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Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

4 Transmission Lines Krishna Naishadham

4.1

Introduction ....................................................................................... 525

Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Massachusetts, USA

4.2

Equivalent Circuit................................................................................ 528

4.3

Alternating Current Analysis.................................................................. 529

4.4 4.5

Smith Chart........................................................................................ 534 Summary ........................................................................................... 536 Appendix A: References ........................................................................ 537

4.1.1 What Is a Transmission Line? . 4.1.2 Lossy Transmission Line 4.2.1 Lossless Line . 4.2.2 Low-Loss Line . 4.2.3 Distortionless Line 4.3.1 Terminated Lossless Line . 4.3.2 Terminated Lossy Transmission Line

4.1 Introduction A transmission line is used to transfer a signal from the generator to the load, as in Figure 4.1, by guiding an electromagnetic (EM) wave between two conductors. In microwave communications, the signal is usually some form of modulation on a high-frequency carrier. The generator circuit represents the Thevenin equivalent (a voltage Vg in series with an impedance Zg ) of the EM wave’s source, such as a radar transmitter, a continuous tone source, or a high-speed data terminal. The load ZL represents the equivalent impedance at the terminals of the receiver (e.g., the transmitting antenna’s input impedance). Figure 4.2 illustrates a cross section of some common transmission lines. The transmission lines are assumed to be uniform along the direction of propagation, so discontinuity effects are not considered in the field distribution. All these lines, except the microstrip, support electric and magnetic field distributions, which are entirely transverse to the direction of propagation. Thus, they can be described mathematically in terms of transverse electromagnetic (TEM) waves. The microstrip line has a small longitudinal field component relative to the transverse component because of flux leakage across the air–dielectric interface. Such a transmission line can be considered as quasi-TEM. The coaxial line and the stripline are completely shielded and do not have any leakage under ideal conditions. Copyright ß 2004 by Academic Press. All rights of reproduction in any form reserved.

A TEM wave has the same phase constant and intrinsic impedance as a plane wave propagating in an unbounded medium and can be uniquely represented by voltages and currents of the form: v(z, t) ¼
i(z, t) ¼

ð

ð

E(r, z)  dl:

(4:1)

CV

H(r, z)  dl:

(4:2)

CI

In equations 4.1 and 4.2, E and H are the electric and magnetic field intensities. CV and CI are arbitrary integration contours, parallel to electric and magnetic flux lines, respectively, as displayed in Figure 4.3 for a coaxial line. The TEM field distribution can be entirely calculated by considering static electric and magnetic fields, namely, an irrotational (or curlfree) electric field and the absence of displacement currents. Hence, the voltages and currents defined in equations 4.1 and 4.2 are independent of the integration contour’s orientation. With reference to Figure 4.3, assuming that the current (I ) flows out of the center conductor and returns into the outer conductor, the contour CV terminates radially on the conductors, while CI follows a closed circular path along a magnetic flux line. 525

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

526

Krishna Naishadham Zg

l

Z0

Vg

Generator

FIGURE 4.1

ZL

Transmission Line

Load

A Transmission Line of Length, ‘, and Characteristic Impedance, Z0 , Connected Between the Generator and the Load

d

2a

2b

2a h

(A) Coaxial Line

(B) Parallel Wires Above a Ground Plane

Dielectric (εr) Dielectric (εr)

(C) Strip Line

(D) Microstrip Line

Examples of Transmission Lines. The dielectric medium has relative permittivity er .

FIGURE 4.2

4.1.1 What Is a Transmission Line? H

I

E

x

−I

FIGURE 4.3 Transverse Electric and Magnetic Field Distribution in a Coaxial Line

A transmission line is a distributed circuit element. Unlike a conventional low-frequency circuit, the voltages and currents on a transmission line vary with longitudinal position because they experience a phase (or time) delay as the wave propagates from one end of the line to the other. This effect becomes important when the line length becomes an appreciable fraction of the wavelength at the operating frequency. Consider the circuit shown in Figure 4.4, where a lossless transmission line of length ‘ is connected between an ideal generator and a load resistance, RL . For simplicity, the load is assumed to be matched to Z 0, the characteristic impedance of the line, so that there is no reflected wave.

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

4

Transmission Lines

527

Z0

Vg

z=0

RL = Z0

l

z=l

FIGURE 4.4 A Matched Transmission Line of Characteristic Impedance, Z0

At time t ¼ 0, let a sinusoidal signal be given by: vg (t) ¼ V0 cos (vt),

so that the voltage wave becomes: (4:3)

