CHAPTER TWO
Review of Waves and
Transmission Lines
2.1 INTRODUCTION
At low RF, a wire or a line on a printed circuit board can be used to connect two
electronic components. At higher frequencies, the current tends to concentrate on the
surface of the wire due to the skin effect. The skin depth is a function of frequency
and conductivity given by
d
s

2
oms

1=2
2:1
where o  2pf is the angular frequency, f is the frequency, m is the permeability,
and s is the conductivity. For copper at a frequency of 10GHz, s  5:8 Â 10
7
S=m
and d
s
 6:6 Â 10
À5
cm, which is a very small distance. The ®eld amplitude decays
exponentially from its surface value according to e
Àz=d
s
, as shown in Fig. 2.1. The
®eld decays by an amount of e
À1

rectangular waveguides will be described.
FIGURE 2.2 The currrent distribution within a wire operating at different frequencies.
FIGURE 2.1 Fields inside the conductor.
2.1 INTRODUCTION
11
2.2 WAVE PROPAGATION
Waves can propagate in free space or in a transmission line or waveguide. Wave
propagation in free space forms the basis for wireless applications. Maxwell
predicted wave propagation in 1864 by the derivation of the wave equations.
Hertz validated Maxwell's theory and demonstrated radio wave propagation in the
FIGURE 2.3 Transmission line and waveguide structures.
12
REVIEW OF WAVES AND TRANSMISSION LINES
TABLE 2.1 Transmission Line and Waveguide Comparisons
Useful Frequency Potential for
Range Impedance Cross-Sectional Power Active Device Low-Cost
Transmission Line (GHz) Range (O) Dimensions Q-Factor Rating Mounting Production
Rectangular waveguide < 300 100±500 Moderate to large High High Easy Poor
Coaxial line < 50 10±100 Moderate Moderate Moderate Fair Poor
Stripline < 10 10±100 Moderate Low Low Fair Good
Microstrip line 100 10±100 Small Low Low Easy Good
Suspended stripline 15020±150Small Moderate Low Easy Fair
Finline 150 20±400 Moderate Moderate Low Easy Fair
Slotline 60 60±200 Small Low Low Fair Good
Coplanar waveguide 6040±150Small Low Low Fair Good
Image guide < 300 30±30 Moderate High Low Poor Good
Dielectric line < 300 20±50 Moderate High Low Poor Fair
13
laboratory in 1886. This opened up an era of radio wave applications. For his work,
Hertz is known as the father of radio, and his name is used as the frequency unit.

E and
~
B are electric and magnetic ®elds,
~
D is the electric displacement,
~
H is
the magnetic intensity,
~
J is the conduction current density, e is the permittivity, and r
is the charge density. The term @
~
D=@t is displacement current density, which was ®rst
added by Maxwell. This term is important in leading to the possibility of wave
propagation. The last equation is for the continuity of ¯ux.
We also have two constitutive relations:
~
D  e
0
~
E 
~
P  e
~
E 2:3a
~
B  m
0
~
H 

0
 8:85Â
10
À12
F=m is the permittivity of vacuum.
With Eqs. (2.2) and (2.3), the wave equation can be derived for a source-free
transmission line (or waveguide) or free space. For a source-free case, we have
~
J  r  0, and Eq. (2.2) can be rewritten as
H Á
~
E  0 2:5a
H Â
~
E Àjom
~
H 2:5b
H Â
~
H  joe
~
E 2:5c
H Á
~
H  0 2:5d
14
REVIEW OF WAVES AND TRANSMISSION LINES
Here we assume that all ®elds vary as e
jot
and @=@t is replaced by jo.

~
E  o
2
me
~
E  0 2:8a
or
H
2
~
E  k
2
~
E  0 2:8b
where k  o

me
p
 propagation constant.
Similarly, one can derive
H
2
~
H  o
2
me
~
H  0 2:9
Equations (2.8) and (2.9) are referred to as the Helmholtz equations or wave
equations. The constant k (or b) is called the wave number or propagation constant,

, and v  c  1=

m
0
e
0
p
 speed of light. Equations (2.8) and
(2.9) can be solved in rectangular, cylindrical, or spherical coordinates. Antenna
radiation in free space is an example of spherical coordinates. The solution in a wave
propagating in the
~
r direction:
~
Er; y; f
~
Ey; fe
Àj
~
kÁ
~
r
2:11
2.2 WAVE PROPAGATION
15
The propagation in a rectangular waveguide is an example of rectangular coordinates
with a wave propagating in the z direction:
~
Ex; y; z
~

2.3 TRANSMISSION LINE EQUATION
The transmission line equation can be derived from circuit theory. Suppose a
transmission line is used to connect a source to a load, as shown in Fig. 2.5. At
position x along the line, there exists a time-varying voltage vx; t and current ix; t.
For a small section between x and x  Dx, the equivalent circuit of this section Dx
can be represented by the distributed elements of L, R, C, and G, which are the
inductance, resistance, capacitance, and conductance per unit length. For a lossless
line, R  G  0. In most cases, R and G are small. This equivalent circuit can be
easily understood by considering a coaxial line in Fig. 2.6. The parameters L and R
are due to the length and conductor losses of the outer and inner conductors, whereas
i(x + ∆ x, t)i(x, t)
i(x, t)
x
v(x, t)
v(x, t)
C∆ xG∆ x
R∆ x
xxx+ ∆ xx + ∆ x
∆ x
xx+ ∆ x
Source Load
=
L∆ x
∆ x
v(x + ∆ x, t
)
FIGURE 2.5 Transmission line equivalent circuit.
2.3 TRANSMISSION LINE EQUATION
17
C and G are attributed to the separation and dielectric losses between the outer and

vx; t
@x
2
ÀR
@ix; t
@x
À L
@
2
ix; t
@x @t
2:20
@
2
ix; t
@t @x
ÀG
@vx; t
@t
À C
@
2
vx; t
@t
2
2:21
FIGURE 2.6 L, R, C for a coaxial line.
18
REVIEW OF WAVES AND TRANSMISSION LINES
By substituting (2.19) and (2.21) into (2.20), one can eliminate @i=@x and @

e
Àgx
 V
À
e
gx
2:26
Equation (2.26) gives the solution for voltage along the transmission line. The
voltage is the summation of a forward wave (V

e
Àgx
) and a re¯ected wave (V
À
e
gx
)
propagating in the x and Àx directions, respectively.
The current Ix can be found from Eq. (2.18) in the frequency domain:
IxI

e
Àgx
À I
À
e
gx
2:27
where
I

R  joL
G  joC

1=2
2:28
2.3 TRANSMISSION LINE EQUATION
19
For a lossless line, R  G  0,wehave
g  jb  jo

LC
p
2:29a
Z
0


L
C
r
2:29b
Phase velocity v
p

o
b
 f l
g

1

characteristic impedance Z
0
.
20
REVIEW OF WAVES AND TRANSMISSION LINES
(Z
L
could be real or complex) is connected to the line as shown in Fig. 2.7b and
Z
L
T Z
0
, there exists a re¯ected wave and the input impedance is no longer equal to
Z
0
. Instead, Z
in
is a function of frequency ( f ), l, Z
L
, and Z
0
. Note that at low
frequencies, Z
in
% Z
L
regardless of l.
In the last section, the voltage along the line was given by
VxV


L

V
À
V

 G0
 reflection coefficient at load 2:31b
Substituting G
L
into Eqs. (2.26) and (2.27), the impedance along the line is given by
Zx
Vx
Ix
 Z
0
e
Àgx
 G
L
e
gx
e
Àgx
À G
L
e
gx
2:32
At x  0, ZxZ

 ZÀlZ
0
e
gl
 G
L
e
Àgl
e
gl
À G
L
e
Àgl
2:35
Substituting (2.34) into the above equation, we have
Z
in
 Z
0
Z
L
 Z
0
tanh gl
Z
0
 Z
L
tanh gl

