frequency components and attenuate high frequency components

Lowpass Filters

8

Linear Plot

Magn

itude

KDC

B

0.707KDC

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Mag

nitu

de

RC circuitR = 1000, C = 10F

Lowpass Filter Example

9

0.05 0.1 0.15 0.2 0.250

5

10

15

20

Time (sec)

Vou

t

0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Time (sec)

Vin

CircuitVin Vout

2

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4

Bode Plots of Lowpass Filters

10

Linear Plot

Magn

itude

KDC

B

0.707KDC

Bode Plot

Magn

itude

(dB)

20log10(KDC)3dB

Example Lowpass Filter Bode Plot

11

Bode Plot

Magn

itude

(dB)

20log10(KDC)3dB

101

102

103

104

105

-80

-60

-40

-20

0

Mag

nitu

de (

dB

)

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5

Passes high frequency components and attenuates low frequency components

Highpass Filter

12

Linear Plot

Magn

itude

vin R+

-vo

C

1+

=

RCj

RCjH )(

Highpass Filter Example

13

0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Time (sec)

Vin

0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Time (sec)

Vou

tCircuitVin Vout

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6

RLC Filters

14

vinR +

-voC

L

+= RCjLCH )()( 21

1

Lowpass Filter

Highpass Filter

+

= RCjLC

LCH )()( 22

1vin

R

+

-vo

C

L

An is a circuit that has a specific shaped frequency response

A passes low frequency component in signals and attenuates high frequency components

A passes high frequency components in signals and attenuates low frequency components

Summary

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Bandpass and Notch Filters

Show schematics and characteristics of notch and bandpass filters

Introduce characteristics of notch and bandpass filters

Lesson Objective

5

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2

Summary of RC Filters

6

vinR +

-voC

vin R+

-vo

C

Summary of RLC Filters

7

vin

R +

-vo

CL

vin

R

+

-vo

C

L

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3

RLC Bandpass Filter

8

+

= RCjLC

RCjH)1(

)(2

Passband

LC1

LR

-3dB

Example Bandpass Filter

9

+

= RCjLC

RCjH)1(

)(2

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4

Notch RLC Filter

10

+

= RCjLC

LCH)1(

1)(

2

2

LC1

Example Notch Filter

11

+

= RCjLC

LCH)1(

1)(

2

2

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Different filter characteristics can be found from RC and RLC circuits

passes frequencies in a bandrejects frequencies in a band

Summary

24

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1



Lab Demo: Guitar String Filtering

Lowpass filtering of the guitar string signal

Tone Control

4

vin

R1

+

-vo

C

R2+

-

R1 = 10kR2 = 47kC = 0.022f

100k pot for tone 100k pot for volume

Stupidly Wonderful Tone Control 2, www.muzique.com/lab/swtc.htm

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2

Input Output Relationship

5

CircuitH()Vin Vout

0 0.05 0.1 0.15 0.2 0.25-2

-1

0

1

2

Time (sec)

v(t)

0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

v(t)

Linear Scale: Ai |H()| = Ao|Input| x |H| = |Output|

Bode Scale: |Input|dB+ |H|dB = |Output|dB

Lab Demo: Guitar String Filtering

6

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Frequency Response of Lowpass Filter

7

Input and Output Spectra

8

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Input/Output Relationship Linear Scale: |Input| x |H| = |Output| Bode Scale: |Input|dB + |H|dB = |Output|dB

First-Order filter: -20dB/dec rolloff Passive filters Made of R, L, and C components Require no power supply

Active filters Made of R, C, and operational amplifiers Require a power supply

Summary

9

Ken Conner from RPI Stupidly Wonderful Tone Control 2,

www.muzique.com/lab/swtc.htm

Credits

10

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1



Module 4 Frequency Analysis Wrap Up

Summary of the Module

Concept Map

3


Reactive Circuits

Frequency Analysis

Power

Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)

RC, RLC circuits

Phasors Impedance AC Circuit

Analysis

Transfer Function

Frequency Response

Filters

Frequency Analysis

9/22/2013

2

Resistive vs Reactive Circuits

4

Time

Volta

ge

Be able to identify sinusoid properties (amplitude, frequency, angular frequency,

period, phase) find phasors of sinusoidal functions add sinusoids using phasors Understand and describe the

properties of sinusoids in capacitors and inductors

Important Concepts and Skills

5

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3

Understand impedance Be able to

calculate impedances of resistors, capacitors, and inductors identify the relationship between

voltage and current based on impedance value


6

Given a source frequency, be able to convert RLC circuits into equivalent circuits with impedances find equivalent impedances for devices in series/parallel solve for voltages and currents using

resistor analysis methods (Ohms Law, KCL, KVL, Mesh, Node, Thvenin, Norton)


7

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4

Know the definition of a transfer function how a linear system responds to a sinusoid in steady state (how the amplitude and

phase change but the frequency stays the same) the meaning of the plot of the transfer function in terms of finding an output

amplitude Be able to

find the transfer functions of simple RL, RC and RLC circuits

sketch the magnitude and angle of the transferfunctions of a first-order system on a linear scale


8

Know the definition of a frequency spectrum

Be able to plot a frequency spectrum of a sum of sinusoids Recognize high and low frequency content in a

signal in both the time domain and in the frequency domain


9

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5

Know the what a frequency response is and understand the graphical features of RC and

RLC circuits when plotted on linear scales and on Bode scales

Be able to sketch a frequency response from a transfer

function on linear scales match time domain and frequency domain inputs

and corresponding outputs for a circuit with a known frequency response


10

Know the motivation for using filters the definition of a filter the frequency response features of lowpass,

highpass, bandpass, and notch filters Be able to

identify RC and RLC circuits as being lowpass, bandpass, highpass, or notch

determine acceptable circuit parameters to achieve desired bandwidth, corner frequencies, and/or passband or rejection frequencies


11

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6

Do all homework for this module Study for the quiz Continue to visit the forum

Reminder

12

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1




Transient response of an RLC circuit

RLC Circuit Schematic

4

vs

20k +

-vc0.01f

3.3mH

+15v

-15v

+

-

9/22/2013

2


5

Low R means low damping and high resonant peak The Bode plot is generated by a sine

sweep Input sinusoids of different frequencies and

calculate the gain (Ao/Ai) and phase for each response

Compute and plot 20*log10(Ao/Ai) vs f Plot phase vs f

Summary

6

Documents

4-2Sinusoids in Circuitsece2040.ece.gatech.edu/videos/Module4Handouts.pdf · 1/5/2014 1 School of Electrical ... Identify how past techniques apply to impedances in AC circuit analysis