Lecture 12

More on Transmission Lines

12.1 Terminated Transmission Lines

Figure 12.1: A schematic for a transmission line terminated with an impedance load ZL at

z = 0.

As mentioned before, transmission line theory is indispensable in electromagnetic engineer-

ing. It is similar to one-dimensional form of Maxwell’s equations, and can be thought of as

Maxwell’s equations in its simplest form. Therefore, it entails a subset of the physics seen in

the full Maxwell’s equations.

For an infinitely long transmission line, the solution consists of the linear superposition

of a wave traveling to the right plus a wave traveling to the left. If transmission line is

terminated by a load as shown in Figure 12.1, a right-traveling wave will be reflected by the

load, and in general, the wave on the transmission line will be a linear superposition of the

left and right traveling waves. We will assume that the line is lossy first and specialize it to

the lossless case later. Thus,

V (z) = a+e−γz + a−eγz = V+(z) + V−(z) (12.1.1)

This is a linear system; hence, we can define the right-going wave V+(z) to be the input, and

that the left-going wave V−(z) to be the output as due to the reflection of the right-going

109

110 Electromagnetic Field Theory

wave V+(z). Or we can define the amplitude of the left-going reflected wave a− to be linearly

related to the amplitude of the right-going or incident wave a+. In other words, at z = 0, we

can let

V−(z = 0) = ΓLV+(z = 0) (12.1.2)

thus, using the definition of V+(z) and V−(z) as implied in (12.1.1), we have

a− = ΓLa+ (12.1.3)

where ΓL is the termed the reflection coefficient. Hence, (12.1.1) becomes

V (z) = a+e−γz + ΓLa+eγz = a+

(e−γz + ΓLeγz ) (12.1.4)

The corresponding current I(z) on the transmission line is given by using the telegrapher’s

equations as previously defined, namely that

I(z) = − 1

Z

dV

dz = a+

Z γ(e−γz − ΓLeγz ) (12.1.5)

where γ = √ZY = √(jωL + R)(jωC + G), and

Z = jωL + R, Y = jωC + G

In the lossless case when R = G = 0, γ = jβ. Hence, Z/γ = √Z/Y = Z0, the characteristic

impedance of the transmission line. Thus, from (12.1.5),

I(z) = a+

Z0

(e−γz − ΓLeγz ) (12.1.6)

Notice the sign change in the second term of the above expression.

Similar to ΓL, a general reflection coefficient (which is a function of z) relating the left-

traveling and right-traveling wave at location z can be defined such that

Γ(z) = V−(z) = a−eγz

V+(z) = a+e−γz = a−eγz

a+e−γz = ΓLe2γz (12.1.7)

Of course, Γ(z = 0) = ΓL. Furthermore, due to the V-I relation at an impedance load, we

must have

V (z = 0)

I(z = 0) = ZL (12.1.8)

or that using (12.1.4) and (12.1.5) with z = 0, the left-hand side of the above can be rewritten,

and we have

1 + ΓL

1 − ΓL

Z0 = ZL (12.1.9)

More on Transmission Lines 111

From the above, we can solve for ΓL in terms of ZL/Z0 to get

ΓL = ZL/Z0 − 1

ZL/Z0 + 1 = ZL − Z0

ZL + Z0

(12.1.10)

Thus, given the termination load ZL and the characteristic impednace Z0, the reflection

coefficient ΓL can be found, or vice versa. Or that given ΓL, the normalized load impedance,

ZL/Z0, can be found. It is seen that ΓL = 0 if ZL = Z0. Thus a right-traveling wave will not

be reflected and the left-traveling is absent. This is the case of a matched load. When there

is no reflection, all energy of the right-traveling wave will be totally absorbed by the load.

