1. ECE410 Spring 2012
Lecture #32
AC Circuits I

2. Homework Due 3/16/2012
Chapter 7 – Problems 33, 35, 48, 50, 55, 70, 73, 74, 78, 89
PSpice – Model the circuit from problem 89 from your homework assignment. This circuit is shown below as well as the input waveform:

3. Homework (cont) Plot voltage v0 vs. time.
Ignore the switch in the circuit and model the input waveform using VPWL, a piecewise linear voltage source. Using this source you can specify the voltage at multiple times (t1,t2,t3,…) and the corresponding voltages at those times (v1, v2, v3, …) and the points will be connected by straight lines. In order to model the vertical transitions use two times that are close together.. (let t1=0, v1=0; t2=1ns, v2=-200mV) to simulate the initial voltage change from 0V to -200mV at time zero.
My suggestion:
Run the simulation for 750ms
Use a LM741 or similar op-amp in your simulation

4. Midterm Exam #3 Friday April 27th Will Cover Chapters 6, 7 and 9
Capacitance
Inductance
RC and RL circuits
Sinusoidal Steady State Analysis
Use 1-3x5 Notecard (both sides)
Can bring a calculator and writing utensil
Be ready to go at the beginning of the class period

5. AC Waveforms
AC waveforms are sinusoidal and can be described by either a sine or cosine.. We will use the cosine convention.
phase
Vm = Amplitude
ω = angular frequency (equal to 2πf or 2π/T)
φ = phase angle

6. Effect of Phase Angle
The phase angle shifts the cosine wave to the right or left in time.
Cosine with non-zero phase angle
Unshifted cosine
Cosine is equal to 1 when the phase is equal to zero. Thus:
The wave with phase angle φ with be shifted in time by:

7. Sine-Cosine Relations
The following are useful relationships for AC circuits:
Shifting a sine by 90 degrees results in ±cosine
Shifting a cosine by 90 degrees results in ±sine

8. RMS Values
AC sinusoids have an average of zero, so we use RMS (root of the mean squared function) to get a measure for effective average value.
For sinusoidal signals

9. RMS calculation for a non-sinusoid
Find RMS value for a Square Wave:
Let Amplitude = Vmax

10. Complex Number Review
Complex numbers involve numbers with imaginary terms (involving j=sqrt(-1))
They can be expressed in polar form or rectangular form
Rectangular x = A+jB
Polar
Complex numbers can be plotted in the complex plane

11. Complex Plane Imaginary
A complex number is a vector in the complex plane
It can be expressed in terms of it’s real and imaginary components… this is the rectangular form:
X = A + jB
Or it can be expressed in terms of it’s angle to the real axis and it’s length… this is the polar form: B C φ
Real A
where

12. Conversion Between Polar and Rectangular Notation
Changing from Polar to Rectangular
Changing from Rectangular to Polar
Given length C and angle φ
Given Real component A and imaginary component B

13. Real and Imaginary Parts
The Real Part of a complex number is it’s vector component in real direction
The Imaginary Part of a complex number is it’s component in the imaginary direction
Imaginary B C φ
Real A

14. Sinusoidal Response
What happens if we drive a circuit with capacitance or inductance with a sinusoidal voltage source?
Full Solution:

15. Characteristics of Response
The transient solution only lasts for a short time
The steady state solution is a sinusoid
The steady state sinusoid has the same frequency as the sinusoid used in the driving voltage.
The Amplitude and phase angle of the steady state response differ from the driving voltage source
Transient Response (dies out with time)
Steady State Response

16. Implications of the Steady State Response
The steady state response of a sinusoidal driving voltage is a sinusoid with the same frequency
We therefor only need to keep track of the amplitudes and phase angles of the voltages and currents in our circuit.
ω is constant in all expressions and can be ignored.

17. Phasors and Phasor Transform
The phasor allows us to simplify dealing with sinusoids by looking at them in the complex domain (also called frequency domain)
Euler’s Identity
The Phasor Representation or Phasor Transform of a Sinusoidal waveform drops the frequency term and the Real Designation:

18. Phasor Continued
A phasor can be expressed in either polar or rectangular form as we showed earlier when discussing complex numbers
People often get tired of writing the exponential form and have developed the following shorthand:

19. Inverse Phasor Transform
The inverse phasor transfrom converts a phasor back into a sinusoidal voltage waveform.
Note that there is nothing in the phasor itself that allows you to know what ω is. It must be independently known for your circuit

20. Usefulness of the Phasor
The phasor transform is useful because it applies directly to the sum of sinusoidal voltages.
If we have a sum of sinusoidal voltages:
We can also represent it as a sum of phasors:

21. Adding Sinusoids with and without phasor
Try adding
Try with trig identities to express this as a single sinusoid
Try doing the same with phasors
I will work through both on the board
Hint… phasors are a LOT easier
Remember our expressions from our complex number review:

22. Hints for using phasors
Phasors are easiest to add in rectangular form
When adding (and subtracting) phasors… convert them all to rectangular format first, it will save you time
Phasor are easiest to multiply and divide in polar form
When converting from a phasor to a sinusoid through the inverse phasor transform, always make sure the phasor is in polar form
Remember, as annoying as it might be to convert back and forth from polar to rectangular form, it is much easier than trying to directly deal with the sinusoidal expression

23. Sample Problems Find the phasor transform of the following:
Find the time-domain expression for the following: