Transmission Line Theory

Electromagnetic wave theory provides the basis for transmission line analysis. Involved are such factors as impedance, impedance matching, loss, velocity of propagation, reflection, and transmission.

Knowledge of all these parameters is necessary to an understanding of electrical signal transmission over metallic wire media.

Transmission networks are made up of resistors, capacitors, and inductors. The characteristics of such components are called lumped constants. A transmission line is an electrical circuit whose constants are not lumped but are uniformly distributed over its length. With care, the theory of lumped constant networks can be applied to trans-mission lines, but lines exhibit additional characteristics which deserve consideration.

The detailed characterization of a given type of cable is dependent on the physical design of the cable. Wire gauge, type of insulation, twisting of the wire pairs, etc., have important effects on attenuation, phase shift, impedance, and other parameters. For the most part, the treatment here pertains to two-wire parallel conductors. However, the analysis is also extended and applied to a coaxial conductor configuration.

5-1 DISCRETE COMPONENT LINE SIMULATION

It is convenient to analyze transmission line characteristics in terms of equivalent discrete-component, four-terminal networks. The conductors of an ideal simple transmission line, evenly spaced and extending over a considerable distance, have self-inductance, L, and resistance, R, which are series-connected elements that must be in-cluded in the discrete component equivalent network. Since the line

115

appears electrically the same when viewed from either end, the equivalent circuit must be symmetrical. Thus, the components of the equivalent network are split and connected as shown in Figure 5-l.

The insulation between the wires is never perfect; there is some leakage between them. The leakage resistance may be very large, as in a dry cable, or it may be fairly small, as in the case of a wet open-wire pair. Hence, the equivalent circuit must contain a conductance, G, in shunt between the line conductors. Also, any two conductors in close proximity to one another have the properties of a capacitor;

therefore, the circuit must have shunt capacitance, C.

These line parameters (resistance, inductance, conductance, and capacitance) are the primary constants and are usually expressed in per-mile values of ohms, millihenries, micromhos, and microfarads.

Derived from these are the characteristic impedance and propagation constant; they are the secondary constants, both of which are func-tions of frequency. Although all primary and secondary constants vary with changes in temperature, they are usually expressed as constants at 68°F with correction factors for small changes in temperature. Both primary and secondary constants are often used to characterize transmission lines or equivalent circuits.

In the discussion of networks in Chapter 4, it was suggested· that any circuit could be simulated by a T structure. It is not surprising then to find that a useful equivalent circuit for a transmission line is the T network shown in Figure 5-1. For convenience, series constants Rand L can be combined as impedance Z A, and shunt constants C and G as impedance Zc as shown in Figure 5-2. For the present, the relationship to line length is ignored.

R/2 L/2 L/2 R/2

c G

Figure 5-1. Primary constants of a section of uniform lina.

Chap. 5 Transmission Line Theory 117

Figure 5-2. Equivalent network of a sedion of uniform line.

The equivalent circuits in Figure 5-1 or 5-2 are poor approxima-tions of a real transmission line because all of the distributed con-stants have been concentrated at one point. The approximation is improved by having two T sections in tandem and, in the ultimate, the best representation is an infinite number of tandem-connected T sections, each having the constants of an infinitely short section of the real line.

Characteristic Impedance

An example of the simulation of a very long uniform transmission line by an infinite number of identical, recurrent, and symmetrical T networks is shown in Figure 5-3. The input impedance at point

a b d

I I I

To infinity

~

I I

Figure 5-3. Uniform line simulated by an infinite number of identical networks.

a is ZOo Now open the line at b and again measure the impedance of the line towards the right. Removing one section from an infinite number of sections produces no effect, so the impedance still measures Zoo

The impedance at point a was Zo when the first section was con-nected to an infinite line presenting impedance Zo at b. If the first section is terminated at b by a discrete-component network of im-pedance Zo, the imim-pedance would still measure Zo at point a. In a transmission line, Zo is called the characteristic impedance. It is re-lated to the equivalent T structure in Figure 5-2 by the expression

ohms. (5-1)

Impedances ZA and Zc contain inductance and capacitance. Since the reactances of inductors and capacitors are functions of frequency, the characteristic impedance of a real or simulated transmission line is also a function of frequency. This property must be recognized when selecting a network which is to terminate a line in its charac-teristic impedance over a band of frequencies.

It is often more convenient to determine Z ⁰ by test than by compu-tation. This can be done by measuring the impedance presented by the line when the far end is open-circuited (Zoe) and when it is short-circuited (Zse). Then, it can be shown that

Zo = yZoeZse ohms, (5-2)

an equation similar to those given for network image impedances, Equations (4-4) and (4-5).

To summarize, every transmission line has a characteristic im-pedance, ZOe It is determined by the materials and physical arrange-ment used in constructing the line. For any given type of line, Zo is by definition independent of the line length but is a function of fre-quency. The input impedance to a line is dependent on line length and on the termination at the far end. As the length increases, the value of the input impedance approaches the characteristic impedance irrespective of the far-end termination.

The term characteristic impedance is properly applied only to uni-form transmission lines. The corresponding property of a discrete

Chap. 5 Transmission Line Theory 119 component network is called image impedance as previously discussed.

However, if the network is symmetrical, it has a single image

imped-'11,1 ance which is analogous to the characteristic impedance of a uniform

line and, as the number of network sections is increased, the imped-ance approaches the characteristic impedimped-ance of the line being

I

I

simulated. .

Attenuation Factor

If a symmetrical T section is terminated in its image impedance (Zo) and voltage E1 is applied to the input terminals, current 11 flows at the input. In general, the output voltage and current, E2 and 12 is less than E1 and 11. Let It/I2 be designated by a; this is the attenua-tion factor for the T secattenua-tion. Also, let lnl It/I21

==

a; this is known as the attenuation constant for the T network. Then,

(5-3)

If a number of symmetrical T sections of image impedance Zo are connected in tandem and terminated in Zo as shown in Figure 5-4, each T section is terminated in Zo, and the ratio of its input current to its output current is a. Thus, from the figure

(5-4) To find the ratio of the input current to the output current for the series, the terms of Equation (5-4) may be multiplied to give

I ~: I = I ~: I . I ~: I . I ~: I . I ~: I =

a'

or, for n sections of identical series-connected T networks terminated in Zo at both ends,

_ _ -an-ena 11

In+1 - - , (5-5)

where an

==

ena is the attenuation factor for the n series-connected T networks.

Figure 5-4. line composed of identical T sections.

Propagation Constant

The ratio of input to output current, It/12, in a symmetrical T section terminated in its image impedance, Zo, is generally a complex number which may be expressed in a number of forms to indicate a change of both magnitude and phase. For the network of Figure 5-2 when it is terminated in Zo at both input and output, the current ratio is

~

^{= 1}

+

^ZA

+

^{Zo = 1}

+

^ZA

+

^JZA

+

^{(ZA)2 (5-6a)}

12 2Zc Zc 2Zc " Zc 2Zc

or, in more general terms,

~ - e^{Y - e(a+jl3)}

12 - - (5-6b)

where 'Y is a complex number called the propagation constant, or complex attenuation constant. I ts real and imaginary parts are defined by

'Y =

ex+i{3,

^(5-7)

where

ex

is the attenuation constant, which represents a change in magnitude, and {3 is the wavelength constant, or phase constant,

Chap. 5 Transmission Line Theory 121 which represents a change in phase. When applied to actual trans-mission lines with distributed parameters, both constants are usually expressed in terms of units of distance; a is usually expressed as

III nepers per mile or dB per mile, and

fJ

is usually expressed as radians ',' per mile or degrees per mile.

For the network of Figure 5-2, the value of 'Y may be computed from Equations (5-6a) and (5-6b) as

(5-8) If the bracketed expression in Equation (5-8) is written in polar form as

A Lf3,

where

A

is the absolute value and

f3

the angle, then 'Y

==

^{In Aej13}

==

^{In A}

+ ^jf3.

^(5-9)

Since the real and imaginary parts of this equation must equal the corresponding parts of Equation (5-7),

I

III IZA/2

+

^Zc

+

^{Zol .}

a

==

In A

==

In J; == In Zc nepers/T sectIOn (5-10a) and

f3

_{== arg} ^ZA/2

+

_Z ^Zc _c

+

^Zo ra lans / sec IOn ^{d' T t'} (5-10b) where arg stands for argument or angle of.

Since the attenuation constant can also be expressed in decibels per T section, Equation (5-10a) may be written

a == InA nepers/section == 8.686 InA dB/section == 20 log A dB/section.

This relationship must not be used indiscriminately; it applies only to an infinite line and a line or discrete-component simulation terminated inZo.

5-2 LINE WITH DISTRIBUTED PARAMETERS

Characterization of transmission lines involves the same electrical parameters (primary and secondary constants) as those used in characterizing discrete component networks. In lines, however, these

parameters are not discrete and concentrated. They are distributed uniformly along the line. Relationships between primary and secon-dary constants and between these parameters and other transmission line characteristics (including velocity of propagation, reflections and reflection loss, standing wave ratios, impedance matching, insertion loss, and return loss) must therefore be established in terms of distributed parameters.

Characteristic Impedance

A single T section may represent a line having distributed elements but at one frequency only. In order to construct an artificial line in simple T-section configurations of discrete ~lements to simulate a real line, it is necessary to construct many tandem-connected T-sections to simulate even very short sections of line. As the number of sections is increased and the elemental length of line is reduced, the artificial line approaches the actual line in its characteristics over a wide band of frequencies.

Consider an elemental length of line, ~l. Let Z be the impedance per unit length along the line and Y be the admittance per unit length across the line. The T section of Figure 5-5 represents the

ZA R +jwL Zill ZA R + jwL Zill

2 2 2 2 2

\,---,

I

^{R/2 L/2} I

, - - - "

I

R/2 L/2

I I L _ _ _ _ _

--l

I I

L _ _ _ _ _ ^~

I

j-. - - ---'Ii

^{1 1}

I

^Zc = y;;-= Yill

I

^{G C}

I .

I I

^{Y c} = G + }WC = YilZ

L _ _ _ _ _ _ -.J

Figure 5-5. T-section equivalent of a short length of line.

Chap. 5 Transmission Line Theory 123 equivalent network of a line having length ill. The value of ZA and Zc of Figure 5-5 may be written

1 Zc

==

Yill.

Substituting these values in Equation (5-1) gives the characteristic impedance for a lumped constant network as

Z -

J

^(Zill)2 +~

0-1 4 Y ^ohms.

For a line with distributed parameters, let ill approach zero; then, the characteristic impedance is

~ ^ZY

lim Zo

==

ill~O

or, in terms of primary constants, Z _ JR+jwL

0-1 G +jwC

Propagation Constant

ohms (5-11a)

ohms. (5-11b)

From the previous derivation of the expression for the propagation constant of an equivalent network, Equation (5-8) may be rewritten

By expanding the terms under the radical sign by the binominal theorem and rearranging terms, this expression may be written

(5-12)

Also, expanding e^Y as a power series yields

e^Y == 1

+ '}' + ~~ + ...

^(5-13)

As shown in Figure 5-5, the terms Z ^A, ZC, and '}' all place Al in the numerators of Equations (5-12) and (5-13). As Al approaches zero, the higher terms of these expansions become insignificant. Combining the two equations and truncating the series yields

This equation may be solved algebraically to give

for the conditions of Figure 5-5 and for length Al. For any length, l, made up of ljAl sections,

'}' == yZY l,

and for a unit length, l == 1, the propagation constant is

(5-14) where a is in nepers or dB per unit length and

f3

is in radians or degrees per unit length. The values of Z and Yare found from the primary constants of the line.

Attenuation Factor

Previously, the attenuation factor for a number of identical, symmetrical T sections having lumped constants was defined for tandem connections of such sections. Here, where the transmission line is made up of distributed parameters, as the length of an ele-mental section Al approaches zero, the attenuation constant becomes a nepers per unit length. Then the expression for the attenuation factor analogous to that of Equation (5-5) is given as

Attenuation factor == eat. (5-15)

1"

I

Chap. 5 Transmission line Theory 125

Velocity of Propagation

The phase shifts represented by the f3 term in Equation (5-6b) express the angular difference in radians between the input and output signals; they imply a time delay between the input signal current (or voltage) and the output signal current (or voltage).

