7. AC ANALYSIS

CIRCUITS by Ulaby & Maharbiz

Overview

Linear Circuits at ac

Objective: To determine the steady state response of a linear circuit to ac signals

Sinusoidal input is common in electronic circuits Any time-varying periodic signal can be represented by a

series of sinusoids (Fourier Series) Time-domain solution method can be cumbersome

Sinusoidal Signals

tVtv cosm

f 2

fT

1

Useful relations

Phase Lead/Lag

Complex Numbers

We will find it is useful to represent sinusoids as complex numbers

jyxz jezzz

1j

Rectangular coordinatesPolar coordinates

sincos je j

Relations based on Euler’s Identity

yz

xz

)Im(

Re

Relations for Complex Numbers

Learn how to perform these with your calculator/computer

Phasor Domain

1. The phasor-analysis technique transforms equationsfrom the time domain to the phasor domain.

2. Integro-differential equations get converted intolinear equations with no sinusoidal functions.

3. After solving for the desired variable--such as a particular voltage or current-- in the phasor domain, conversion back to the time domainprovides the same solution that would have been obtained had the original integro-differential equations been solved entirely in the time domain.

Phasor Domain

Phasor counterpart of

Time and Phasor Domain

It is much easier to deal with exponentials in the phasor domain than sinusoidal relations in the time domain.

You just need to track magnitude/phase, knowing that everything is at frequency .w

Phasor Relation for Resistors

Time Domain Frequency Domain tRIiRv cosm

Current through a resistor

tIi cosm

Time domain

Phasor Domain

Phasor Relation for Inductors

Time Domain

Current through inductor in time domain

Time domain

Phasor Domain tIi cosm

Phasor Relation for Capacitors

Time Domain

Voltage across capacitor in time domain is

Time domain

Phasor Domain

tVv cosmdt

dvCi

Summary of R, L, C

ac Phasor Analysis General Procedure

Using this procedure, we can apply our techniques from dc analysis

Example 1-4: RL Circuit

Cont.

Example 1-4: RL Circuit cont.

Impedance and Admittance

R = resistance = Re(Z)

Impedance is voltage/current

X = reactance = Im(Z)

Resistor

Inductor

Capacitor

RZ

LjZ

Cj/1Z

R/1Y

Lj/1Y

CjY

G = conductance = Re(Y)

Admittance is current/voltage

B = susceptance = Im(Y)

Impedance Transformation

Voltage & Current Division

Cont.

Example 7-6: Input Impedance (cont.)

Example 7-9: Thévenin Circuit

Linear Circuit Properties

Thévenin/Norton and Source Transformation Also Valid

Phasor Diagrams

Phase-Shift Circuits

Example 7-11: Cascaded Phase Shifter

Solution leads to:

Node 1 Cont.

(cont.)

Cont.

(cont.)

Example 7-14: Mesh Analysis by Inspection

Example 7-16: Thévenin Approach

Example 7-16: Thévenin Approach (Cont.)

Example 7-16: Thévenin Approach (Cont.)

Power Supply Circuit

Ideal Transformer

Half-Wave Rectifier

Full-Wave Rectifier

Current flow during first half of cycle

Current flow during second half of cycle

Smoothing RC Filter

Complete Power Supply

Example 7-20: Multisim Measurement of Phase Shift

Example 7-20 (cont.)

Using Transient Analysis

Summary