L
.
The power transmitted and re¯ected can be calculated by the following:
Incident power  P
in

jV

j
2
Z
0
2:38
Reflected power  P
r

jV
À
j
2
Z
0

jV

j
2
jG
L
j

, a re¯ected wave exists, and the incident and re¯ected waves interfere to
produce a standing-wave pattern along the line. The voltage at point x along the
lossless line is given by
VxV

e
Àjbx
 V
À
e
jbx
 V

e
Àjbx
1  G
L
e
2jbx
2:41
Substituting G
L
jG
L
je
jf
into the above equation gives the magnitude of Vx as
jVxj  jV

j1 jG

j when sinbx 
1
2
f1 (or bx 
1
2
f  mp À
1
2
p). Figure 2.8
shows the pattern that repeats itself every
1
2
l
g
.
22
REVIEW OF WAVES AND TRANSMISSION LINES
The ®rst maximum voltage can be found by setting x Àd
max
and n  0.We
have
2bd
max
 f 2:43
The ®rst minimum voltage, found by setting x Àd
min
and m  0, is given by
2bd
min

FIGURE 2.8 Pattern of voltage magnitude along line.
2.5 VOLTAGE STANDING-WAVE RATIO
23
line, a re¯ectometer, or a network analyzer. Figure 2.9 shows a nomograph of the
VSWR. The return loss and power transmission are de®ned in the next section. Table
2.2 summarizes the formulas derived in the previous sections.
Example 2.1 Calculate the VSWR and input impedance for a transmission line
connected to (a) a short and (b) an open load. Plot Z
in
as a function of bl.
FIGURE 2.9 VSWR nomograph.
24
REVIEW OF WAVES AND TRANSMISSION LINES
Solution (a) A transmission line with a characteristic impedance Z
0
is connected to
a short load Z
L
as shown in Fig. 2.10. Here, Z
L
 0, and G
L
is given by
G
L

Z
L
À Z
0

 P
in
Transmitted power 1 ÀjG
L
j
2
P
in
 0
TABLE 2.2 Formulas for Transmission Lines
Quantity General Line Lossless Line
Propagation constant, g  a  jb

R  joLG  joC
p
jo

LC
p
Phase constant, b Img o

LC
p

w
v

2p
l
Attenuation constant, a Reg 0

sin bl
Z
0
cos bl  jZ
L
sin bl
Impedance of shorted line Z
0
tanh gljZ
0
tan bl
Impedance of open line Z
0
coth gl ÀjZ
0
cot bl
Impedance of quarter-wave line Z
0
Z
L
sinh al  Z
0
cosh al
Z
0
sinh al  Z
L
cosh al
Z
2

0
Z
L
 Z
0
Voltage standing-wave ratio
(VSWR)
1 jG
L
j
1 ÀjG
L
j
1 jG
L
j
1 ÀjG
L
j
2.5 VOLTAGE STANDING-WAVE RATIO
25
The input impedance is calculated by Eq. (2.37),
Z
in
 Z
0
Z
L
 jZ
0

L
 Z
0
 1  1e
j0

jG
L
je
jf
Therefore, jG
L
j1 and f  0

. Again,
VSWR 
1 jG
L
j
1 ÀjG
L
j
I and P
r
 P
in
The input impedance is given by
Z
in
 Z

and P
2
are the two power levels being compared. If power level P
2
is
higher than P
1
, the decibel is positive and vice versa. Since P  V
2
=R, the voltage
de®nition of the decibel is given by
Voltage ratio in dB  20log
10
V
2
V
1
2:48
The decibel was originally named for Alexander Graham Bell. The unit was used as
a measure of attenuation in telephone cable, that is, the ratio of the power of the
signal emerging from one end of a cable to the power of the signal fed in at the other
end. It so happened that one decibel almost equaled the attenuation of one mile of
telephone cable.
2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSS
27
2.6.1 Conversion from Power Ratios to Decibels and Vice Versa
One can convert any power ratio (P
2
=P
1

