E246: Electronics & Instrumentation

Lecture 5: AC Steady-State Analysis

Plan

Sinusoidal Sources and Sinusoidal ResponseReview of Complex NumbersPhasorImpedance and AdmittanceKirchhoff’s Laws Using PhasorsCircuit Analysis in Frequency DomainSections in textbook: 4.1 – 4.3, 4.5, 4.7-4.12

Introduction

We now discuss sources that vary sinusoidallySinusoidal sources are important because

Generation, transmission, and consumption of power occur under sinusoidal steady-state conditionsSteady-state sinusoidal behavior often simplifies the design of electrical systems

Circuit analysis and simplifications techniques of previous chapters work for circuits with steady-state sinusoidal sources

Introduction

A sinusoidal voltage source (dependent or independent) produces a voltage that varies sinusoidally with timeA sinusoidal current source (dependent or independent) produces a current that varies sinusoidally with timeWe can express a sinusoidally varying function with either a sine or a cosine function.

Sinusoidal Sources

Signal )sin()( φω += tVtv m

Features of the curve

It is periodic (the time, T, required to repeat is called the period);The frequency is f=1/T (1 cycle per second is 1Hz);We define the angular frequency as

Vm is the maximum amplitude of the sinusoidal voltageThe angle is the phase angle and determines the value of the voltage at t=0.If is positive, the curve shifts to the left; if it is negative, it shifts to the right.Note that should carry same units.

radians/s /22 Tf ππω ==

φ

φ

)sin()( φω += tVtv m

and φωt

Phase Relationship

If the voltage across an element is

and a current flows through the element is

then we say “the current leads the voltage by radians”.

tVtv m ωsin)( =

)sin()( φω += tIti m

φ

Example:

Determine the phase relationship between the voltage and current.

A )103sin(2)(V 3cos3)(

o+−==

ttittv

Sinusoidal Signals

If a signal has the form

it can also be represented by tBtAv ωω sincos +=

.0 when tan180

;0 when tan where

)cos(

1

1

22

<+=

>=

−+=

−

−

AAB

AAB

tBAv

oθ

θ

θω

Sinusoidal Responses

Applying a sinusoidal voltageto the RL circuit, regardless of the initial value, the response will become purely sinusoidal as t tends to infinity. We are interested in finding the steady-state response.

tVv ms ωcos=

The key points of the steady-state solution areIt varies sinusoidally;The frequency of the response is identical to the frequency of the source;The maximum amplitude of the response is different from the maximum amplitude of the source and depends on the circuit parameters;The response is out of phase with the source by a phase angle that also depends on the circuit parameters.

Review of Complex Numbers

For the complex number

x is the real part of z while y is the imaginary partz can also be written r is the magnitude of z and is the phase of z

1-j , =+= jyxz

φφ ∠== rrez j

φ

z can be represented three ways

The relationship between the rectangular and polar forms are

form lexponentia formpolar

formr rectangula

φ

φ

∠==

+=

rzrez

jyxzj

φφ

φ

sin,cos

tan, 122

ryrxxyyxr

==

=+= −

φφ

φ

sin,cos

tan, 122

ryrxxyyxr

==

=+= −

Mathematical operations of complex numbers are summarized in Appendix B in the textbook.Addition and subtraction of complex numbers are better performed in rectangular form while multiplication and division are best performed in polar form

Phasor

Phasors provide a simple means of analyzing linear circuits excited by sinusoidal sourcesA phasor is a complex number that represents the amplitude and phase of a sinusoidTo understand phasors, we must understand complex numbers

Phasor

Given a sinusoid

we define the phasor V to be

In the phasor notation, the term ejwt is always implied; always remember the frequency of the phasor or you will make mistakes

{ } { }φωφωφω jtjm

tjmm eeVeVtVtv ReRe)cos()( )( ==+= +

φφ ∠== mj

m VeVV

Phasor

To get the phasor corresponding to a sinusoid, we express the sinusoid in the cosine form, (so that it can be expressed as the real part of a complex number), take out the ejwt, and what ever is left is the phasorBy suppressing the time factor, we have transformed the sinusoid from the time domain to the phasordomainThough the frequency is suppressed, the response still depends on the frequency. Therefore, the phasordomain is also known as the frequency domain.

Phasor

The phasor concept may be used when the circuits is linear, the steady-state response is sought, and all independent sources are sinusoidal and have the same frequency.

domain) (frequency domain) (time VV )cos()( m φφω ∠=⇔+= tVtv m

Phasor

The differences between v(t) and V must be emphasized

v(t) is the instantaneous or time domain representation while V is the frequency domain representation.v(t) is time-dependent; V is not.v(t) is always real with no complex term; V is generally complex.Phasor analysis applies only when frequency is constant and applies in manipulating two or more sinusoids only if they are of the same frequency.

Phasor

The phasor method uses the transformation from the time domain to the frequency domain to more easily obtain the sinusoidal steady-state solution of the differential equation.Examples on RL and RC circuits.

Impedance

The impedance of an element is the ratio of the phasor voltage to the phasor current:

Impedance is the ratio of two phasors, but it is not a phasor itself. It is a complex number that relates one phasor to the other.Impedance has no meaning in the time domain.

IVZ =

Impedance in the frequency domain is the quantity analogous to resistance, inductance, and capacitance in the time domain.The imaginary part of the impedance is called reactance.Impedance is measured in ohms.

formpolar || form lexponentia ||

formr rectangula

θ

θ

∠==

+=

ZZeZZjXRZj

Impedance of R,L,C:

CjZC

LjZLRZR

ω

ω1:

::

=

==

Admittance

The reciprocal of impedance is called the admittance:

ZY 1=

Kirchhoff’s Laws Using Phasors

Kirchhoff’s voltage and current laws hold in the frequency domain. KVL: phasor voltages in a closed path is zero.

KCL: phasor currents at a node is zero.

021 =+++ nVVV L

021 =+++ nIII L

Impedances in Series

neq ZZZZ +++= L21

Admittance in Parallel

neq YYYY +++= L21

neq ZZZZ1111

21

+++= L

Circuit Analysis Methods

All the techniques of analysis we developed for resistive circuits hold for phasor currents and voltages, ie. Principle of superposition, source transformations, Thevenin and Norton’s equivalent circuits, node voltage and mesh current analysis. (All these methods apply as long as the circuit is linear.)

Source transformations

General procedure for circuit analysis:

The steady-state response of a linear circuit with a sinusoidal input is obtained:

Transform the circuit into frequency domain using phasors and impedances;Represent the frequency domain circuit by algebraic equations, e.g, mesh or node equations;Solve the algebraic equations to obtain the response of the circuit;Transform the response into the time domain using phasors.