7. AC Analysis CIRCUITS by Ulaby & Maharbiz
Linear Circuits at ac Objective:Todetermine 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
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
ac Phasor Analysis General Procedure Using this procedure, we can apply our techniques from dc analysis
Example 1-4: RL Circuit Cont.
Resistor Inductor Capacitor 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)
Linear Circuit Properties Thévenin/Norton and Source Transformation Also Valid
Example 7-11: Cascaded Phase Shifter Solution leads to:
Node 1 Cont.
Full-Wave Rectifier Current flow during first half of cycle Current flow during second half of cycle
Example 7-20 (cont.) Using Transient Analysis