ELEC 225
Circuit Theory I
Fall 2010
Review Topics for Exam #2
The following is a list of topics that could appear in one form or another on the exam. Not all of
these topics will be covered, and it is possible that an exam problem could cover a detail not
specifically listed here. However, this list has been made as comprehensive as possible. You
should be familiar with the topics on the previous review sheet in addition to those listed below.
Sinusoidal voltages and currents (usually called AC)
- standard cosine form: v(t) = Vm cos (t + )
- Vm = amplitude or magnitude (in units of Vpk, if voltage)
- relationship of Vpk (peak) to Vpp (peak-to-peak) units
-  = radian frequency, in rad/s
- f = linear or cyclic frequency, in Hz (cycles/s)
-  = phase (in degrees or radians, but note that t is in radians)
- T = period, in s (time duration of one full cycle)
-  = 2f, T = 1/f
Sinusoidal steady-state analysis
- A sinusoidal source (stimulus) causes all of the other voltages and currents in a circuit
(response) to be sinusoidal at the same frequency, but they will not generally have the
same magnitude and phase as the source.
- Sinusoidal steady-state analysis applies to the steady-state (long-term) response of a
circuit to an applied voltage or current; the transient (short-term) response is ignored.
- advantages of/reasons to study AC include:
o Electrical power worldwide is generated almost exclusively in AC form
(usually at 50 or 60 Hz; 400 Hz in aircraft).
o All radio/wireless devices use AC to generate/detect electromagnetic waves.
o Many signals produced by sensors (such as microphones) are AC in nature at
one frequency, multiple frequencies, or over a continuum of frequencies.
Inductors
- time-varying magnetic field causes voltage to appear across terminals (this voltage is
sometimes called the “back emf,” where emf stands for “electromotive force”)
- unit is the Henry (H)
-
-
circuit symbol
current-voltage relationships
o passive sign convention – use pos. form if i flows into pos. side of v
di t 
o vt    L
dt
1 t
o i t     v( )d  i t o 
L t0
voltage leads current by 90° (ELI in “ELI the ICE man”); the voltage peaks occur 90°
before the current peaks in a plot of voltage and current vs. time
equivalent inductance formulas
o series: Leq  L1  L2    L N
o parallel: Leq 
1
1
1
1


L1 L2
LN
1
Capacitors
- unit is the Farad (F)
-
-
Phasors
-
circuit symbol
current-voltage relationships
o passive sign convention – use pos. form if i flows into pos. side of v
dvt 
o i t   C
dt
1 t
o vt     i ( )d  vt o 
C to
voltage lags current by 90° (ICE in “ELI the ICE man”); the voltage peaks occur 90°
after the current peaks in a plot of voltage and current vs. time
equivalent capacitance formulas
1
o series: Ceq 
1
1
1


C1 C 2
CN
o parallel: C eq  C1  C 2    C N
definition, voltage example: v(t) = Re{Vejt}; V is the phasor
by convention in EE, a phasor represents a cosine (not a sine) function;
for example, V = Vm L v ↔ Vm cos (t + v)
magnitude (amplitude) of cosine function = magnitude (modulus) of phasor
phase of cosine function in time domain (everything added to t) = phase of phasor
A given phasor representation (its complex numerical value) is valid only at a single
frequency; however, phasors can be expressed as functions of frequency.
in EE, square root of –1 is j, not i
Phasors can only be used to evaluate the sum (or difference) of two or more sinusoids
at the same frequency, but not their product (or ratio).
Although impedances are complex numbers, they are not phasors, because they do
not represent sinusoidal signals.
representations of phasors:
o polar form (using the angle symbol); example: V  0.330 V
o polar form (complex exponential); example: V  0.3e j 30  0.3e j 6 V
o rectangular form; example: V  0.3cos 30  j sin 30  0.26  j 0.15 V
o phasor diagram (vector in complex plane) can also be used
- conversion from one phasor representation to another
Sinusoidal steady-state AC circuit analysis using phasors
- also known as frequency-domain analysis
- impedance, Z:
o resistor: Z = R
o inductor: Z = jL
1
o capacitor: Z 
jC
o general complex impedance: Z = R + jX
o real part (R) is resistance
o imaginary part (X) is reactance
2
-
-
-
reactance (X)
o reactance itself is a real quantity; impedance is (in general) complex
o example 1: the reactance of a 5-mH inductor at 1000 Hz is 31.4 , but its
impedance is j31.4 
o example 2: the reactance of a 1-F capacitor at 1000 Hz is −159 , but its
impedance is −j159 
o example 3: a given complicated arrangement of resistors, inductors, and
capacitors has an equivalent impedance of 4120 – j1800 . Its equiv.
resistance is 4120 , and its equiv. reactance is 1800 .
o inductive reactance is positive (impedance of inductor is pos. imaginary)
o capacitive reactance is negative (impedance of capacitor is neg. imaginary)
Ohm’s Law for impedances:
o V = IZ
o voltage and current are phasors (indicated by boldface or by tilde ~)
o impedance, although complex, is not a phasor (so not boldface/no tilde)
o note that the use of boldface and/or tilde is not a widespread standard; must
pay attention to context
equivalent impedance formulas
o series: Z eq  Z 1  Z 2    Z N
1
1
1
1


Z1 Z 2
ZN
o same rules as those used for resistors
circuit analysis techniques applicable in frequency domain:
o Ohm’s law
o KVL, KCL
o voltage-divider formula, current-divider formula
o nodal analysis, mesh analysis
o Thévenin equivalent circuits, Norton equivalent circuits
o source transformations
o superposition and linearity
o parallel: Z eq 
-
Relevant course material:
HW:
Labs:
Textbook:
Supplements:
#4-#5
#4-#5
Secs. 7-1 through 7-5
none
3