where v ¼ 2pf is the radian frequency. Let this signal be applied to the generator located at z ¼ 0. If the line is an air line, the signal travels at a velocity vp ¼ 3  108 m=sec and reaches the load at time t ¼ ‘=vp . Thus, the signal experiences a time delay proportional to the distance that the wave has traveled along the line. At the load, the signal waveform is given by:
 2p‘ vL (t) ¼ V0 cos [v(t
t)] ¼ V0 cos vt
, (4:4) l where vp ¼ lf is used to obtain the second equality. It is evident from equation 4.4 that the delay tracks with the distance traveled, normalized to the wavelength l. The larger the fraction ‘=l gets, the longer is the delay. For conventional circuits operated at a low frequency, the fraction ‘=l is small; hence, the phase delay along the wire can be neglected. At higher frequencies, the wavelength gets shorter; hence, ‘=l becomes larger for a fixed ‘. An important effect of the phase delay is that the line voltage varies with position along the line because the wave requires a finite time to travel from the source location to the measurement location. At low frequencies, these local potential differences are negligible because of very short transit times. This is not valid, however, when the line length becomes an appreciable fraction of the operating wavelength. Equation 4.4 can be generalized to any position z along the line by writing: h zi v(z, t) ¼ V0 cos [v(t
tz )] ¼ V0 cos vt
2p , (4:5) l where vtz is the phase delay incurred over a transit distance z, measured relative to the origin z ¼ 0 at the generator-end of the line. The electrical distance may be written in terms of the phase constant: b¼

2p (rad=m), l

(4:6)

v(z, t) ¼ V0 cos [vt
bz)]:

(4:7)

Equation 4.7 represents an incident wave traveling from the generator toward the load, and it remains valid on an infinitely long line, in which case there is no reflected wave. We can obtain some insight into wave propagation along the line by plotting the voltage as a function of phase delay bz for a fixed time variable vt. Figure 4.5 shows three progression stages of the voltage waveform. As time progresses from t ¼ 0 to t ¼ t1 and then to t ¼ t2, the reference voltage sample at the waveform’s peak moves along the positive z direction from location z ¼ 0 to z1 to z2 , for example. This is true for every point on the waveform. In fact, there is a constant phase delay, bDz ¼ vDt, between successive snapshots. Since v and b are constant, it means that as time increases by Dt, the position of the reference sample is proportionately displaced along positive z direction

V(z,t )

V0

0

π

π 2 t=0

2π

βz

t = t2 >t1

t = t1 > 0

FIGURE 4.5 Progression of the Voltage Waveform on a Lossless Transmission Line

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

528

Krishna Naishadham

by a distance (v=b)Dt. The displacement per unit time is defined as the velocity of propagation or the phase velocity: vp ¼

v ¼ f l: b

(4:8)

4.2 Equivalent Circuit

The current on the incident wave is given by: v(z, t) V0 i(z, t) ¼ ¼ cos [vt
bz]: Z0 Z0

(4:9)

Equation 4.9 depicts only a change in amplitude relative to the voltage in equation 4.7. Since the phase is unchanged, the conclusions on the distributed nature, discussed earlier for the voltage, are also valid for the current.

4.1.2 Lossy Transmission Line On a lossy transmission line the voltage and current waveforms for a wave traveling along the z direction are given by:

i(z, t) ¼

The next section shows that these two parameters, Z 0 and g, are complex because of series resistive loss in the conductors and shunt conductive loss in the dielectric medium.

v(z, t) ¼ V0 e
az cos [vt
bz];

(4:10)

v(z, t) V0
az ¼ e cos [vt
bz]: Z0 Z0

(4:11)

In addition to the phase delay linearly proportional to the distance traveled, the envelope of the wave pattern attenuates in amplitude exponentially according to e
az , as shown in Figure 4.6. In this case, the propagation constant is g ¼ a þ jb, where a is the attenuation constant (measured in nepers per meter) and b is the phase constant. In general, the characteristic impedance Z 0 is also complex. For practical lowloss lines, however, Z 0 can be considered approximately real.

The TEM mode on a transmission line can be represented by the equivalent circuit, shown in Figure 4.7, for a length Dz. Physically, this is a distributed circuit because the circuit elements are not lumped at discrete locations as in a conventional lowfrequency circuit but distributed uniformly along the length of the line. Thus, the elements are defined on per-unit-length basis. The series resistance per unit length, R, is the combined resistance of all the conductors in the line, and it accounts for power dissipation in the conductors. As exemplified in Figure 4.3 for a coaxial line, current flow along the conductor’s surface is accompanied by a circumferential magnetic field. The linkage of this magnetic flux with the current produces a combined inductance per unit length, L, which accounts for flux linkages both internal and external to the conductors. Positive and negative charges of equal magnitude are deposited on the two conductors (Figure 4.3). This charge separation creates a potential difference between the conductors, which accounts for the capacitance per unit length, C. The shunt conductance per unit length, G, accounts for the power dissipation in the non-ideal dielectric medium between the conductors. Application of Kirchhoff ’s voltage and current laws to the circuit in Figure 4.7 leads to the system of coupled differential equations for the phasor voltage and current in the limit Dz ! o: dV (z) ¼
(R þ jvL)I(z), dz dI(z) ¼
(G þ jvC)V (z): dz

V(z,t )

(4:12)

Eliminating the current variable in the former and the voltage variable in the latter of equation 4.12 leads to the uncoupled system of second-order differential equations:

V0 e−αz

I(z)

R∆z

I(z+∆z)

L∆z

+

0

π 2

π

2π

βz V (z)

t=0

+

t = t2> t1 t = t1> 0

C∆z

−

G∆z

V (z+∆z)

− ∆z

FIGURE 4.6 Progression of the Voltage Waveform on a Lossy Transmission Line

FIGURE 4.7 Lumped Parameter Equivalent Circuit of a General Transmission Line

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

4 Transmission Lines

529

d 2 V (z)
g2 V (z) ¼ 0, dz 2 d 2 I(z)
g2 I(z) ¼ 0: dz 2

(4:13)

A lossless line has these properties: (a) it does not dissipate any power, (b) it is non-dispersive (i.e., the phase constant varies linearly with frequency v, or the velocity vp ¼ v=b is independent of frequency), and (c) its characteristic impedance Z 0 is real.

