transmission lines 1

8/12/2019 Transmission Lines 1

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TRANSMISSION LINE THEORY

I. The Transmission Line Model:

Consider the following repeating (uniform) sequence of "lumped" circuit elements:

Applying elementary circuit analysis to each node of such a "discrete" transmission line we

may write a set of basic circuit equations.

vn+1t( ) = vn t( ) R sin +1t( ) Ls

d

dtin+1t( ) [ I-1a ]

in +1 t( ) =in t( ) Gp vn t( ) Cp

d

dtv

n t( ) [ I-1b ]

The crucial matter is that the voltage and current vary both in time and space! To obtain a

solution, we first deal with the time dependence by making use of the "phasor" concept --

i.e. we replace the time dependent variables with their Fourier Transforms

vn t( ) = V

n( ) exp j t[ ]d

+

and in t( ) = In ( ) expj t[ ]d

+

[ I-2]

or in the language of circuit analysis

vn t( ) = Vn ( ) exp j t[ ]{ } =Vn ( ) cos t+ V( ) [ I-3a ]

in t( ) = In ( ) expj t[ ]{ } = In ( ) cos t+ I( ) [ I-3b ]


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TRANSMISSION LINE THEORY PAGE 2

R. V. Jones, October 23, 2002

Thus, the set of differential circuit equations for a discrete, uniform transmission line

becomes a huge set of algebraic equations -- viz.

Vn+1 ( ) =Vn ( ) Zs ( )In +1 ( ) [ I-4a ]

In +1 ( ) =In ( ) Yp ( ) Vn ( ) [ I-4b ]

where Zs ( ) = R s +j L s and Yp ( ) = Gp +j Cp are, respectively, these ri es

impedance and theshunt (paral lel) admittance of the transmission line.

II. Exact Solutions of Transmission Line Equations:

Our task is to solve Eqs. [ I-4 ]. To that end, we first cast this array of coupled

inhomogeneousequations in the form of a set of coupled, homogeneous algebraic

equations -- viz.

Zs ( ) Yp ( ) Vn ( ) =Vn +1 ( ) +Vn1 ( ) 2 Vn ( ) [ II-1a ]

Zs ( ) Yp ( )In ( ) =In +1 ( ) +In 1 ( ) 2In ( ) [ II-1b ]

Fortunately, here is an amazingly simple set of solutions for this enormous set of algebraic

equations. These solutions may be written in the form

Vn ( ) = a complex constant{ } exp j n ( )[ ] [ II-2a ]

In ( ) = another complex constant{ } exp j n ( )[ ] [ II-2b ]

We might characterize these solutions as constant phase solutions in the sense that the

solution at a given node along transmission line is identical to the solution at an adjacent

node except for constant phase factor. If these constant phase solutions are to be valid

solutions of Eqs. [ II-1 ], the phase constant ( ) must satisfy the equation

Zs ( ) Yp ( ) exp j n ( )[ ] = exp j n +1( ) ( )[ ] + exp j n 1( ) ( )[ ] 2 expj n ( )[ ] [ II-3 ]


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Canceling the common exp j n ( )[ ] factor on both sides of the equation, we obtain

Zs ( ) Yp ( ) = exp j ( )[ ] + exp j ( )[ ] 2

= expj ( ) 2[ ] exp j ( ) 2[ ]{ }

2

= 2j sin ( ) 2[ ]{ }

2 [ II-4 ]

Thus, we have obtained an extremely important result which we will, hereafter, refer to as

the dispersion relationship for a discrete, uniform transmission line-- viz.

2j sin ( ) 2[ ] = Zs ( ) Yp ( )[ II-5 ]

Important Special Cases:

1. The "Ideal" or "Lossless" LC-Transmission Line:

If we take Z s ( ) =j L s and Yp ( ) =j C p , then Eq [ II-5 ] becomes

sin ( ) 2( ) = Ls Cp 2 [ II-6 ]

which is the dispersion relationship of a discrete, uniform, ideal trans-

mission line (Note that the discrete ideal line is, effectively, a low-pass filter.).


