ELEC 390
Theory and Applications of Electromagnetics
Spring 2008
Review Topics for Exam #1
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.
Electromagnetic spectrum
- allocation of frequencies is determined by international agreement
- choice of frequency for radio service can be dictated by:
o antenna size (which in turn depends on wavelength)
o absorption properties of atmosphere
o other propagation conditions
o capabilities of current technology
o government regulations and/or treaty obligations
Permittivity of free space (o) = 8.854  10-12 F/m
Permeability of free space (o) = 4  10-7 H/m
Phasors
- complex numbers that represent sinusoidal functions in time
~
-
one standard definition (the one we use): A(t )  Re{ A e jt }
in the textbook, phasors are represented by the tilde symbol (~) over the variable
all phasors are generally complex, but not all complex numbers are phasors (e.g.,
impedance is not a phasor quantity)
-
if A | A | e j , then A(t ) | A | cos( t   )
~
~
~
~
~
~
A(t )
 2 A(t )
2
2
 j A and

(
j

)
A



A
t
t 2
- addition and subtraction in time domain are equivalent to addition and subtraction in
phasor domain
- multiplication and division in time domain are not equivalent to multiplication and
division in phasor domain
- phasors can be functions of any variable (usually spatial variables such as x, y, and/or
z) except time (normally)
Complex arithmetic
- addition and subtraction, multiplication and division
- complex conjugate
- polar form vs. rectangular form, and conversion between forms
- identification of magnitude and phase
- identification of real part and imaginary part
- graphical representation of complex nos. on the complex plane (Im part vs. Re part)
- Euler’s identity: e j  cos  j sin 
-
-
z   z z * (z is a complex number)
-
j  e j ( / 2 2n ) where n  0, 1, 2, ...
 j  e j (  / 2 2n ) and  1  e  j (  2n )
1
Transmission line fundamentals
- what constitutes a TEM line? (two long, parallel conductors separated by a dielectric;
there must be two conductors across which a voltage can be measured; current must
be able to flow outward along one conductor and back via the other conductor)
- when is transmission line analysis necessary? (line length is a “significant” fraction of
a wavelength or longer), and when it is not (line length is a “negligible” fraction of a
wavelength)
- characteristic impedance determines initial ratio of V to I (before wave reaches load)
and is dependent on geometry and material properties
- concepts of “source” (or “generator”) and “load”
- concept of “lumped element”
- lumped-element model (R′, L′, G′, and C′) – but you do not need to memorize the
formulas in Table 2-1
- L′ and C′ are directly proportional to  and , respectively, regardless of line type
- common types of transmission lines (basic geometries; see Fig. 2-4)
o coaxial cable
o two-wire line
o microstrip line
o parallel-plate line
- transmission line (or telegrapher’s) equations – time domain form and phasor form
- wave equation for voltage and/or current along transmission lines
- complex propagation constant ()
- solutions to the wave equation for both voltage and current:
~
o V ( z)  Vo e z  Vo ez
~
o I ( z)  I o e z  I o ez
- meanings of Vo+ and Vo- (and of Io+ and Io-); all four quantities can be complex
- how to find voltage and/or current in time domain from phasor representation
- cosine representation of waves in time domain
- phasor representation of waves in frequency domain
- linearity and superposition (two or more waves traveling over the same transmission
line simply add together)
- characteristic impedance (Zo)
o how it is derived (where it comes from); what is represents
o formula in terms of R′, G′, C′, and L′
o dependence on xmsn line’s geometry and material properties
- meanings of real and imaginary parts of propagation constant:     j
o  = attenuation constant
o  = phase constant (or wavenumber)
- permeability of all parts of almost all xmsn lines is free space value (i.e.,  = o)
- definitions of:
o T = period (measured in s)
o  = wavelength (m)
o f = frequency (Hz, or 1/s)
o  = radian frequency (rad/s)
o  = phase constant, or wavenumber (rad/m)
o  = attenuation constant (Np/m)
o up = phase velocity (m/s)
o  = phase (rad or deg)
2
determination of up =  /, if y(z,t) = A cos (t – z + )
up =  / (i.e., up is positive and wave travels in +z direction) if one of the signs
before t and z is positive and the other is negative
- up = − / (i.e., up is negative and wave travels in −z direction) if signs before t and
z are either both positive or both negative
- for a lossless transmission line, Zo = (L′/C′)1/2 and  = 0
- phase velocity along line [up =  / = c/r-1/2 = (v.f.)c = ()-1/2], where v.f. = velocity
factor (v.f. = r-1/2)