The propagation constant is defined by: g¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (R þ jvL)(G þ jvC):

4.2.2 Low-Loss Line (4:14)

Solution of the differential equations 4.13 results in the timeinstantaneous traveling wave representation: v(z, t) ¼ V0þ e
az cos [vt
bz] þ V0
e az cos [vt þ bz], (4:15) i(z, t) ¼

V0þ
az Z0

e

cos [vt
bz]


V0
az Z0

e

cos [vt þ bz]: (4:16)

Equations 4.15 and 4.16 use the following equation: f (z, t) ¼ R e[F(z)e jvt ],

(4:17)

to convert the voltage and current phasors, V (z) and I(z), into the time-domain. The first term in each equation corresponds to a wave traveling along the þz direction, while the second corresponds to a wave traveling along the opposite direction. Section 4.3 analyzes the transmission line circuit shown in Figure 4.1 using equations 4.15 and 4.16 and also determines the wave amplitudes V0þ and V0
. The characteristic impedance Z 0 can be calculated using: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R þ jvL Z0 ¼ : G þ jvC

(4:18)

The per-unit-length elements R, L, G, and C depend on the resistivity of the conductors, cross-sectional dimensions and shape of the transmission line, and the permittivity of the dielectric medium. Expressions of these elements for some practical transmission lines can be found in any standard electromagnetics text, such as in Ulaby (1999), listed in Appendix A. In summary, the propagation constant g and the characteristic impedance Z 0 for the TEM mode depend only on the transmission line’s geometrical and physical parameters.

4.2.1 Lossless Line For the lossless line R ¼ 0 ¼ G; hence, the attenuation constant a ¼ 0, and the characteristic impedance Z 0 is real. In this case, these equations apply: pffiffiffiffiffiffi g ¼ jb ¼ jv LC : rffiffiffiffi L : Z0 ¼ C

(4:19) (4:20)

A low-loss line is one for which R  vL and G  vC. In this case, g and Z 0 can be approximated from equations 4.14 and 4.18, respectively, as: rffiffiffiffi rffiffiffiffi C G L : þ L 2 C pffiffiffiffiffiffi b ffi v LC : rffiffiffiffi L R G Z0 ffi , 2v 
: C L C R affi 2

(4:21) (4:22) (4:23)

It is emphasized that a low-loss line is non-dispersive subject to the approximations used to derive equation 4.22.

4.2.3 Distortionless Line A distortionless line is truly non-dispersive, exhibits a real characteristic impedance Z 0 , and satisfies the condition RC ¼ LG. Thus, despite being dissipative, it possesses two of the three desirable characteristics of the lossless line. In this case, the propagation parameters are exactly given by: rffiffiffiffi C a¼R : L pffiffiffiffiffiffi b ¼ v LC : rffiffiffiffi L : Z0 ¼ C

(4:24) (4:25) (4:26)

4.3 Alternating Current Analysis This section focuses on the steady-state alternating current (ac) analysis of transmission lines using the basic configuration in Figure 4.8, which shows a line connected between a generator of impedance Zg and a load of impedance ZL . This discussion begins with the general solution in equations 4.15 and 4.16, which contain two unknown voltage amplitudes, V0þ and V0
. The boundary condition is applied at the load to introduce the reflection coefficient, which relates V0þ to V0
. The condition at the generator-end of the line gives an expression for V0
. The standing wave properties of terminated lossless lines as well as lossy transmission lines are presented with examples.

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

530

Krishna Naishadham Zg

z=0

Z0, γ

Vg

ZL

l

Zin(0)

FIGURE 4.8 A Terminated Transmission Line of Characteristic Impedance, Z0 , and Propagation Constant, g

4.3.1 Terminated Lossless Line Reflection Coefficient For a lossless transmission line, the voltage and current phasors can be calculated from equations 4.15 and 4.16 subject to a ¼ 0 and are given by:

Input Impedance The voltage to current ratio for any z, called the input impedance of the line, can be calculated using equations 4.27 and 4.28 as: Zin (z) ¼ Z 0

V (z) ¼

V0þ e
jbz

þ

V0
e jbz

¼

V0þ e
jbz [1

þ G(z)],

(4:27)

 V þ e
jbz 1  þ
jbz I(z) ¼ V0 e
V0
e jbz ¼ 0 [1
G(z)]: (4:28) Z0 Z0 In these equations, the voltage reflection coefficient has been introduced as: G(z) ¼

V0
j2bz e : V0þ

(4:29)

Specializing equations 4.27 and 4.28 to the load position z ¼ ‘ and taking the ratio of voltage to current yields: V (‘) 1 þ G(‘) ¼ ZL ¼ Z 0 : I(‘) 1
G(‘)

(4:30)

Equation 4.30 defines the load reflection coefficient: GL  G(‘) ¼

ZL
Z 0 : ZL þ Z 0

(4:31)