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2. The "Lossy" RC-Transmission Line:

If we take Z s ( ) = R s and Yp ( ) = j Cp , then Eq [ II-5 ] becomes

sin ( ) 2( ) = 2j[ ]1

j R sCp [ II-7 ]

We have a problem! What, in heavens name, do we mean by the square root of

j (i.e. the fourth root of 1)? To interpret what is meant by j , note that

j =1 +j

2

2

so that

j = 1+j

2

[ II-8 ]

Therefore, the dispersive relationship for a "lossy" RC-transmission line -- i.e. Eq.

[ II-7 ] becomes


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sin2

= sin

+j 2

= sin

2

cos

j 2

+ cos

2

sin

j 2

= 1j{ }

2

R sCp

2

[ II-9 ]

which yields, upon equating real and imaginary parts,

sin

2

cosh

2

=

1

2

R sCp2

[ II-10a ]

cos

2

sinh

2

= m

1

2

R sCp2

. [ II-10b ]

For ease of interpretation, we make the small argument approximation so that

R sCp

2 Phase shift per section [ II-11a ]

= m

R sC p2

Attenuation per section [ II-11b ]


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To first-order in z, we obtain the famous inhomogeneous "telegrapher equations"fora lossless transmission line -- viz.

zv z, t( ) ls

ti z, t( ) where ls = lim

z 0

L s

z

[ III-4a ]

z

i z, t( ) cpt

vz ,t( ) where cp = limz 0Cp

z

[ III-4b ]

which, in turn, yields the even more famous homogeneous "wave equations"

2

z2vz, t( ) = ls

z

t

i z, t( ) = lst

z

i z, t( ) = lscp 2

t2vz, t( )

or

2

z2v z, t( ) = lscp

2

t2vz, t( ) [ III-5a ]

and 2

z2i z, t( ) = lscp

2

t2i z, t( ) [ III-5b ]

The truly remarkable point is that any old function of the form vz, t( ) = fz v t( ) and/or of

the form vz, t( ) = gz + v t( ) will satisfy the Telegrapher and Wave equations!!!!


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zc =1

cp v=

lscp

is the characteristic impedance [ III-8b ]

General Uniform, Continuous Transmission Line:

We now turn to the more general case represented by Eqs. [ I-4 ]. Again, we develop a

"continuous approximation" of these equations by making use of a Taylor expansion for

small spatial separation of the nodes. Thus, Eqs. [ I-4 ] become

Vn ( ) + Zs ( )In ( ) =Vn1 ( ) Vn ( ) zz

Vn ( ) [ III-9a ]

In

( ) Yp ( ) Vn ( ) =In +1 ( ) In ( ) + zz

In

( ) [ III-9b ]

Once again, to first-order in z , these gyrations lead to a more general version of theinhomogeneousTelegrapherequations -- viz.

z

V z,( ) zsI z,( ) where zs = lim z0Zsz

[ III-10a ]

z

I z,( ) ypV z,( ) where yp = limz 0Yp

z

[ III-10b ]

and to a more general version of the homogeneous Helmhotz equation(s) -- viz.

2

z2V z,( ) = zsypV z,( ) [ III-11a ]

2

z2I z,( ) = zsypI z,( ) [ III-11b ]

Drawing on our experience above in the analysis of the discrete case, we now look for

solutions in the form


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V z,( ) =Vo exp ( )z[ ] and I z,( ) =Io exp ( )z[ ] [ III-12 ]

so that

( ) V z,( ) = zs

I z,( ) and ( )I z,( ) = yp

V z,( ) [ III-13 ]

and

2 ( ) V z,( ) = zsypV z,( ) and 2 ( )I z,( ) = zsypI z,( ) [ III-14 ]

where ( ) is, in general, complex -- i.e. ( ) = ( ) +j ( ) . Therefore, if the

proposed solution is to valid we must have

2 ( ) = ( ) +j ( )[ ]2

= zs ( ) yp ( ) = rs +j ls[ ] gp +j cp[ ] [ III-15 ]

On equating real and imaginary parts of this expression, we obtain

2 ( ) 2 ( ) = rs gp 2

ls cp [ III-16a ]

and 2 ( ) ( ) = rs cp +g p ls[ ] . [ III-16b ]

In the small attenuation approximation, we see that the phase shift per unit is given by

