T RANSMISSION L INE T HEORY I. The Transmission Line Modelpeople.seas.harvard.edu/~jones/es151/lectures/transmission_lines_1.… · T RANSMISSION L INE T HEORY P AGE 2 R. V. Jones,

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+1 t( ) = vn t( ) − R s in +1 t( ) − Ls

ddt

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

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

ddt

vn 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( ) = Vn ω( ) exp j ω t[ ] dω−∞

+∞

∫ and in t( ) = In ω( ) exp j ω t[ ] dω−∞

+∞

∫ [ I-2]

or in the language of circuit analysis

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

in t( ) = ℜ In ω( ) exp j ω t[ ]{ } = In ω( ) cos ω t + ϕ I( ) [ I-3b ]

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

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

where Zs ω( ) = R s + j ωL s and Yp ω( ) = Gp + j ωCp are, respectively, the series

impedance and the shunt (parallel) 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

inhomogeneous equations in the form of a set of coupled, homogeneous algebraic

equations -- viz.

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

Zs ω( ) Yp ω( ) In ω( ) = In +1 ω( ) + In −1 ω( ) − 2 In ω( ) [ 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 ω( ) = a complex constant{ } exp j n φ ω( )[ ] [ II-2a ]

In ω( ) = another complex constant{ } exp j n φ ω( )[ ] [ 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 φ ω( ) must satisfy the equation

Zs ω( ) Yp ω( ) exp j nφ ω( )[ ] = exp j n +1( )φ ω( )[ ] + exp j n −1( )φ ω( )[ ]− 2 exp j n φ ω( )[ ] [ II-3 ]



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

Zs ω( ) Yp ω( ) = exp j φ ω( )[ ] + exp − j φ ω( )[ ] − 2

= exp j φ ω( ) 2[ ]− exp − j φ ω( ) 2[ ]{ }2= 2 j sin φ ω( ) 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.

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

Important Special Cases:

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

If we take Zs ω( ) = j ωL s and Yp ω( ) = j ωCp , then Eq [ II-5 ] becomes

sin φ ω( ) 2( ) = ω 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.).



2. The "Lossy" RC-Transmission Line:

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

sin φ ω( ) 2( ) = 2 j[ ]−1j ω 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



sinφ2

= sin

′ φ + j ′ ′ φ 2

= sin

′ φ 2

cos

j ′ ′ φ 2

+ cos

′ φ 2

sin

j ′ ′ φ 2

= ±1− j{ }

2

ω R sCp

2

[ II-9 ]

which yields, upon equating real and imaginary parts,

sin′ φ

2

cosh

′ ′ φ 2

= ±

12

ω R sCp

2[ II-10a ]

cos

′ φ 2

sinh

′ ′ φ 2

= m

12

ω R sCp

2 . [ II-10b ]

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

′ φ ≈ ±ω R sCp

2 ← Phase shift per section [ II-11a ]

′ ′ φ = m

ω R sCp

2 ← Attenuation per section [ II-11b ]



III. Continuous Transmission Lines

An Approximate Solution of Transmission Line Equations:

In most instances, we are interested in continuous rather than discrete transmission lines.

To obtain a representation of the voltage across and current along a continuous line, we

develop a "continuous approximation" of the basic circuit equations by making use of a

Taylor expansion for small node spatial separation. Before looking at the most general

case, it is useful to first look at the lossless or ideal case. From Eqs. [ I-1 ] we may write

the basic circuit equations

vn+1 t( ) = vn t( ) − Ls

ddt

in+1 t( ) [ III-1a ]

in +1 t( ) = in t( ) − Cp

ddt

vn t( ) [ III-1b ]

Using the Taylor expansion for small node spatial separation, ∆z, we obtain

vn+1 t( ) ≈ vn t( ) + zn+1 − zn( ) ∂

∂zvn t( ) = vn t( ) + ∆z

∂∂z

vn t( ) [ III-2a ]

in +1 t( ) ≈ in t( ) + zn+1 − zn( ) ∂∂z

in t( ) = in t( ) + ∆z∂∂z

in t( ) [ III-2b ]

Therefore Eqs [ III-1a ] and [ III-1b ] become

vn t( ) + ∆z

∂∂z

vn t( ) ≈ vn t( ) − L s

ddt

in t( ) + ∆z∂∂z

in t( )

[ III-3a ]

in t( ) + ∆z

∂∂z

in t( ) ≈ in t( ) − Cp

ddt

vn t( ) [ III-3b ]



To first-order in ∆z, we obtain the famous inhomogeneous "telegrapher equations" for

a lossless transmission line -- viz.

