Lecture 11

Transmission Lines

11.1 Transmission Line Theory

Figure 11.1:

Transmission lines were the first electromagnetic waveguides ever invented. The were driven

by the needs in telegraphy technology. It is best to introduce transmission line theory from

the viewpoint of circuit theory. This theory is also discussed in many textbooks and lecture

101

102 Electromagnetic Field Theory

notes. Transmission lines are so important in modern day electromagnetic engineering, that

most engineering electromagnetics textbooks would be incomplete without introducing the

topic [29, 31, 38, 47, 48, 59, 71, 75, 77].

Circuit theory is robust and is not sensitive to the detail shapes of the components in-

volved such as capacitors or inductors. Moreover, many transmission line problems cannot

be analyzed with the full form of Maxwell’s equations,1 but approximate solutions can be

obtained using circuit theory in the long-wavelength limit. We shall show that circuit theory

is an approximation of electromagnetic field theory when the wavelength is very long: the

longer the wavelength, the better is the approximation [47].

Examples of transmission lines are shown in Figure 11.1. The symbol for a transmission

line is usually represented by two pieces of parallel wires, but in practice, these wires need

not be parallel.

Figure 11.2: Courtesy of slides by A. Wadhwa, A.L. Dal, N. Malhotra [78].

Circuit theory also explains why waveguides can be made sloppily when wavelength is long

or the frequency low. For instance, in the long-wavelength limit, we can make twisted-pair

waveguides with abandon, and they still work well (see Figure 11.2). Hence, we shall first

explain the propagation of electromagnetic signal on a transmission line using circuit analysis.

11.1.1 Time-Domain Analysis

We will start with performing the time-domain analysis of a simple, infinitely long trans-

mission line. Remember that two pieces of metal can accumulate attractive charges between

them, giving rise to capacitive coupling, electric field, and hence stored energy in the electric

field. Moreover, a piece of wire carrying a current generates a magnetic field, and hence,

yielding stored energy in the magnetic field. These stored energies are the sources of the

capacitive and inductive effects. But these capacitive and inductive effects are distributed

over the spatial dimension of the transmission line. Therefore, it is helpful to think of the

two pieces of metal as consisting of small segments of metal concatenated together. Each of

these segments will have a small inductance, as well as a small capacitive coupling between

them. Hence, we can model two pieces of metal with a distributed lumped element model as

shown in Figure 11.3. For simplicity, we assume the other conductor to be a ground plane,

so that it need not be approximated with lumped elements.

1Usually called full-wave analysis.

Transmission Lines 103

In the transmission line, the voltage V (z, t) and I(z, t) are functions of both space z and

time t, but we will model the space variation of the voltage and current with discrete step

approximation. The voltage varies from node to node while the current varies from branch

to branch of the lump-element model.

Figure 11.3:

First, we recall that the V-I relation of an inductor is

V0 = L0

dI0

dt (11.1.1)

where L0 is the inductor, V0 is the time-varying voltage drop across the inductor, and I0 is

the current through the inductor. Then using this relation between node 1 and node 2, we

have

V − (V + ∆V ) = L∆z ∂I

∂t (11.1.2)

The left-hand side is the voltage drop across the inductor, while the right-hand side follows

from the aforementioned V-I relation of an inductor, but we have replaced L0 = L∆z. Here,

L is the inductance per unit length (line inductance) of the transmission line. And L∆z is

the incremental inductance due to the small segment of metal of length ∆z. Then the above

can be simplified to

∆V = −L∆z ∂I

∂t (11.1.3)

Next, we make use of the V-I relation for a capacitor, which is

I0 = C0

dV0

dt (11.1.4)

where C0 is the capacitor, I0 is the current through the capacitor, and V0 is a time-varying

voltage drop across the capacitor. Thus, applying this relation at node 2 gives

−∆I = C∆z ∂

∂t (V + ∆V ) ≈ C∆z ∂V

∂t (11.1.5)

104 Electromagnetic Field Theory

where C is the capacitance per unit length, and C∆z is the incremental capacitance between

the small piece of metal and the ground plane. In the above, we have used Kirchhoff current

law to surmise that the current through the capacitor is −∆I, where ∆I = I(z+∆z, t)−I(z, t).