( ) ls cp 1rs

ls

gp

cp

[ III-17a ]

and the attenuation per unit length by

( ) 1

2rs

cp

ls+ gp

lscp

[ III-17b ]

If we consider, once again, the all important special case of a "lossless" LC transmission

line -- i.e. where gp rs 0, we see that


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( ) = ( ) +j ( ) = 0 j ls cp [ III-18 ]

Thus, for a general uniform, continuous transmission line, the linear combination

of two independent frequency domainsolutions may be written

V z,( ) =V+ ( ) exp ( )z[ ] +V ( ) exp + ( )z[ ] [ III-19 ]

where ( ) = zs ( ) yp ( ) . In the time domain for a single frequency, we have

vz,t( ) = V+ ( ) exp ( )z +j t[ ] +V ( ) exp + ( )z +j t[ ]{ } [ III-20 ]

To be concrete and for ease of interpretation, we discuss in much of what follows a general

lossless or non-attenuating transmission line so that the time domain solution becomes

vz, t( ) = V+ ( ) exp j t ( )z[ ]{ } +V ( ) exp j t+ ( )z[ ]{ }( )=V+ ( ) cos t ( )z + +[ ] +V ( ) cos t+ ( )z + [ ]

[ III-21 ]

As in earlier discussions, we interpret

V+ ( ) cos t ( )z + +[ ]

as a continuous wave propagating to the right (positive z-direction) and

V ( ) cos t+ ( )z + [ ]as a continuous wave propagating to the left (negative z-direction). From one of the

Telegrapher equations -- viz. Eq. [ III-10a ] -- we see that

I z,( ) = 1

zs ( )z

V z,( ) [ III-22a ]

so that


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I z,( ) =I+ ( ) exp ( )z[ ] +I ( ) exp + ( )z[ ]

=1

Zc ( ) V+ ( ) exp ( )z[ ] -V ( ) exp + ( )z[ ]{ }

[ III-22b ]

where Zc ( ) = zs ( ) ( ) = zs ( ) yp ( ) [ III-23 ]

is the characteristic impedance of the transmission line. For a "lossless" LC

transmission line, we see that

Zc ( ) = Zc = ls cp [ III-24 ]

Finally, we introduce the extreme important (but rather confusing) notion of a spatially

varying wave impedance which is define as

Z z,( ) V z,( )I z,( )

= Zc ( ) V+ ( ) exp ( )z[ ] +V ( ) exp + ( )z[ ]

V+ ( ) exp ( )z[ ] -V ( ) exp + ( )z[ ]

= Zc ( )1 +V ( ) V+ ( )[ ] exp +2 ( )z[ ]1 - V ( ) V+ ( )[ ] exp +2 ( )z[ ]

= Zc ( )

1 +V z,( )1 - V z,( )

[ III-25 ]

where Vz,( ) V ( ) V+ ( )[ ] exp +2 ( )z[ ] [ III-26 ]

is the spatial varying voltage reflection coefficient. Thus, we may write the general

solution in the compact form

V z,( ) =V+ ( ) exp ( )z[ ] 1 +Vz,( ){ [ III-27a ]

I z,( ) Zc ( ) =V+ ( ) exp ( )z[ ] 1 - Vz,( ){ [ III-27b ]


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IV. Transmission Lines Terminations:

It remains for us to determine the spatial varying wave impedance and reflection coefficient

which must satisfy the dual equations

Z z,( ) = Zc ( )1 + Vz,( )1 Vz,( )

[ IV-1a ]

Vz,( ) =Z z,( ) Zc ( )Z z,( ) + Zc ( ) [ IV-1b ]

at every point along the transmission line. To that end we must consider the effect of

transmission line terminations.

LOADED TRANSMISSION LINE

Consider first some simple, but important cases.