∂∂z

v z,t( ) ≈ −ls

∂∂t

i z,t( ) where ls = lim∆z→ 0

L s

∆z

[ III-4a ]

∂∂z

i z,t( ) ≈ −cp

∂∂t

v z,t( ) where cp = lim∆z→ 0

Cp

∆z

[ III-4b ]

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

∂ 2

∂z2 v z,t( ) = −ls

∂∂z

∂∂t

i z,t( ) = −ls

∂∂t

∂∂z

i z,t( ) = lscp

∂ 2

∂t2 v z,t( )

or

∂ 2

∂z2 v z,t( ) = lscp

∂ 2

∂t2 v z,t( ) [ III-5a ]

and∂ 2

∂z2 i z,t( ) = lscp

∂2

∂t2 i z,t( ) [ III-5b ]

The truly remarkable point is that any old function of the form v z,t( ) = f z − vt( ) and/or of

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



Wave equation(s):

∂ 2


∂ 2

∂t2 v z,t( ) → ′ ′ f z − vt( ) = lscp −v( )2 ′ ′ f z − v t( ) [ III-6a ]

∂ 2


∂ 2

∂t2 v z,t( ) → ′ ′ g z + v t( ) = lscp +v( )2 ′ ′ g z + vt( ) [ III-6b ]

Telegrapher equations:

∂∂z

i z,t( ) = −cp

∂∂t

v z,t( ) = −cp −v( ) ′ f z − v t( ) → i+ z,t( ) = cp v f z − vt( ) [ III-7a ]

∂∂z

i z,t( ) = −cp

∂∂t

v z,t( ) = −cp +v( ) ′ g z + v t( ) → i− z,t( ) = −cp v g z + v t( ) [ III-7b ]

so that

v =1

lscp

is the wave velocity [ III-8a ]



zc =1

cp v=

ls

cp

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 ω( ) + Zs ω( ) In ω( ) = Vn−1 ω( ) ≈ Vn ω( ) − ∆z∂∂z

Vn ω( ) [ III-9a ]

In ω( ) − Yp ω( ) Vn ω( ) = In +1 ω( ) ≈ In ω( ) + ∆z∂∂z

In ω( ) [ III-9b ]

Once again, to first-order in ∆z , these gyrations lead to a more general version of the

inhomogeneous Telegrapher equations -- viz.

∂∂z

V z,ω( ) ≈ −zs I z,ω( ) where zs = lim∆ z→0

Zs

∆z

[ III-10a ]

∂∂z

I z,ω( ) ≈ −yp V z,ω( ) where yp = lim∆z→ 0

Yp

∆z

[ III-10b ]

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

∂ 2

∂z2 V z,ω( ) = zsyp V z,ω( ) [ III-11a ]

∂ 2

∂z2 I z,ω( ) = zsyp I z,ω( ) [ III-11b ]

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

solutions in the form



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

so that

γ ω( ) V z,ω( ) = −zs I z,ω( ) and γ ω( ) I z,ω( ) = −yp V z,ω( ) [ III-13 ]

and

γ 2 ω( ) V z,ω( ) = zsyp V z,ω( ) and γ 2 ω( ) I z,ω( ) = zsyp I z,ω( ) [ III-14 ]

where γ ω( ) is, in general, complex -- i.e. γ ω( ) = α ω( ) + jβ ω( ) . Therefore, if the

proposed solution is to valid we must have

γ 2 ω( ) = α ω( ) + j β ω( )[ ]2= zs ω( ) yp ω( ) = rs + j ωls[ ] gp + j ω cp[ ] [ III-15 ]

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

α 2 ω( ) − β2 ω( ) = rs gp − ω2 ls cp [ III-16a ]

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

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

β ω( ) ≈ ω ls cp 1−rs

ω ls

gp

ω cp

[ III-17a ]

and the attenuation per unit length by

α ω( ) ≈12

rs

cp

ls

+ gp

ls

cp

[ 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



γ ω( ) = α ω( ) + j β ω( ) = 0 ± j ω ls cp [ III-18 ]

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

of two independent frequency domain solutions may be written

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

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

v z,t( ) = ℜ V+ ω( ) exp −γ ω( ) z + j ω t[ ] +V− ω( ) exp +γ ω( ) z + j ω 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

v z,t( ) = ℜ V+ ω( ) exp j ω t −β ω( )z[ ]{ } +V− ω( ) exp j ω t +β ω( )z[ ]{ }( )= V+ ω( ) cos ω t −β ω( )z + φ+[ ] + V− ω( ) cos ω t + β ω( )z + φ−[ ]

[ III-21 ]

As in earlier discussions, we interpret

V+ ω( ) cos ω t −β ω( )z + φ+[ ]as a continuous wave propagating to the right (positive z-direction) and