We can eliminate V0
=V0þ from equation 4.29 by specializing it to z ¼ ‘ and the translation formula, which relates the reflection coefficient for any z to that at the load: G(z) ¼ GL e j2b(z
‘) :

(4:32)

1 þ G(z) : 1
G(z)

(4:33)

After substituting the translation formula 4.32 for G(z) into equation 4.33, using equation 4.31 and the identity e ju ¼ cos u þ j sin u, equation 4.33 can be rewritten as: Zin (z) ¼ Z 0

ZL þ jZ 0 tan [b(‘
z)] : Z 0 þ jZL tan [b(‘
z)]

(4:34)

At the input terminal z ¼ 0, the following equation is obtained: Zin (0) ¼ Z 0

ZL þ jZ 0 tan b‘ : Z 0 þ jZL tan b‘

(4:35)

In particular, for a short circuit (ZL ¼ 0), Zin (0) ¼ jZ 0 tan b‘; for an open circuit (ZL ! 1), Zin (0) ¼
jZ 0 cot b‘. Clearly, the input impedance is periodic in position with a period of a half wavelength. It can be seen from equation 4.32 that the reflection coefficient also has the same periodicity. Thus, both the input impedance and the reflection coefficient attain identical values at points separated by a distance of l=2. This property, known as the replication property, has been used in designing a graphical tool called the Smith chart and considerably simplifies the analysis of lossless transmission lines (see Appendix A, Smith [1969], for more information and further reading). We conclude this discussion now with some examples. Figure 4.9 shows the input impedance of a reactively termin-

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

4

Transmission Lines

531 XOC

XSC

0

λ 4

5λ 4 l

3λ 4 λ 2

λ

λ 2

0 λ 4

(A) Short-Circuited Line

λ 3λ 4

l 5λ 4

(B) Open-Circuited Line

FIGURE 4.9 The Input Reactance of Lossless Lines

ated line (open or short) plotted as a function of its length. The half wavelength periodicity is evident. Depending on the line’s electrical length, the input reactance is either capacitive (X < 0) or inductive (X > 0). This suggests that a discrete capacitor or inductor can be constructed from an appropriate length of shorted or open-circuited transmission line. In practice, at microwave frequencies, it is necessary to have a small footprint and reduced parasitic effects for the element. This precludes the use of conventional transmission lines. Instead, for example, an RF inductor is made of a planar spiral etched on a printed circuit board, and a miniature capacitor is constructed in parallel plate configuration using a material with high er and metallized pads for the contacts (see Wadell [1991] in Appendix A for further reading). Quarter Wavelength Transformer The input impedance of a l=4 long line can be calculated from equation 4.35 as: Zin (0) ¼

Z 02 , ZL

Standing Waves From equations 4.27, 4.28, and 4.32 the voltage and current on a lossless line are given by: V (z) ¼ V0þ e
jbz [1 þ GL e j2b(z
‘) ] I(z) ¼

Z0

e

[1
GL e j2b(z
‘) ]:

At the generator end z ¼ 0, this equation results:   V (0) ¼ V0þ 1 þ GL e
j2b‘ :

(4:37) (4:38)

(4:39)

Applying voltage division at the generator-end of the line (see Figure 4.8), the following is calculated: V (0) ¼ Vg

Zin (0) , Zg þ Zin (0)

(4:40)

where, using equations 4.32 and 4.33:

(4:36)

which indicates that a l=4 line can be used as an impedance inverter, (i.e., a high impedance at the load can be converted to a low impedance at the input and vice versa). Using equation 4.36, it follows p that a l=4 line with characteristic impedance ffiffiffiffiffiffiffiffiffiffiffiffiffi given by Z 0 ¼ Z01 Z02 provides matched impedance when connected between two transmission lines of characteristic impedances Z01 and Z02 . Such a line, known as a quarterwave impedance transformer, is frequency-selective (i.e., the mismatch becomes significant as the operating frequency deviates from the narrow band around the center frequency at which the line length ‘ ¼ l=4).

V0þ
jbz

Zin (0) ¼ Z 0

1 þ GL e
j2b‘ : 1
GL e
j2b‘

(4:41)

From equations 4.39 through 4.41, the unknown amplitude V0þ is calculated as: V0þ ¼ Vg

Z0 1 : Zg þ Z 0 1
Gg GL e
j2b‘

(4:42)

Therefore, the voltage and current on a lossless transmission line are given by: V (z) ¼ Vg

Z 0 e
jbz [1 þ GL ej2b(z
‘) ] , Zg þ Z 0 1
Gg GL e
j2b‘

(4:43)

Krishna Naishadham 1 e
jbz [1
GL ej2b(z
‘) ] : Zg þ Z 0 1
Gg GL e
j2b‘

(4:44)

