ECE410 Spring 2012 Lecture #32 AC Circuits I

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:

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

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

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

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:

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

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

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

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

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

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

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

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

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

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.

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:

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:

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

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:

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:

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

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