1. A "shorted" transmission line -- i. e. Z zL ,( ) = 0 so that V zL ,( ) = 1.


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At other points along the line

V z,( ) = V ( ) V+ ( )[ ] exp +2 ( )z[ ] = VzL ,( ) exp +2 ( ) z zL[ ]{ } [ IV-2 ]

so that in this case

Vz,( ) = exp 2 ( ) zL z[ ]{ } [ IV-3a ]

and

V z,( ) =V+ ( ) exp ( )z[ ] 1 exp 2 ( ) zL z[ ]{ }( )

=V+ ( ) exp ( )z ( ) zL z[ ][ ] exp + ( ) zL z[ ]{ } exp ( ) zL z[ ]{ }( )

=V+ ( ) exp ( )zL[ ] exp + ( ) zL z[ ]{ } exp ( ) zL z[ ]{ }( )

[ IV-3b ]

When the attenuation is zero

V z,( ) = 2j V+ ( ) exp j ( )zL[ ] sin ( ) zL z[ ]{ } [ IV-4a ]

Zc ( )I z,( ) = 2V+ ( ) exp j ( )zL[ ] cos ( ) zL z[ ]{ } [ IV-4b ]

Z z,( ) =V z,( )I z,( )

=j Zc ( ) tan ( ) zL z[ ]{ } [ IV-4c ]

Such a solution is called a "pure" standing wave. It is a spatial varying

voltage oscillation which may be observed with an oscilloscope. The pattern

that would be observed is graphed below as it would be seen at 16 distinct

times equally spaced at 1/16 of a period. The voltage across the short is, of

coarse, zero at all times! There is, of course, another voltage node whenever

( ) zL

z[ ] = 2zL z[ ] ( ) = integer[ ] .


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VOLTAGE ACROSS A SHO RT ED TRANSMISSION LINE

2. An "open" transmission line -- i. e. Z zL

,( ) = so that VzL ,( ) = +1.

Vz,( ) = +exp 2 ( ) zL z[ ]{ } [ IV-5a ]

and


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VSWR V z,( )

Max

V z,( )Min

=1 + V ( )1 - V ( )

[ IV-10 ]

Note that for a pure running wave solution VSWR=1 and for a pure standing

wave solution VSWR= .

We have one last really important task -- viz. establishing the all important

wave impedance transformation. To that end we see from Eqs. [ IV-1 ]

that

Z z,( ) = Zc ( )1 +Vz,( )1 - Vz,( )

= Zc ( )1 +

V

zL

,( )

exp 2 ( ) zL

z[ ]{ }1 - VzL ,( ) exp 2 ( ) zL z[ ]{ }

= Zc ( )exp + ( ) zL z[ ]{ } +VzL ,( ) exp ( ) zL z[ ]{ }exp + ( ) zL z[ ]{ } - V zL ,( ) exp ( ) zL z[ ]{ }

[ IV-11 ]

But VzL ,( ) =Z z

L,( ) Zc ( )

Z zL

,( ) + Zc ( )

so that

Z z,( ) = Zc ( )

exp + ( ) zL z[ ]{ } +

Z zL,( ) Zc ( )

Z zL,( ) + Zc ( )exp ( ) z

L z[ ]{ }

exp + ( ) zL z[ ]{ }-Z z

L,( ) Zc ( )

Z zL ,( )+ Zc ( )exp ( ) zL z[ ]{ }

= Zc ( )Z zL ,( ) exp + ( ) z L z[ ]{ } + exp ( ) zL z[ ]{ }[ ] + Zc ( ) exp + ( ) zL z[ ]{ } exp ( ) zL z[ ]{ }[ ]Z zL ,( ) exp + ( ) zL z[ ]{ } exp ( ) zL z[ ]{ }[ ] +Z c ( ) exp + ( ) zL z[ ]{ } + exp ( ) zL z[ ]{ }[ ]

[ IV-12a ]


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Zmatchz ,( ) = Zc ( )Z c ( ) +j Zc ( ) tan ( ) zL z[ ]{ }Zc ( )+ j Z c ( ) tan ( )zL z[ ]{ }

= Zc ( )

[ IV-14c ]

d. Quarter wavelength matching transformer:

Z 4 z,( ) = Zc ( ) Z zL,( ) + j Z c ( ) tan 2{ }Z c ( ) +j Z zL ,( ) tan 2{ }

= Zc ( )[ ]2

Z zL

,( )

[ IV-14d ]

Matched if Zc ( ) = Z 4 z,( )Z zL ,( ) !!!

V. Parameters of a Coaxial Transmission Line:

We now look to Maxwell's Equations (in integral form) for values of the line parameters of

a coaxial line of inner radius a and outer radius b :

We first make use of the Gaussian law of electrostatics to obtain the capacitance of the line.