The voltage or current magnitude describes the standing wave nature of the field on a terminated transmission line. Standing waves result from reflections along the line, and they describe mismatch between load or source termination and the line. From equations 4.37 and 4.42, these equations are calculated:    jV (z)j ¼ V0þ 1 þ jGL je j[uL þ2b(z
‘)]  (4:45)   1=2 ¼ V0þ  1 þ jGL j2 þ 2jGL j cos u ,

and occurs for cos u ¼ 1 in equation 4.45 or at: zmax ¼ ‘ þ

 
1=2 Z 0  , (4:46) 1 þ jGg GL j2
2jGg GL j cos f  Zg þ Z 0

where u ¼ uL 2b(z
‘), f ¼ ug þ uL
2b‘, GL ¼ jGL jffuL , and Gg ¼ jGg jffug . The voltage magnitude in equation 4.45 depends on the position as 2bz and, thus, replicates every half wavelength. The phase can be calculated from equation 4.37 and shown to replicate at full wavelength spacing. The magnitude of the current is derived from equation 4.37 as:  jV0þ j  1
jGL je[uL þ2b(z
‘)]  Z0 1=2 jV0þ j  ¼ 1 þ jGL j2
2jGL j cos u : Z0

  jV jmin ¼ V0þ (1
jGL j),

(4:50)

occurs for cos u ¼
1 in (4.45) or at: l zmin ¼ zmax : 4

(4:51)

The ratio of maximum to minimum voltage (or current) is called the standing wave ratio (SWR) and is given by: S¼

jV jmax 1 þ jGL j : ¼ jV jmin 1
jGL j

(4:52)

This factor defines the mismatch along the line and can range from unity for a matched load to infinity for an ideal short or open termination. When the load termination is resistive, equation 4.52 yields: 

jI(z)j ¼

S¼

(4:47)

It can be observed from equations 4.45 and 4.47 that the voltage is maximum when the current is minimum and vice versa. The maximum voltage is given by:   jV jmax ¼ V0þ (1 þ jGL j), (4:48)

(4:49)

Likewise, the voltage minimum of:

and   þ    V  ¼ Vg  0 

nl uL l
, n ¼ 0, 1, . . . N : 2 4p

RL =Z 0 , Z 0 =RL ,

RL > Z 0 . RL < Z 0 :

(4:53)

When RL > Z 0 (RL < Z 0 ), a voltage maximum (minimum) occurs at the load position. Figure 4.10 displays the magnitude of the voltage and current for a terminated transmission line with Vg ¼ 10ff45 V , Z 0 ¼ 50V, Zg ¼ 75V, ZL ¼ 100V, and ‘ ¼ l. The voltage and current magnitudes replicate every half wavelength. The 200

200

5.5

0.11

150

150

0.1

100

100

50

50

0

0

5 4.5

0.09

4

0.08

3.5

0.07

3

0.06

Voltage Current

2.5 2

0

0.25 0.5 0.75 Normalized position z/λ

(A) Magnitude

FIGURE 4.10

1

Voltage Phase [degrees]

0.12

Current magnitude [A]

6

−50

−50

−100

−100

0.05

−150

0.04

−200

Voltage Current 0

−150

0.25 0.5 0.75 Normalized position z/λ

(B) Phase

Voltage and Current on a Resistively Terminated Lossless Line

1

−200

Current phase [degrees]

I(z) ¼ Vg

Voltage magnitude [V]

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

532

Copyright © 2005. Academic Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

4

Transmission Lines

533

there are no standing waves (S ¼ 1) because GL ¼ 0. In this case, the standing wave pattern looks flat as shown in Figure 4.11.

|V(z)| λ 2

2 |V 0+|

0

λ 4

λ 2

3λ 4

λ

z

Power Flow A transmission line is intended for power transfer between the generator and the load. The average power flow along a lossless line can be calculated using:

(A) Short Circuit

1 Pav ¼ Re½V (z)I(z) , 2

|V(z)| λ 2

2 |V 0+|

0

λ 4

λ 2

3λ 4

λ

z

where the voltage and current are given by equations 4.37 and 4.38, respectively, and the asterisk denotes a complex conjugate. On a lossless line, since there is no dissipation, the average power is independent of position z. Therefore, the voltage and current in equation 4.57 can be calculated at any point along the line. Using the conditions at the load in equations 4.37 and 4.38, it follows that:

(B) Open Circuit

Pav ¼

|V(z)|

|V 0+|

0

λ 4

λ 2

3λ 4

λ

z

(C) Matched Termination

FIGURE 4.11 Line

The Voltage Standing Wave Patterns on a Lossless

 1 jV0þ j2 1
jGL j2 : 2Z 0


   þ  2p    (z
‘ , jV (z)j ¼ 2 V0 sin l

(4:54)

and for the open circuit, the pattern is as follows: 
   þ  2p    jV (z)j ¼ 2 V0 cos (z
‘) : l

(4:58)

The first term in equation 4.58 corresponds to the incident wave, and the second corresponds to the reflected wave. The magnitude jV0þ j can be computed using equation 4.46. For a matched load, GL ¼ 0 (ZL ¼ Z 0 ); therefore, the power transferred to the load is just the first term in equation 4.58. This is not the maximum power transfer, however, for there is still a mismatch at the generator-end of the line because Zg 6¼ Zin ¼ Z 0 (Gg 6¼ 0). The maximum power transfer on a lossless line occurs for a conjugate match at the generator-end, Zin ¼ Zg , irrespective of the load impedance, and is given by: Pav

standing wave ratio is observed to be 5:7=2:85 ¼ 2, which agrees with equation 4.53. Likewise, Figure 4.11 shows the voltage standing wave patterns for a short circuit, an open circuit, and a matched termination, with ‘ ¼ l. For the short circuit, equation 4.45 yields:

