Download review for elec 105 midterm exam #1 (fall 2001)

ELEC 225
Circuit Theory I
Fall 2007
Review Topics for Final Exam
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 sheets in addition to those listed below.
This list is not guaranteed to be free of errors. In a case of conflicting information, the text book
will almost always be considered to be the final authority.
Capacitors and inductors in DC (non-time-varying) circuits
- capacitors act as open circuits
- inductors act as short circuits
Phasors
- definition, voltage example: v(t) = Re{Vejt}
- 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 quotient)
- although impedances are complex numbers, they are not phasors, because they do not
represent sinusoidal signals
- representations of phasors:
o polar form using complex exponential function (mag. and phase)
o polar form using the angle symbol (mag. and phase)
o rectangular form (real and imaginary parts)
- 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
o capacitor: Z = 1/jC
- general complex impedance: Z = R + jX
o real part (R) is resistance
o imaginary part (X) is reactance
- 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 
1
-
-
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. imag.)
o capacitive reactance is negative (impedance of capacitor is neg. imag.)
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
o parallel: Z eq 
1
1
1
1


Z1 Z 2
ZN
o same rules as those used for resistors
- admittance, Y:
o resistor: Y = G = 1/R
o inductor: Y = 1/jL
o capacitor: Y = jC
- general complex admittance: Y = G + jB
- B = susceptance
- inductive susceptance is negative
- capacitive susceptance is positive
- Ohm’s Law for admittances: V = I/Y or I = VY
- equivalent admittance formulas
1
o series: Yeq 
1 1
1
 
Y1 Y2
YN
o parallel: Yeq  Y1  Y2    YN
o reverse rules of those used with 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
Resonant circuits
- resonant frequency of series or parallel LC resonant circuit is given by 2 = 1/LC
- series resonance: C and L in series; capacitive reactance equal and opposite to
inductive reactance; equivalent impedance is zero; equiv. admittance is infinity
- parallel resonance: C and L in parallel; capacitive reactance equal and opposite to
inductive reactance; equivalent impedance is infinity; equiv. admittance is zero
2
-
voltages can be extremely high in series resonant circuits, even though total across L
and C together is near zero
- currents can be extremely high in parallel resonant circuits (“circulating” currents),
even though total flowing into/out of L and C combination is near zero
Thévenin and Norton equivalent circuits
- use same rules for finding open-ckt voltage and short-ckt current as with DC circuits
- equivalent impedance instead of equivalent resistance
- three methods for finding equivalent impedance are conceptually the same as with
DC circuits:
o Zth = Voc/Isc (can use only if there are independent sources)
o Zth = Zeq (can use if there are no dependent sources)
o Zth = Vt/It (“test source” method; can always use)
RMS voltage and current values
- rms = “root mean square” (the square root of the time average power over one cycle;
voltage is proportional to square root of power)
- for sinusoids, peak and rms values differ by square root of two factor
- conversion from rms to peak to pp (and back)
- rms value is the “effective” power, which is used in energy transfer calculations
Power calculations in AC circuits
- real, or time-average, power (P); unit is the Watt (W)
- reactive power (Q); unit is the Volt-Ampere reactive (VAR)
- P represents power that either does useful work or is wasted as heat
- Q represents power that is stored and later returned to the source or to other energystorage devices (capacitors or inductors) in the circuit
- complex power (* indicates complex conjugation):
o S = P + jQ
o S = 0.5VI*, if magnitudes of V and I are expressed as peak values
o S = VI*, if magnitudes of V and I are expressed as rms values
o S = 0.5|I|2Z (peak magnitudes) or |I|2Z (rms magnitudes)
o S = 0.5|V|2Y (peak magnitudes) or |V|2Y (rms magnitudes)
- power factor (pf)
o cosine of the power angle (power angle = v – i)
o also equal to cosine of phase of impedance
o lagging vs. leading power factor (lagging if i lags v; leading if i leads v)
o lagging pf indicates an inductive load (pos. reactance); leading pf indicates a
capacitive load (neg. reactance)
o mnemonic: ELI the ICE man
o pf correction capacitors often added to inductive loads to bring pf to 1
- magnetizing VARs
o refers to reactive power, Q
o magnetizing VARs “absorbed” by inductive loads (Q is pos.)
o magnetizing VARs “delivered” by capacitive loads (Q is neg.)
Relevant course material:
HW:
Labs:
Textbook:
#7-#9
#4-#6
Secs. 9.3-9.9, 10.1-10.5
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