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 sin(wt +f)
- A cos(wt +f)
A sin(wt +f)
A cos(wt +f)
A cos(wt + f - 90o)
A sin(wt + f + 180o )
Or
A sin(wt + f - 180o )
A cos(wt + f + 180o )
Or
A cos(wt + f - 180o )
A sin (wt + f - 360o)
Or
A sin (wt + f + 360o)
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)
Impedances
Admittances
Phasor Notation
ZR = R = 1/G ZR = R  0o
Phasor Notation
YR = 1/R = G YR = G 0o
ZL = jwL
ZL = wL90o
ZC = -j/(wC)
ZC = 1/(wC) -90o YC = jwC
YL =-j/(wL)
YL = 1/(wL) -90o
YC = wC 90o
Ohm’s Law in Phasor Notation
V=IZ
I = V/Z
V = I/Y
I=VY
Equivalent Impedances
In Series:
Zeq = Z1 + Z2 + Z3….+ Zn
In Parallel:
Equivalent Admittances
In Series:
Yeq = [1/Y1 +1/Y2 +1/Y3….+ 1/Yn] -1
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.