(4:57)

 2 Vg  ¼ : 4Rg

(4:59)

4.3.2 Terminated Lossy Transmission Line The voltage and current on a lossy transmission line are given by: V (z) ¼ V0þ e
az e
jbz þ V0
e az e jbz ¼ V0þ e
az e
jbz [1 þ G(z)]:  1  þ
az
jbz V0 e e
V0
e az e jbz Z0 V þ e
az e
jbz [1
G(z)]: ¼ 0 Z0

(4:60)

I(z) ¼ (4:56)

These waves are known as complete standing waves. The minimum value is zero, the nulls are sharper than those in Figure 4.10 for the resistive load, and the SWR becomes infinite. For a matched load ZL ¼ Z 0 , equation 4.52 shows that

(4:61)

The translation formula for the reflection coefficient follows from: G(z) ¼ GL e 2a(z
‘) e j2b(z
‘)

(4:62)

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534

Krishna Naishadham

The magnitude of the reflection coefficient is given by: jG(z)j ¼ jGL je 2a(z
‘) ,

(4:63)

which is a function of both frequency and position, unlike the constant reflection coefficient magnitude: jG(z)j ¼ jGL j:

(4:64)

Equation 4.64 is for a lossless line. The input impedance at z ¼ 0 is as follows: ZL þ Z 0 tanh g‘ Zin (0) ¼ Z 0 : Z 0 þ ZL tanh g‘

4.4 Smith Chart (4:65)

The impedance at any position z is obtained by replacing ‘ ! (‘
z). It should be noted that the characteristic impedance Z 0 for a lossy line is complex. For practical low-loss lines, however Z 0 may be considered as real. Because of attenuation, it is observed from equations 4.63 and 4.65 that neither the reflection coefficient nor the input impedance exhibit the replication property. Also varying with position along the line is the standing wave ratio: 1 þ jGL je 2a(z
‘) S¼ : 1
jGL je 2a(z
‘)

Again, the first term denotes the power loss in the incident wave, the second term denotes the power loss in the reflected wave, and the third term represents power loss due to interaction between the two. As a increases, the power loss in the incident and reflected waves as well as in the interaction term increases. In the limiting case of a lossless line, there should not be any dissipation. Clearly, the interaction term vanishes because u0 ¼ 0, and the power dissipation in the first two terms reduces to zero.

(4:66)

The power flow is calculated from equation 4.57 using the voltage and current from equations 4.60 and 4.61, respectively. Denoting GL ¼ jGL jffuL and Z 0 ¼ jZ 0 jffu0 , the following is obtained:

The Smith chart is a graphical tool for determination of the reflection coefficient and impedance along a transmission line. It is an integral part of microwave circuit performance visualization, modern computer-aided design (CAD) tools, and RF/ microwave test instrumentation. Basically, a Smith chart is a polar graph of normalized line impedance in the complex reflection coefficient plane. Let Z ¼ R þ jX be the impedance at some location along a lossless line. The reflection coefficient is given by: G¼

Z
Z0 ¼ Gr þ jGi : Z þ Z0

If the impedance Z is normalized with respect to Z 0 and z  Z=Z 0 ¼ r þ jx is written in terms of the reflection coefficient (z in this section should not be confused with the position variable z used elsewhere), the following equation is obtained:

2

Pav (z) ¼


  jV0þ j
2az e cos u0 1
jGL j2 e 4a(z
‘) 2jZ 0 j 2 jV0þ j

jZ 0 j

r þ jx ¼ (4:67)

jGL je
2a‘ sin u0 sin½uL þ 2b(z
‘) :

The first term in 4.67 computes the incident power, the second term corresponds to the reflected power, and the third term represents the interference between these two waves. Note that for a lossless line, since u0 ¼ 0, the interaction term will vanish, and the power transferred to the load reduces to the difference between incident and reflected contributions. The power dissipated along the line can be computed by subtracting the power at the load, Pav (‘), from the power at the input, Pav (0), and is given by: 2

 jV0þ j cos u0 1
e
2a‘ 2jZ 0 j 
2a‘

Pdiss ¼ Pav (0)
Pav (‘) ¼  þjGL j2 e
2a‘ 1
e þ

2 2jV0þ j

jZ 0 j

jGL je
2a‘ sin u0 sin (b‘) cos (uL
b‘):

(4:68)

(4:69)

(1 þ Gr ) þ jGi : (1
Gr )
jGi

(4:70)

The key to understanding the Smith chart is the realization that equation 4.70 corresponds to two families of circles given by: r¼

1
G2r
G2i , (1
Gr )2 þ G2i

(4:71)

x¼

2Gi : (1
Gr )2 þ G2i

(4:72)

For a given r, 4.71 represents a circle in the complex G-plane that is centered at the point [r=(1 þ r), 0] and has a radius of 1=(1 þ r). For a given x, equation 4.72 yields a circle centered at [1, 1=x] with a radius 1=jxj. A few of the former circles are shown as solid lines and the latter circles as dashed lines in Figure 4.12. The reactance is inductive in the upper half of the plane and capacitive in the lower half. Viewed in terms of impedance coordinates, the Smith chart displays the normalized impedance z ¼ r þ jx at the intersection of an r
x circle