In general, we can define a generalized impedance at z 6 = 0 to be

Z(z) = V (z)

I(z) = a+(e−γz + ΓLeγz )

1

Z0 a+(e−γz − ΓLeγz )

= Z0

1 + ΓLe2γz

1 − ΓLe2γz = Z0

1 + Γ(z)

1 − Γ(z) (12.1.11)

or

Z(z)/Z0 = 1 + Γ(z)

1 − Γ(z) (12.1.12)

where Γ(z) is as defined in (12.1.7). Conversely, one can write the above as

Γ(z) = Z(z)/Z0 − 1

Z(z)/Z0 + 1 = Z(z) − Z0

Z(z) + Z0

(12.1.13)

Usually, a transmission line is lossless or has very low loss, and for most practical purpose,

γ = jβ. In this case, (12.1.11) becomes

Z(z) = Z0

1 + ΓLe2jβz

1 − ΓLe2jβz (12.1.14)

From the above, one can show that by setting z = −l, using (12.1.10), and after some algebra,

Z(−l) = Z0

ZL + jZ0 tan βl

Z0 + jZL tan βl (12.1.15)

112 Electromagnetic Field Theory

12.1.1 Shorted Terminations

Figure 12.2: The input reactance (X) of a shorted transmission line as a function of its length

l.

From (12.1.15) above, when we have a short such that ZL = 0, then

Z(−l) = jZ0 tan(βl) = jX (12.1.16)

Hence, the impedance remains reactive (pure imaginary) for all l, and can swing over all

positive and negative imaginary values. One way to understand this is that when the trans-

mission line is shorted, the right and left traveling wave set up a standing wave with nodes

and anti-nodes. At the nodes, the voltage is zero while the current is maximum. At the

anti-nodes, the current is zero while the voltage is maximum. Hence, a node resembles a

short while an anti-node resembles an open circuit. Therefore, at z = −l, different reactive

values can be observed as shown in Figure 12.2.

When β  l, then tan βl ≈ βl, and (12.1.16) becomes

Z(−l) ∼ = jZ0βl (12.1.17)

After using that Z0 = √L/C and that β = ω√LC, (12.1.17) becomes

Z(−l) ∼ = jωLl (12.1.18)

The above implies that a short length of transmission line connected to a short as a

load looks like an inductor with Leff = Ll, since much current will pass through this short

producing a strong magnetic field with stored magnetic energy. Remember here that L is the

line inductance, or inductance per unit length.

More on Transmission Lines 113

12.1.2 Open terminations

Figure 12.3: The input reactance (X) of an open transmission line as a function of its length

l.

When we have an open circuit such that ZL = ∞, then from (12.1.15) above

Z(−l) = −jZ0 cot(βl) = jX (12.1.19)

Again, as shown in Figure 12.3, the impedance at z = −l is purely reactive, and goes through

positive and negative values due to the standing wave set up on the transmission line.

Then, when βl  l, cot(βl) ≈ 1/βl

Z(−l) ≈ −j Z0

βl (12.1.20)

And then, again using β = ω√LC, Z0 = √L/C

Z(−l) ≈ 1

jωCl (12.1.21)

Hence, an open-circuited terminated short length of transmission line appears like an effective

capacitor with Ceff = Cl. Again, remember here that C is line capacitance or capacitance

per unit length.

But the changing length of l, one can make a shorted or an open terminated line look

like an inductor or a capacitor depending on its length l. This effect is shown in Figures 12.2

and 12.3. Moreover, the reactance X becomes infinite or zero with the proper choice of the

length l. These are resonances or anti-resonances of the transmission line, very much like an

LC tank circuit. An LC circuit can look like an open or a short circuit at resonances and

depending on if they are connected in parallel or in series.

114 Electromagnetic Field Theory

12.2 Smith Chart

In general, from (12.1.14) and (12.1.15), a length of transmission line can transform a load ZL

to a range of possible complex values Z(−l). To understand this range of values better, we

can use the Smith chart (invented by P.H. Smith 1939 before the advent of the computer) [79].

The Smith chart is essentially a graphical calculator for solving transmission line problems.

Equation (12.1.13) indicates that there is a unique map between the normalized impedance

Z(z)/Z0 and reflection coefficient Γ(z). In the normalized impedance form where Zn = Z/Z0,

from (12.1.11) and (12.1.13)

Γ = Zn − 1

Zn + 1 , Zn = 1 + Γ

1 − Γ (12.2.1)

Equations in (12.2.1) are related to a bilinear transform in complex variables [80]. It is a kind

of conformal map that maps circles to circles. Such a map is shown in Figure 12.4, where lines

on the right-half of the complex Zn plane are mapped to the circles on the complex Γ plane.