This can be used to compute the velocity of propagation through the network or transmission line. The velocity is given by

l1J

V

== -rJ'

^(5-16)

typically expressed as miles per second.

Equation (5-16) shows that the velocity of propagation is a function of frequency, since ^ill

==

^{211'1. However, f3} is also frequency-dependent since it is made up of reactances derived from ZA and Zc of Equation (5-8). Thus, while a transmission line having either discrete or distributed elements tends to introduce delay distortion (a non-linear phase/frequency characteristic) because the velocity of propagation is different at different frequencies, it is theoretically possible to design a line having no delay distortion by designing f3 to be directly proportional to w.

Reflections

Only lines which are uniform and which are terminated in their characteristic impedance have so far been considered. As long as the signal is presented with the same impedance at all points in a connection, the only loss is attenuation.

However, if one line with characteristic impedance Za is joined to a second line with characteristic impedance ZL, an additional transmission loss is observed. While the signal is traveling in the first line, it has a voltage-to-current ratio Ea/la

==

Za. Before the signal can enter the second line, it must adjust to a new voltage-to-current ratio E L/ I L

==

^{Z L.} In making this adj ustment, a portion of the signal is reflected back towards the sending end of the connection.

It is not surprising that there should be a reflection at an abrupt change in the electrical characteristics of a line. There is a dis-turbance in any form of wave energy at a discontinuity in the

transmission medium. For example, sound is reflected from a cliff;

light is reflected by a mirror. These conditions are equivalent to a line terminated in either an open circuit or a short circuit; all of the energy in the incident wave is reflected. A less abrupt change in impedance would cause a partial reflection. For example, a landscape is mirrored in the surface of a pool of water because part of the light falling on the water is reflected. The bottom of the pool is also visible if the pool is not too deep, since part of the light falling on the water passes through the diSCONtinuity of the air-water junction and illumi-nates the bottom. Such a partial reflection occurs when two circuits with different impedances are joined, as in Figure 5-6. The power,

ZG = IZG I L()G

x

I

I

I

I I

P ----I111!---j.~ I P^L Pr ... - _ " )

Z G ' - ---+ ZL

Figure 5-6. Reflection at an impedance discontinuity.

P, in the signal arriving at junction x is divided. A portion of the signal, PL, is transmitted through the junction to the load ZL. The remainder of the signal, Pr, is reflected and travels back towards the source. If ZG and ZL were equal, the power, P, would all be delivered to Z L; there would be no reflection.

Reflection Loss. A concept frequently used to describe the effect of reflections is that of reflection loss, which is defined as the difference in dB between the power that is actually transferred from one circuit to the next and the power that would be transferred if the impedance of the second circuit were identical to that of the first.

Chap. 5 Transmission Line Theory 127 Consider the circuit of Figure 5-6 where the impedances do not match. The current in the circuit is I

==

^{E / (ZG}

+

^Z

d.

The power deli vered to Z ^L, the load, is

where RL is the resistive component of the load impedance. If impedance ZL matched ZG, that is, ZL

==

^ZG, the current in the circuit would be 1m

==

E/2ZG, and the power delivered to the load would be

The ratio of the two values of power is

Thus, the reflection loss may be written Reflection loss

==

^{10 log}

~; ==

^{20 log}

~ ~:

1

ZG

+

^{ZL 1}

J

^{cos ()G}

==

20 log 2 yZGZL

+

20 log" cos ()L dB.

(5-17) Thus, the reflection loss has two components. The first is related to the inverse of the reflection factor, defined as K, where

K _-

_I

² _{ZG+ZL .}

^yZ;;Z;: 1

(5-18) The second is dependent on the angular relationships between the two mismatched circuits.

It is possible to have negative reflection loss, or reflection gain.

This does not mean that power can be generated at an impedance discontinuity. It results from the choice of identical impedances as the reference condition for zero reflection loss which is not the con-dition for maximum power transfer, as pointed out in Chapter 4.

Standing Wave Ratio. A second concept that is useful in describing the effect of reflections is that of a standing wave ratio. This concept may be approached from the point of view of a theoretically lossless transmission line.

Equation (5-11) gives the expression for the characteristic im-pedance of a line as

z - J

R +jwL

Dans le document I 0 i 0 (Page 135-148)