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4 Transmission Lines

535 Γi

x=1

x=2

x = 0.5

r=0 r = 0.5 SC

x=0 O

r=1

OC

Γr

r=2

x = −0.5 x = −1

FIGURE 4.12

x = −2

Basic Construction of the Smith Chart

pair. Because the underlying polar grid is a reflection coefficient plane, at any point on the chart the reflection coefficient can be read by measuring the radius and angle of the point (polar coordinates) on an appropriate scale. The outermost circle (OC) corresponds to the unity reflection coefficient. The open circuit lies at (1,0) in the reflection coefficient plane, and the short circuit (SC) is at (
1, 0). Figure 4.13 displays a typical commercially available Smith chart. The basic operations of the chart can be understood by an example to be discussed next. For better clarity and practice, it is highly recommended to follow this example by repeating the graphical solution taps on a new Smith chart. Suppose that a (1=3)l-long, 50V line is connected to a load impedance of (25 þ j25) V. The input impedance and reflection coefficient can be determined by using the Smith chart. The load impedance is normalized, and the point (0:5 þ j0:5) is plotted and shown as point A in Figure 4.13. This point can be translated to the input by moving (1=3)l toward the generator. Two scales on the Smith chart’s outer periphery indicate movement in wavelengths either toward the generator (clockwise) or toward the load (counterclockwise). One complete rotation around the

0.088 λ TG

A

C

S = 2.65

B

|Γ| = 0.45

0.421 λ TG

FIGURE 4.13

Smith Chart with Solution to an Example Problem

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536

Krishna Naishadham

chart amounts to a half wavelength traversal. The relative position of a point on the transmission line can be determined by extending the radius of the point to intersect one of these two scales and reading its value in l. Thus, point A reads 0:088l toward generator (TG). If the radius of point A is projected onto the reflection coefficient, E or I, scale at the bottom of the chart, the measurement is jGL j ¼ 0:45. The angle of GL is measured using the corresponding angle of reflection coefficient scale on the periphery of the unit circle as 116:5 . Therefore, GL ¼ 0:45ff116:5 . The input terminal is at a distance of 0:333l TG from the load, located at 0:088l. Thus, the position of the input terminal is at 0:088 þ 0:333 ¼ 0:421l TG. Since jGj is constant for a lossless transmission line, the reflection coefficient at the input, denoted by point B on the 0:421l TG radial line, has the same radius as point A ( ¼ 0:45). Thus, we obtain the input reflection coefficient Gin ¼ 0:45ff
123 . For convenience, jGj ¼ 0:45 circle is drawn in Figure 4.13 and allows the determination of the reflection coefficient at any point along the line. The normalized impedance at point B is read as 0:47
j0:44, which, after denormalization (multiplication by 50), yields (23:5
j22) V. The standing wave ratio can be directly read from the scale at the bottom of the chart by projecting the radius of the reflection coefficient circle. It is instructive, however, to determine the point at which the normalized impedance equals SWR. From equation 4.48 there is a voltage maximum of:   jV jmax ¼ V0þ (1 þ jGL j), (4:73) which occurs at points along the line where the incident and reflected waves add in phase (constructive interference). Likewise, from equation 4.50 there is a voltage minimum of:   jV jmin ¼ V0þ (1
jGL j), (4:74) which occurs when the incident and reflected waves add out of phase (destructive interference). Because the current reflection coefficient is negative of the voltage reflection coefficient, at a voltage maximum location the current is a minimum and is given by: jIjmin ¼

jV0þ j Z0

(1
jGL j):

(4:75)

Dividing equation 4.75 into 4.73 and normalizing with respect to Z 0 results in the impedance:   Vmax  1 þ jGL j D  ¼ zmax ¼  ¼ S: (4:76) Imin  1
jGL j Thus, at the position of a voltage maximum (or a current minimum), the normalized impedance is a maximum and equals S. Similarly, it can be shown that at the position of a voltage minimum (or a current maximum), the normalized impedance is a minimum and equals:

  Vmin  1
jGL j D 1 ¼ ¼ : zmin ¼  Imax  1 þ jGL j S

(4:77)

These two points can be easily obtained on the Smith chart as the two points where the real axis intersects the constant jGj circle. From Figure 4.13, S ¼ 2:65 at the intersection of the jGj circle and the positive real axis. This agrees with the value read from the SWR scale at the bottom of the chart. In addition, the voltage maximum nearest to the load is at a distance of 0:25
0:088 ¼ 0:162l from the load. The Smith chart can be used to calculate admittances, a feature very useful in designing impedance matching circuits. The normalized admittance occurs on the reflection coefficient circle, diametrically opposite to the normalized impedance. Corresponding to the r and x circles, respectively, the admittance coordinates are the constant conductance and constant susceptance circles. The upper half of the plane yields positive susceptances, and the lower half of the plane yields negative susceptances. The normalized input admittance for the example in Figure 4.13 is observed (at point C) as (1:1 þ j1:08). The net admittance follows from: Yn ¼

y 1:1 þ j1:08 (mhos): ¼ Z0 50

(4:78)

The Smith chart is used in several applications, including microwave measurements, instrument displays, and computeraided design of microwave circuits. For example, impedance matching networks such as single-stub and double-stub tuners or quarter-wave transformers can be conveniently designed using the Smith chart (see Cheng [1989] in Appendix A for further reading). For space limitations, these applications are not presented in this chapter. The reader is referred to Appendix A at the end of the chapter for several useful references illustrating microwave circuit design utilizing the Smith chart.