V− ω( ) cos ω t + β ω( )z + φ−[ ]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,ω( ) = −1

zs ω( )∂∂ z

V z,ω( ) [ III-22a ]

so that



I z,ω( ) = I+ ω( ) exp −γ ω( ) z[ ] + I− ω( ) exp +γ ω( ) z[ ]=

1

Zc ω( ) V+ ω( ) exp −γ ω( )z[ ] -V− ω( ) exp +γ ω( )z[ ]{ } [ III-22b ]

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

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

transmission line, we see that

Zc ω( ) = 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,ω( ) ≡V z,ω( )I z,ω( )

= Zc ω( ) V+ ω( ) exp −γ ω( )z[ ] +V− ω( ) exp +γ ω( ) z[ ]V+ ω( ) exp −γ ω( ) z[ ] -V− ω( ) exp +γ ω( ) z[ ]

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

= Zc ω( ) 1 +ΓV z,ω( )

1 - ΓV z,ω( )

[ III-25 ]

where ΓV z,ω( ) ≡ V− ω( ) V+ ω( )[ ] exp +2γ ω( ) z[ ] [ III-26 ]

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

solution in the compact form

V z,ω( ) = V+ ω( ) exp −γ ω( )z[ ] 1 +ΓV z,ω( ){ } [ III-27a ]

I z,ω( ) Zc ω( ) = V+ ω( ) exp −γ ω( )z[ ] 1 - ΓV z,ω( ){ } [ III-27b ]



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,ω( ) = Zc ω( ) 1 + ΓV z,ω( )1− ΓV z,ω( )

[ IV-1a ]

ΓV z,ω( ) =Z z,ω( ) − Zc ω( )Z z,ω( ) + Zc ω( ) [ 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 ,ω( ) = 0 so that ΓV zL ,ω( ) = −1.



At other points along the line

ΓV z,ω( ) = V− ω( ) V+ ω( )[ ] exp +2γ ω( ) z[ ] = ΓV zL ,ω( ) exp +2γ ω( ) z − zL[ ]{ } [ IV-2 ]

so that in this case

ΓV z,ω( ) = −exp −2γ ω( ) zL − z[ ]{ } [ IV-3a ]

and

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

[ IV-3b ]

When the attenuation is zero

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

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

Z z,ω( ) =V z,ω( )I z,ω( ) = j Zc ω( ) tan β ω( ) 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

β ω( ) zL − z[ ] = 2π zL − z[ ] λ ω( ) = π integer[ ] .



VOLTAGE ACROSS A SHORTED TRANSMISSION LINE

2. An "open" transmission line -- i. e. Z zL ,ω( ) = ∞ so that ΓV zL ,ω( ) = +1.

ΓV z,ω( ) = +exp −2γ ω( ) zL − z[ ]{ } [ IV-5a ]

and



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

[ IV-5b ]


V z,ω( ) = 2 V+ ω( ) exp − j β ω( ) zL[ ] cos β ω( ) zL − z[ ]{ } [ IV-6a ]

Zc ω( ) I z,ω( ) = − j 2 V+ ω( ) exp − j β ω( ) zL[ ] sin β ω( ) zL − z[ ]{ } [ IV-6b ]

Z z,ω( ) =V z,ω( )I z,ω( ) = j Zc ω( ) cot β ω( ) zL − z[ ]{ } [ IV-6c ]

Again as above, the solution is a "pure" standing wave. However, the

"open" line is the dual of the "shorted" line in the sense that the role of current

and voltage are reversed.

3. A "matched" transmission line -- i. e. Z zL ,ω( ) = Zc ω( ) so that ΓV zL ,ω( ) = 0.

ΓV z,ω( ) = 0 [ IV-7a ]

and V z,ω( ) = V+ ω( ) exp −γ ω( )z[ ] [ IV-7b ]


V z,ω( ) = V+ ω( ) exp − jβ ω( ) z[ ] [ IV-8a ]

I z,ω( ) = Zc ω( )V+ ω( ) exp − jβ ω( )z[ ] [ IV-8b ]

Z z,ω( ) =V z,ω( )I z,ω( ) = Zc ω( ) [ IV-8c ]



Such a solution is called a "pure" running wave. In such cases, the average

value (or the amplitude) of the current and voltage are spatially uniform.