Since straight lines on the complex Zn plane are circles with infinite radii, they are mapped

to circles on the complex Γ plane. The Smith chart allows one to obtain the corresponding Γ

given Zn and vice versa as indicated in (12.2.1), but using a graphical calculator.

Notice that the imaginary axis on the complex Zn plane maps to the circle of unit radius on

the complex Γ plane. All points on the right-half plane are mapped to within the unit circle.

The reason being that the right-half plane of the complex Zn plane corresponds to passive

impedances that will absorb energy. Hence, such an impedance load will have reflection

coefficient with amplitude less than one, which are points within the unit circle.

On the other hand, the left-half of the complex Zn plane corresponds to impedances with

negative resistances. These will be active elements that can generate energy, and hence,

yielding |Γ| > 1, and will be outside the unit circle on the complex Γ plane.

Another point to note is that points at infinity on the complex Zn plane map to the point

at Γ = 1 on the complex Γ plane, while the point zero on the complex Zn plane maps to

Γ = −1 on the complex Γ plane. These are the reflection coefficients of an open-circuit load

and a short-circuit load, respectively. For a matched load, Zn = 1, and it maps to the zero

point on the complex Γ plane implying no reflection.

More on Transmission Lines 115

Figure 12.4: Bilinear map of the formulae Γ = Zn−1

Zn+1 , and Zn = 1+Γ

1−Γ . The chart on the right,

called the Smith chart, allows the values of Zn to be determined quickly given Γ, and vice

versa.

The Smith chart also allows one to quickly evaluate the expression

Γ(−l) = ΓLe−2jβl (12.2.2)

and its corresponding Zn. Since β = 2π/λ, it is more convenient to write βl = 2πl/λ, and

measure the length of the transmission line in terms of wavelength. To this end, the above

becomes

Γ(−l) = ΓLe−4jπl/λ (12.2.3)

For increasing l, one moves away from the load to the generator, l increases, and the phase

is decreasing because of the negative sign. So given a point for ΓL on the Smith chart, one

has negative phase or decreasing phase by rotating the point clockwise. Also, due to the

exp(−4jπl/λ) dependence of the phase, when l = λ/4, the reflection coefficient rotates a half

circle around the chart. And when l = λ/2, the reflection coefficient will rotate a full circle,

or back to the original point.

Also, for two points diametrically opposite to each other on the Smith chart, Γ changes

sign, and it can be shown easily that the normalized impedances are reciprocal of each other.

Hence, the Smith chart can also be used to find the reciprocal of a complex number quickly.

A full blown Smith chart is shown in Figure 12.5.

116 Electromagnetic Field Theory

Figure 12.5: The Smith chart in its full glory. It was invented in 1939 before the age of digital

computers, but it still allows engineers to do mental estimations and rough calculations with

it, because of its simplicity.

12.3 VSWR (Voltage Standing Wave Ratio)

The standing wave V (z) is a function of position z on a terminated transmission line and it

is given as

V (z) = V0e−jβz + V0ejβz ΓL

= V0e−jβz (1 + ΓLe2jβz )

= V0e−jβz (1 + Γ(z)) (12.3.1)

More on Transmission Lines 117

where we have used (12.1.7) for Γ(z) with γ = jβ. Hence, V (z) is not a constant or indepen-

dent of z, but

|V (z)| = |V0||1 + Γ(z)| (12.3.2)

In Figure 12.6, the relationship variation of 1 + Γ(z) as z varies is shown.

Figure 12.6: The voltage amplitude on a transmission line depends on |V (z)|, which is pro-

portional to |1 + Γ(z)| per equation (12.3.2). This figure shows how |1 + Γ(z)| varies as z

varies on a transmission line.

Using the triangular inequality, one gets

|V0|(1 − |Γ(z)|) ≤ |V (z)| ≤ |V0|(1 + |Γ(z)|) (12.3.3)

But from (12.1.7) and that γ = jβ, |Γ(z)| = |ΓL|; hence

Vmin = |V0|(1 − |ΓL|) ≤ |V (z)| ≤ |V0|(1 + |ΓL|) = Vmax (12.3.4)

The voltage standing wave ratio, VSWR is defined to be

VSWR = Vmax

Vmin

= 1 + |ΓL|

1 − |ΓL| (12.3.5)

Conversely,one can invert the above to get

|ΓL| = VSWR − 1

VSWR + 1 (12.3.6)

Hence, the knowledge of voltage standing wave pattern, as shown in Figure 12.7, yields the

knowledge of |ΓL|. Notice that the relations between VSWR and |ΓL| are homomorphic to

those between Zn and Γ. Therefore, the Smith chart can also be used to evaluate the above

equations.