In the last approximation in (11.1.5), we have dropped a term involving the product of ∆z

and ∆V , since it will be very small or second order in magnitude.

In the limit when ∆z → 0, one gets from (11.1.3) and (11.1.5) that

∂V (z, t)

∂z = −L ∂I(z, t)

∂t (11.1.6)

∂I(z, t)

∂z = −C ∂V (z, t)

∂t (11.1.7)

The above are the telegrapher’s equations. They are two coupled first-order equations, and

can be converted into second-order equations easily. Therefore,

∂2V

∂z2 − LC ∂2V

∂t2 = 0 (11.1.8)

∂2I

∂z2 − LC ∂2I

∂t2 = 0 (11.1.9)

The above are wave equations that we have previously studied, where the velocity of the wave

is given by

v = 1

√LC (11.1.10)

Furthermore, if we assume that

V (z, t) = f+(z − vt) (11.1.11)

a right-traveling wave, and substituting it into (11.1.6) yields

−L ∂I

∂t = f ′

+(z − vt) (11.1.12)

Substituting V (z, t) into (11.1.7) yields

∂I

∂z = Cvf ′

+(z − vt) (11.1.13)

The above implies that

I =

√ C

L f+(z − vt) (11.1.14)

Consequently,

V (z, t)

I(z, t) =

√ L

C = Z0 (11.1.15)

Transmission Lines 105

where Z0 is the characteristic impedance of the transmission line. The above ratio is only

true for one-way traveling wave, in this case, one that propagates in the +z direction.

For a wave that travels in the negative z direction, i.e.,

V (z, t) = f−(z + vt) (11.1.16)

with the corresponding I(z, t) derived, one can show that

V (z, t)

I(z, t) = −

√ L

C = −Z0 (11.1.17)

Time-domain analysis is very useful for transient analysis of transmission lines, especially

when nonlinear elements are coupled to the transmission line. Another major strength of

transmission line model is that it is a simple way to introduce time-delay in a circuit. Time

delay is a wave propagation effect, and it is harder to incorporate into circuit theory or a

pure circuit model consisting of R, L, and C. In circuit theory, Laplace’s equation is usually

solved, which is equivalent to Helmholtz equation with infinite wave velocity, namely,

lim

c→∞ ∇2Φ(r) + ω2

c2 Φ(r) = 0 =⇒ ∇2Φ(r) = 0 (11.1.18)

Hence, events in Laplace’s equation happen instantaneously.

11.1.2 Frequency-Domain Analysis

Frequency domain analysis is very popular as it makes the transmission line equations very

simple. Moreover, generalization to a lossy system is quite straight forward. Furthermore, for

linear time invariant systems, the time-domain signals can be obtained from the frequency-

domain data by performing a Fourier inverse transform.

For a time-harmonic signal on a transmission line, one can analyze the problem in the

frequency domain using phasor technique. A phasor variable is linearly proportional to a

Fourier transform variable. The telegrapher’s equations (11.1.6) and (11.1.7) then become

d

dz V (z, ω) = −jωLI(z, ω) (11.1.19)

d

dz I(z, ω) = −jωCV (z, ω) (11.1.20)

The corresponding Helmholtz equations are then

d2V

dz2 + ω2LCV = 0 (11.1.21)

d2I

dz2 + ω2LCI = 0 (11.1.22)

The general solutions to the above are

V (z) = V+e−jβz + V−ejβz (11.1.23)

I(z) = I+e−jβz + I−ejβz (11.1.24)

106 Electromagnetic Field Theory

where β = ω√LC. This is similar to what we have seen previously for plane waves in the

one-dimensional wave equation in free space, where

Ex(z) = E0+e−jk0z + E0−ejk0z (11.1.25)

where k0 = ω√μ00. We see a much similarity between (11.1.23), (11.1.24), and (11.1.25).

To see the solution in the time domain, we let V± = |V±|ejφ± , and the voltage signal

above can be converted back to the time domain as

V (z, t) = <e{V (z, ω)ejωt} (11.1.26)

= |V+| cos(ωt − βz + φ+) + |V−| cos(ωt + βz + φ−) (11.1.27)

As can be seen, the first term corresponds to a right-traveling wave, while the second term is

a left-traveling wave.