As one can see from these results, the use of decibels is very convenient to represent
a very large or very small number. To convert from decibels to power ratios, the
following equation can be used:
Power ratio  10
dB=10
2:49
2.6.2 Gain or Loss Representations
A common use of decibels is in expressing power gains and power losses in the
circuits. Gain is the term for an increase in power level. As shown in Fig. 2.12, an
ampli®er is used to amplify an input signal with P
in
 1 mW. The output signal is
200 mW. The ampli®er has a gain given by
Gain in ratio 
output power
input power
 200 2:50a
Gain in db  10log
10
output power
input power
 23 dB 2:50b
Now consider an attenuator as shown in Fig. 2.13. The loss is the term of a decrease
in power. The attenuator has a loss given by
Loss in ratio 
input power
output power
 2 2:51a
Loss in db  10log
10

 L
3
ÁÁÁ 2:52
FIGURE 2.12 Ampli®er circuit.
FIGURE 2.14 Cascaded circuit.
FIGURE 2.13 Attenuator circuit.
30
REVIEW OF WAVES AND TRANSMISSION LINES
2.6.3 Decibels as Absolute Units
Decibels can be used to express values of power. All that is necessary is to establish
some absolute unit of power as a reference. By relating a given value of power to this
unit, the power can be expressed with decibels.
The often-used reference units are 1 mW and 1 W. If 1 milliwatt is used as a
reference, dBm is expressed as decibels relative to 1 mW:
Pin dBm10log Pin mW2:53
Therefore, the following results can be written:
1mW 0dBm
10mW  10dBm
1W 30dBm
0:1mWÀ10dBm
1 Â 10
À7
mW À70dBm
If 1 W is used as a reference, dBW is expressed as decibels relative to 1 W. The
conversion equation is given by
Pin dBW10log Pin W2:54
From the above equation, we have
1W 0dBW 10W  10dBW 0:1WÀ10dBW
Now for the system shown in Fig. 2.14, if the input power P
in

2
 4000 W
2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSS
31
2.6.4 Insertion Loss and Return Loss
Insertion loss, return loss, and VSWR are commonly used for component speci®ca-
tion. As shown in Fig. 2.15, the insertion loss and return loss are de®ned as
Insertion loss  IL  10log
P
in
P
t
2:55a
Return loss  RL  10log
P
in
P
r
2:55b
Since P
r
jG
L
j
2
P
in
, Eq. (2.55b) becomes
RL À20log jG
L

in
P
r
À20log jG
L
j13:98 dB
FIGURE 2.15 Two-port component.
32
REVIEW OF WAVES AND TRANSMISSION LINES
The transmitted power is calculated from Eq. (2.55a):
IL  10log
P
in
P
t
 0:5dB P
t
 0:89P
in
Assuming the input power is split into three output ports equally, each output port
will transmit 29.7% of the input power. The input mismatch loss is 4% of the input
power. Another 7% of power is lost due to the output mismatch and circuit losses.
j
2.7 SMITH CHARTS
The Smith chart was invented by P. H. Smith of Bell Laboratories in 1939. It is a
graphical representation of the impedance transformation property of a length of
transmission line. Although the impedance and re¯ection information can be
obtained from equations in the previous sections, the calculations normally involve
complex numbers that can be complicated and time consuming. The use of the
Smith chart avoids the tedious computation. It also provides a graphical representa-

1  Gx
1 À Gx
2:59
Therefore

Rxj

Xx
1  G
r
 jG
i
1 À G
r
À jG
i
2:60
By multiplying both numerator and denominator by 1 À G
r
 jG
i
, two equations are
generated:
G
r
À

R
1 



2
2:61b
In the G
r
À G
i
coordinate system, Eq. (2.61a) represents circles centered at
(

R=1 

R; 0) with a radii of 1=1 

R. These are called constant

R circles.
Equation (2.61b) represents circles centered at (1; 1=X ) with radii of 1=X . They
are called constant

X circles. Figure 2.17 shows these circles in the G
r
À G
i
plane.
The plot of these circles is called the Smith chart. On the Smith chart, a constant jGj
is a circle centered at (0, 0) with a radius of jGj. Hence, motion along a lossless
transmission line gives a circular path on the Smith chart. From Eq. (2.31), we know
for a lossless line that
GxG

l
g
along the transmission line.
4. The same chart can be used for reading admittance.
34
REVIEW OF WAVES AND TRANSMISSION LINES