118 Electromagnetic Field Theory

Figure 12.7: The voltage standing wave pattern as a function of z on a load-terminated

transmission line.

The phase of ΓL can also be determined from the measurement of the voltage standing

wave pattern. The location of ΓL in Figure 12.6 is determined by the phase of ΓL. Hence,

the value of d1 in Figure 12.6 is determined by the phase of ΓL as well. The length of the

transmission line waveguide needed to null the original phase of ΓL to bring the voltage

standing wave pattern to a maximum value at z = −d1 is shown in Figure 12.7. Hence, d1 is

the value where the following equation is satisfied:

|ΓL|ejφL e−4πj(d1/λ) = |ΓL| (12.3.7)

Thus, by measuring the voltage standing wave pattern, one deduces both the amplitude and

phase of ΓL. From the complex value ΓL, one can determine ZL, the load impedance.

From the above, one surmises that measuring the impedance of a device at microwave

frequency is a tricky business. At low frequency, one can use an ohm meter with two wire

probes to do such a measurement. But at microwave frequency, two pieces of wire become

inductors, and two pieces of metal become capacitors. More sophisticated ways to measure

the impedance need to be designed as described above.

In the old days, the voltage standing wave pattern was measured by a slotted-line equip-

ment which consists of a coaxial waveguide with a slot opening as shown in Figure 12.8. A

field probe can be put into the slotted line to determine the strength of the electric field inside

the coax waveguide.

More on Transmission Lines 119

Figure 12.8: A slotted-line equipment which consists of a coaxial waveguide with a slot

opening at the top to allow the measurement of the field strength and hence, the voltage

standing wave pattern in the waveguide (courtesy of Microwave101.com).

A typical experimental setup for a slotted line measurement is shown in Figure 12.9. A

generator source, with low frequency modulation, feeds microwave energy into the coaxial

waveguide. The isolator, allowing only the unidirectional propagation of microwave energy,

protects the generator. The attenuator protects the slotted line equipment. The wavemeter

is an adjustable resonant cavity. When the wavemeter is tuned to the frequency of the

microwave, it siphons off some energy from the source, giving rise to a dip in the signal of the

SWR meter (a short for voltage-standing-wave-ratio meter). Hence, the wavemeter measures

the frequency of the microwave.

The slotted line probe is usually connected to a square law detector that converts the

microwave signal to a low-frequency signal. In this manner, the amplitude of the voltage

in the slotted line can be measured with some low-frequency equipment, such as the SWR

meter. Low-frequency equipment is a lot cheaper to make and maintain. That is also the

reason why the source is modulated with a low-frequency signal. At low frequencies, circuit

theory prevails, engineering and design are a lot simpler.

The above describes how the impedance of the device-under-test (DUT) can be measured

at microwave frequencies. Nowadays, automated network analyzers make these measurements

a lot simpler in a microwave laboratory. More resource on microwave measurements can be

found on the web, such as in [81].

Notice that the above is based on the interference of the two traveling wave on a ter-

minated transmission line. Such interference experiments are increasingly difficult in optical

frequencies because of the much shorter wavelengths. Hence, many experiments are easier to

perform at microwave frequencies rather than at optical frequencies.

Many technologies are first developed at microwave frequency, and later developed at

optical frequency. Examples are phase imaging, optical coherence tomography, and beam

steering with phase array sources. Another example is that quantum information and quan-

tum computing can be done at optical frequency, but the recent trend is to use artificial atoms

working at microwave frequencies. Engineering with longer wavelength and larger component

is easier; and hence, microwave engineering.

Another new frontier in the electromagnetic spectrum is in the terahertz range. Due to

120 Electromagnetic Field Theory

the dearth of sources in the terahertz range, and the added difficulty in having to engineer

smaller components, this is an exciting and a largely untapped frontier in electromagnetic

technology.

Figure 12.9: An experimental setup for a slotted line measurement (courtesy of Pozar and

Knapp, U. Mass [82]).

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