Study Guide

Final

Final

• Closed book/Closed notes• Bring a calculator or use a mathematical

program on your computer

• The length of the exam is the standard 2 hours for a final.– Hopefully, you will not take that long.

Conversions

• Sinusoids – Sine and Cosine• Know the transformation between angular frequency

(w) and frequency (f)• Know how to calculate the period

• Phasor Notation• Exponential Form• Rectangular Coordinates

Conversions for Sinusoids

A sin(wt +f) A cos(wt + f - 90o)- A sin(wt +f) A sin(wt + f + 180o )

Or A sin(wt + f - 180o )

- A cos(wt +f) A cos(wt + f + 180o )Or

A cos(wt + f - 180o )

A sin(wt +f) A sin (wt + f - 360o)Or

A sin (wt + f + 360o)

A cos(wt +f) A cos (wt + f - 360o)Or

A cos (wt + f + 360o)

Sinusoid to Phasor

• The sinusoid should be written as a cosine• Amplitude or magnitude of the cosine should

be positive– This becomes the magnitude of the phasor

• Angle should be between +180o and -180o.– This becomes the phase angle of the phasor.

Phasor to Exponential Form

• The magnitude of the phasor is the magnitude of the coefficient.

• The phase angle multiplied by j is the exponent of e.

B f = B ejf

Rectangular to Phasor

• Amplitude of phasor is equal to the square root of the sum of the square of the real component and the square of the imaginary component.

• Angle of the phasor is equal to the inverse tangent of the imaginary component divided by the real component

Phasor to Rectangular

• Real component is the product of – the amplitude of the phasor and the cosine of the

phase angle.• Imaginary component is the product of – the amplitude of the phasor and the sine of the

phase angle.

Steps to Perform Before Comparing Angles between Signals

• The comparison can only be done if the angular frequency of both signals are equal.

• Express the sinusoidal signals as the same trig function (either all sines or cosines).

• If the magnitude is negative, modify the angle in the trig function so that the magnitude becomes positive.

• If there is more than 180o difference between the two signals that you are comparing, rewrite one of the trig functions

• Subtract the two angles to determine the phase angle.

Lagging or Leading

v1(t) = Vm1 sin(wt + f1)

v2(t) = Vm2 sin(wt + f2)

If f1 > f2, v1(t) leads v2(t) and v2(t) lags v1(t)

If f1 < f2, v2(t) leads v1(t) and v1(t) lags v2(t)

ImpedancesPhasor Notation

AdmittancesPhasor Notation

ZR = R = 1/G ZR = R 0o YR = 1/R = G YR = G 0o

ZL = jwL ZL = wL90o YL =-j/(wL) YL = 1/(wL) -90o

ZC = -j/(wC) ZC = 1/(wC) -90o YC = jwC YC = wC 90o

Ohm’s Law in Phasor Notation

V = I Z V = I/YI = V/Z I = V Y

Equivalent Impedances Equivalent Admittances

In Series: In Series:

Zeq = Z1 + Z2 + Z3….+ Zn Yeq = [1/Y1 +1/Y2 +1/Y3….+ 1/Yn] -1

In Parallel: In Parallel:

Zeq = [1/Z1 +1/Z2 +1/Z3….+ 1/Zn] -1 Yeq = Y1 + Y2 + Y3….+ Yn

Voltage Division:Impedances and Admittances

The voltage associated with one component in a chain of multiple components in series is

Vz = [Z/Zeq] Vtotal

Vz = [Yeq/Y] Vtotal

where Vtotal is the total of the voltages applied across the resistors.

Current Division:Impedances and Admittances

The current associated with one component in parallel with one or more components is

IZ = [Zeq/Z] Itotal

IZ = [Y/Yeq] Itotal

where Itotal is the total of the currents entering the node shared by the resistors in parallel.

Transformations to Simplify Circuits

• The relationship between the sources used in Thévenin and Norton transformations is:– Vth = Zth IN

• If Zth has an non-zero phase angle, then the phase angle of Vth is equal to the sum of the phase angle of the impedance plus the phase angle of the Norton current source.

Circuit Analysis

• The other circuit analysis techniques that were discussed in the first half of the course can be applied when a circuit contains resistors, capacitors, and/or inductors.– These do not have to be applied when calculating

the currents and voltages in the circuits on the final. You should be able to find thes values of these parameters using equivalent impedance, voltage and/or current division, and Thévenin and/Norton transformations.