Furthermore, if we assume only a one-way traveling wave to the right by letting V− =

I− = 0, then it can be shown that, for a right-traveling wave

V (z)

I(z) = V+

I+

=

√ L

C = Z0 (11.1.28)

In the above, the telegrapher’s equations, (11.1.19) or (11.1.20) have been used to find a

relationship between I+ and V+.

Similarly, applying the same process for a left-traveling wave only, by letting V+ = I+ = 0,

then

V (z)

I(z) = V−

I−

= −

√ L

C = −Z0 (11.1.29)

11.2 Lossy Transmission Line

Figure 11.4:

The strength of frequency domain analysis is demonstrated in the study of lossy transmission

lines. The previous analysis, which is valid for lossless transmission line, can be easily gen-

eralized to the lossy case. In using frequency domain and phasor technique, impedances will

become complex numbers as shall be shown.

Transmission Lines 107

To include loss, we use the lumped-element model as shown in Figure 11.4. One thing

to note is that jωL is actually the series line impedance of the transmission line, while

jωC is the shunt line admittance of the line. First, we can rewrite the expressions for the

telegrapher’s equations in (11.1.19) and (11.1.20) in terms of series line impedance and shunt

line admittance to arrive at

d

dz V = −ZI (11.2.1)

d

dz I = −Y V (11.2.2)

where Z = jωL and Y = jωC. The above can be generalized to the lossy case as shall be

shown.

The geometry in Figure 11.4 is homomorphic2 to the lossless case in Figure 11.3. Hence,

when lossy elements are added in the geometry, we can surmise that the corresponding tele-

grapher’s equations are similar to those above. But to include loss, we generalize the series

line impedance and shunt admittance from the lossless case to lossy case as follows:

Z = jωL → Z = jωL + R (11.2.3)

Y = jωC → Y = jωC + G (11.2.4)

where R is the series line resistance, and G is the shunt line conductance, and now Z and

Y are the series impedance and shunt admittance, respectively. Then, the corresponding

Helmholtz equations are

d2V

dz2 − ZY V = 0 (11.2.5)

d2I

dz2 − ZY I = 0 (11.2.6)

or

d2V

dz2 − γ2V = 0 (11.2.7)

d2I

dz2 − γ2I = 0 (11.2.8)

where γ2 = ZY , or that one can also think of γ2 = −β2. Then the above is homomorphic to

the lossless case except that now, β is a complex number, indicating that the field is decaying

as it propagates. As before, the above are second order one-dimensional Helmholtz equations

where the general solutions are

V (z) = V+e−γz + V−eγz (11.2.9)

I(z) = I+e−γz + I−eγz (11.2.10)

2A math term for “similar in structure”. The term is even used in computer science describing a emerging

field of homomorphic computing.

108 Electromagnetic Field Theory

and

γ = √ZY = √(jωL + R)(jωC + G) = jβ (11.2.11)

Hence, β = β′ − jβ′′ is now a complex number. In other words,

e−γz = e−jβ′z−β′′z

is an oscillatory and decaying wave. Or focusing on the voltage case,

V (z) = V+e−β′′z−jβ′z + V−eβ′′z+jβ′z (11.2.12)

Again, letting V± = |V±|ejφ± , the above can be converted back to the time domain as

V (z, t) = <e{V (z, ω)ejωt} (11.2.13)

= |V+|e−β′′z cos(ωt − β′z + φ+) + |V−|eβ′′z cos(ωt + β′z + φ−) (11.2.14)

The first term corresponds to a decaying wave moving to the right while the second term is

also a decaying wave moving to the left. When there is no loss, or R = G = 0, and from

(11.2.11), we retrieve the lossless case where β′′ = 0 and γ = jβ = jω√LC.

Notice that for the lossy case, the characteristic impedance, which is the ratio of the

voltage to the current for a one-way wave, can similarly be derived using homomorphism:

Z0 = V+

I+

= − V−

I−

=

√ L

C =

√

jωL

jωC → Z0 =

√ Z

Y =

√

jωL + R

jωC + G (11.2.15)

The above Z0 is manifestly a complex number. Here, Z0 is the ratio of the phasors of the

one-way traveling waves, and apparently, their current phasor and the voltage phasor will not

be in phase for lossy transmission line.

In the absence of loss, the above again becomes

Z0 =

√ L

C (11.2.16)

the characteristic impedance for the lossless case previously derived.

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