Assume a Gaussian surface which is an imaginary coaxial cylinder which has a radius r in

the range a, b[ ]and a length l so that

r

E n dAS

= 10 dVV

[ V-1 ]


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leads to

Er

2r l[ ] = dV

dr2r l[ ] =

Q

0[ V-2 ]

or

dV =1

2 0

Q

l

1

r

V=

1

2 0

Q

l

ln

b

a

[ V-3 ]

Therefore, the capacitance per unit line lengthis

cp = Q l( )

V=

2 0ln b a( )

[ V-4 ]

Obtaining the inductance of the line is a bit more complicated. We make use of Ampre's

law to find the magnetic field and then use Faraday's law to find the induced emf associated

with a time varying current. We apply the integral form of Ampre's to a circular loop of

radius r which is coaxial with the inner conductor so that

rB dl

L

= 0rJ n dA

S

[ V-5 ]

leads to

B 2r[ ] = 0I or B = 02

I

r

[ V-6 ]

We use this expression for the field to find the changing magnetic flux through a loop in the

median plane of the coaxial line.


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emf =r

E dlL

= tr

B n dAS

= t l B dra

b

= 02 l

dr

ra

b

It=

02 l ln b a( ){ }

It

[ V-7 ]

Therefore, the inductance per unit line lengthis

ls =02

ln b a( ) [ V-8 ]

Therefore, Maxwell's equations give us expressions for the all important transmission line

parameters of a coaxial line -- viz.

v =1

ls cp=

20 ln b a( )

ln b a( )

2 0

=

1

0 0= phase velocity [ V-9a ]

Zc = ls cp =0 ln b a( )

2

ln b a( )

2 0

=

ln b a( )

2

00

= characteristic impedance [ V-

9b ]

VI. Jones on Smith Charts:

Let us examine a very important property of the pair of equations [ IV-1a ] and [ IV-1b ].

Recall that

Z z,( ) = Zc ( )1 + V z,( )1 - V z ,( )

[ VI-1a ]

or


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z =1 + 1 -

where z =Z z,( ) Zc ( ) [ VI-1b ]

Writing this expression in terms of real and imaginary parts we see that

zr +j zi =1 + r +j i1 - r j i

= 1 + r +j i1 - r ji

1 - r +j i1 - r + ji

=1 - r

2 i2 +j 2i

1 - r( )2 + i( )

2

[ VI-2 ]

Equating real and imaginary components on either side of the equation

zr =1 - r

2 i2

1 - r( )2

+ i( )2 [ VI-3a ]

zi =2i

1 - r( )2

+ i( )2 [ VI-3b ]

we obtain

1 2r +r2 +i

2 =1

zr

1 - r2 i

2{ } [ VI-4a ]

1 2r +r2

+i2

=1

zi2i{ } [ VI-4b ]

or with further messaging

r2 zr +1

zr

2r + i2 zr +1

zr

=1 zrz r

[ VI-5a ]

r2 2r +1 +i

2 2izi

= 0 [ VI-5b ]


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Completing the square in both cases

r2

2zr

zr +1+

zr

zr +1

2

+i2 =

zr

zr + 1

2

+1 zrzr +1

=1

zr +1

2

[ VI-6a ]

r2 2r +1{ } + i

2 2izi

+1

zi

2

=1

zi

2

[ VI-6b ]

Therefore, the loci of constant zr and constant zi in the r , i[ ] plane are equations for

circles -- viz.


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CIRCLES OF CONSTANT "RESISTANCE" CIRCLES OF CONSTANT "REACTANCE"

r zr

z

r

+ 1

2

+ i2 =

1

z

r

+1

2

[ VI-7a ]

r 1{ }2

+ i 1

z

i

2

=1

z

i

2

[ VI-7b ]

radius =1

1 +z r; center=

zr

1 +zr,0

radius =

1

zi; center = 1,

1

zi

1r =

r

i

zr = 1z =r 0

z r 1

1r =

r

i

These isoresistance and isoreactance curves are the basis for the famous

and very useful Smith charts.1

1 P. H. Smith,Electronics12 , 29 (1939); 17 , 130 (1944)

transmission lines 1

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