- derivation of phase velocity: using cos(t   x)  0 , or using
t
cos(t  x)  cost  t    x  x  constant, where x is the distance traveled
in time t by any single point on the cosine wave
- units of sinusoidal quantity vs. units of argument (e.g., the sinusoidal function might
describe the behavior of a current wave, which would be measured in mA, while
distance along transmission line might be measured in m, and the term z might be
measured in radians)
Loaded transmission lines
- total V-I relationship at load vs. total V-I relationship along line (fwd + ref)
- Ohm’s law must be satisfied at load
- voltage reflection coefficient ()
o complex-valued (how to find magnitude and phase r)
o how to calculate  from load and characteristic impedances
o equal to Vo / Vo+
o special cases (short-ckt load, open-ckt load, ZL = Zo, ZL purely real or reactive)
o phase of reflection coefficient usually defined in the range −  r  
- standing waves
~
~
o derivations of expressions for V ( x) and I ( x )
-
o
o
o
o
o
o
o
voltage magnitude maxima and minima (Vmax and Vmin)
current magnitude maxima and minima (Imax and Imin)
voltage and current maxima and minima repeat every /2
voltage standing wave ratio (VSWR, or just S) and its formula
VSWR, Vmax, and Vmin on perfectly matched line (ZL = Zo)
VSWR often easier and/or cheaper to measure than 
VSWR can be the same for many different load impedances (because VSWR
is a function only of ||, not of the full complex quantity)
o return loss: R.L. = 20 log || (sometimes expressed as a positive value)
Transmission line impedance calculations
- electrical length vs. physical length of transmission lines
- input impedance of lossless line with load:
Z  jZ o tan(  l )
o Z in (l )  Z o L
Z o  jZ L tan(  l )
1  e  j 2  l
o Z in (l )  Z o
1  e  j 2  l
o open-circuit load (ZL = ±): Z in (l )   jZ o cot(  l )
o short-circuit load (ZL = 0): Z in (l )  jZ o tan(  l )
3
-
-
o matched line (ZL = Zo): Z in (l )  Z o
o quarter-wave matching section (length = /4): Zin = Zo2 / ZL
o half-wave section (length = n/2): impedances repeat every /2 along line
o purely reactive loads (ZL = jX): || = 1; Zin is purely imaginary
o “electrically short” lines (i.e., if l << , then Zin ≈ ZL)
o difference between Zo and Zin
o difference between Zo and ZL
o VSWR along line (especially along matching sections)
~
calculation of Vo+ and/or Vin , and the meanings of the two quantities
1
~  Z in 
~
~  Z in 

Vo  V g 
V

V
and


in
g
 Z  Z  e j l  e  j l 

in 
 g
 Z g  Z in 
~
~
phase relationship between V fwd z  and Vref z  at a given location z along line
use of short- or open-circuited line (or line loaded by a pure reactance) to obtain a pure
reactance at the input of the line (e.g., “wave trap” design)
determination of time-domain voltage or current at a specific location along line
cascaded line sections (one follows another) of different characteristic impedances
equivalent impedance at junction of two or more line sections (i.e., “shunt” connections)
Relevant homework, readings, and other resources:
HW:
Textbook:
Web Links:
Mathcad:
Matlab:
1-3
All of Chap. 1; Sections 2-1 through 2-7
NTIA Frequency Allocation Chart
Decaying Sinusoid
WVBU Wave Trap Example
Animation of Forward and Reflected Waves on a Xmsn Line
4