CIRCUITS by Ulaby & Maharbiz 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 Useful relations

Phase Lead/Lag

Complex Numbers We will find it is useful to represent sinusoids as complex numbers Rectangular coordinates Polar coordinates Relations based on Euler’s Identity

Relations for Complex Numbers Learn how to perform these with your calculator/computer

Phasor Domain 1. The phasor-analysis technique transforms equations from the time domain to the phasor domain. 2. Integro-differential equations get converted into linear 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 domain provides 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 Current through a resistor Time domain Time Domain Frequency Domain Phasor Domain

Phasor Relation for Inductors Current through inductor in time domain Time domain Phasor Domain Time Domain

Phasor Relation for Capacitors Voltage across capacitor in time domain is Time domain Time Domain Phasor Domain

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 Impedance is voltage/current Admittance is current/voltage R = resistance = Re(Z) G = conductance = Re(Y) X = reactance = Im(Z) B = susceptance = Im(Y) Resistor Inductor Capacitor

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