4.5 Summary This chapter was devoted to the fundamentals of frequencydomain analysis of transmission lines with an emphasis on physical concepts rather than detailed mathematical derivations. A distributed circuit model was employed to derive the transmission line equations, whose solution describes the wave behavior of voltages and currents along the line. Initially, we considered an infinite line to emphasize the concepts of phase delay and spatial dependence, which account for the distributed nature of voltages and currents along a transmission line. The propagation characteristics of lossless and lossy lines were discussed, with the latter specialized to low-loss and distortionless cases. Properties of the standing waves along a terminated transmission line, such as impedance, reflection coefficient, voltage and current distribution, were discussed in detail. Mathematical expressions were derived for power flow, including

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4 Transmission Lines power loss due to interference between incident and reflected waves on a lossy line and the condition of conjugate matching for maximum power transfer. Finally, the basic operations of the Smith chart to characterize standing waves on a terminated line were explained with an example.

Appendix A: References The technical literature on transmission lines is pervasive with several articles scattered in books and journals, ranging from introductory material to advanced topics and covering both theoretical and practical aspects. The basic background material introduced in this chapter can be supplemented with further reading from the references listed below that are arranged by topics. A few books are devoted exclusively to transmission lines. Transmission line theory, however, is covered exhaustively in several electromagnetics textbooks. The list is not intended to be exhaustive. Transmission Lines Dworsky, L.N. (1979). Modern transmission line theory and applications. New York: John Wiley & Sons. Freeman, J.C. (1996). Fundamentals of microwave transmission lines. New York: John Wiley & Sons. Gardiol, F.E. (1987). Lossy transmission lines. Norwood, MA: Artech House. Granzow, K.D. (1998). Digital transmission lines: Computer modelling and analysis. New York: Oxford University. Johnson, W.C. (1950). Transmission lines and networks. New York: McGraw-Hill. Magnusson, P.C., Alexander, G.C., and Tripathi, V.K. (1992). Transmission lines and wave propagation. Boca Raton, FL: CRC Press. Matick, R.E. (1995). Transmission lines for digital and communication networks. New York: IEEE Press. Smith, P.H. (1969). Electronic applications of the smith chart in waveguide: Circuit and Component Analysis. New York: McGraw-Hill. Wadell, B.C. (1991). Transmission line design handbook. Norwood, MA: Artech House.

Introductory Electromagnetics Cheng, D.K. (1989). Field and wave electromagnetics. Reading, MA: Addison-Wesley.

537 Hayt, W.H. Jr. (1989). Engineering electromagnetics. New York: McGraw-Hill. Iskander, M.F. (1992). Electromagnetic fields and waves. Upper Saddle River, NJ: Prentice Hall. Johnk, C.T.A. (1988). Engineering electromagnetic fields and waves. New York: John Wiley & Sons. Paul, C.R., Whites, K.W., and Nasar, S.A. (1988). Introduction to electromagnetic fields. New York: McGraw-Hill. Rao, N.N. (1987). Elements of engineering electromagnetics. Upper Saddle River, NJ: Prentice Hall. Ulaby, F. (1999). Fundamentals of applied electromagnetics. Upper Saddle River, NJ: Prentice Hall.

Advanced Electromagnetics Balanis, C.A. (1989). Advanced engineering electromagnetics. New York: John Wiley & Sons. Harrington, R.F. (1961). Time-harmonic electromagnetic fields. New York: McGraw-Hill. King, R.W.P., and Prasad, S. (1986). Fundamental electromagnetic theory and applications. Englewood Cliffs, NJ: Prentice Hall. Kong, J.A. (1990). Electromagnetic wave theory. New York: John Wiley & Sons. Kraus, J.D., and Fleisch, D.A. (1999). Electromagnetics with applications. New York: McGraw-Hill. Wait, J.R. (1985). Electromagnetic theory. New York: Harper and Row.

Microwave Circuits Collin, R.E. (1992). Foundations for microwave engineering. New York: McGraw-Hill. Edwards, T. (1992). Foundations for microstrip circuit design. New York: John Wiley & Sons. Gupta, K.C., Garg, R., and Chadha, R. (1981). Computer-aided design of microwave circuits. Norwood, MA: Artech House. Howe, H. (1973). Stripline circuit design. Dedham, MA: Artech House. Matthaei, G.L., Young, L., and Jones, E.M.T. (1965). Microwave filters, impedance matching networks, and coupling structures. New York: McGraw-Hill. Pozar, D.M. (1998). Microwave engineering. New York: John Wiley & Sons. Ramo, S., Whinnery, J.R., and van Duzer, T. (1994). Fields and waves in communication electronics. New York: John Wiley & Sons. Sander, K.F., and Reed, G.A.L. (1978). Transmission and propagation of electromagnetic waves. Cambridge: Cambridge University Press.

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