VOLTAGE ACROSS A MATCHED TRANSMISSION LINE

4. The general case -- i. e. Z zL ,ω( ) ≠ any special value .

In these cases the solutions are combinations of standing and running

waves! For nonattenuating lines the, so called, voltage-standing-wave-

ratio or VSWR is an important measure of the character of the solution. From

Eq. [ III-27a ] we see that

V z,ω( ) = V+ ω( ) 1 +ΓV z,ω( ) [ IV-9a ]

where

ΓV z,ω( ) ≡ V− ω( ) V+ ω( )[ ] exp +2 j β ω( ) z[ ] = ΓV ω( ) exp j φΓVz,ω( ) z[ ] [ IV-9b ]

Therefore, we have the definition



VSWR ≡V z,ω( )

Max

V z,ω( )Min

=1 + ΓV ω( )1 - ΓV ω( ) [ 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,ω( ) = Zc ω( ) 1 +ΓV z,ω( )1 - ΓV z,ω( )

= Zc ω( )1 +ΓV zL ,ω( ) exp −2γ ω( ) zL − z[ ]{ }1 - ΓV zL ,ω( ) exp −2γ ω( ) zL − z[ ]{ }

= Zc ω( ) exp +γ ω( ) zL − z[ ]{ } +ΓV zL ,ω( ) exp −γ ω( ) zL − z[ ]{ }exp +γ ω( ) zL − z[ ]{ } - ΓV zL ,ω( ) exp −γ ω( ) zL − z[ ]{ }

[ IV-11 ]

But ΓV zL ,ω( ) =Z zL ,ω( ) − Zc ω( )Z zL ,ω( ) + Zc ω( )

so that

Z z,ω( ) = Zc ω( )exp +γ ω( ) zL − z[ ]{ } +

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

exp +γ ω( ) zL − z[ ]{ } -Z zL ,ω( ) − Zc ω( )Z zL ,ω( ) + Zc ω( ) exp −γ ω( ) zL − z[ ]{ }

= Zc ω( )Z zL ,ω( ) exp +γ ω( ) zL − z[ ]{ } + exp −γ ω( ) zL − z[ ]{ }[ ] + Zc ω( ) exp +γ ω( ) zL − z[ ]{ } − exp −γ ω( ) zL − z[ ]{ }[ ] Z zL ,ω( ) exp +γ ω( ) zL − z[ ]{ }− exp −γ ω( ) zL − z[ ]{ }[ ] +Z c ω( ) exp +γ ω( ) zL − z[ ]{ } + exp −γ ω( ) zL − z[ ]{ }[ ]

[ IV-12a ]



or more compactly

Z z,ω( ) = Zc ω( )Z zL ,ω( ) + Zc ω( ) tanh γ ω( ) zL − z[ ]{ }

Zc ω( ) + Z zL ,ω( ) tanh γ ω( ) zL − z[ ]{ }

[ IV-12b]

For non attenuating lines, this expression reduces to the famous impedance

transformation formula -- viz.

Z z,ω( ) = Zc ω( )Z zL ,ω( ) + j Zc ω( ) tan β ω( ) zL − z[ ]{ }

Zc ω( ) + j Z zL ,ω( ) tan β ω( ) zL − z[ ]{ }

[ IV-13 ]

Applications of the famous impedance transformation formula:

a. "Shorted" transmission line:

Zshort z ,ω( ) = Z c ω( ) 0( ) + j Z c ω( ) tan β ω( ) zL − z[ ]{ }

Zc ω( ) + j 0( ) tan β ω( ) zL − z[ ]{ }

= j Zc ω( ) tan β ω( ) zL − z[ ]{ }[ IV-14a ]

b. "Open" transmission line:

Zopen z,ω( ) = Z c ω( ) ∞( ) + j Z c ω( ) tan β ω( ) zL − z[ ]{ }

Zc ω( ) + j ∞( ) tan β ω( ) zL − z[ ]{ }

= − j Z c ω( ) cot β ω( ) zL − z[ ]{ }[ IV-14b ]

c. "Matched" transmission line:



Zmatch z ,ω( ) = Zc ω( )Z c ω( ) + j Zc ω( ) tan β ω( ) zL − z[ ]{ }

Z c ω( ) + j Z c ω( ) tan β ω( ) zL − z[ ]{ }

= Z c ω( )[ IV-14c ]

d. Quarter wavelength matching transformer:

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

Z c ω( ) + j Z zL ,ω( ) tan π 2{ }

= Z c ω( )[ ]2Z zL ,ω( )

[ IV-14d ]

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

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 δ l so that

r E ⋅ ˆ n dA

S

∫∫ =1

ε0

ρ dVV

∫∫∫ [ V-1 ]



leads to

Er 2π r δ l[ ] = −

dV

dr2π r δ l[ ] =

Q

ε0

[ V-2 ]

or

dV =−

1

2πε 0

Q

δ l

1

r

→ V =

1

2πε 0

Q

δ l

ln

b

a

[ V-3 ]

Therefore, the capacitance per unit line length is

cp =

Q δ l( )V

=2πε 0

ln b a( ) [ V-4 ]

Obtaining the inductance of the line is a bit more complicated. We make use of Ampère'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 Ampère's to a circular loop of

radius r which is coaxial with the inner conductor so that

r B ⋅d ˆ l

L

∫ = µ0

r J ⋅ ˆ n dA

S

∫∫ [ V-5 ]

leads to

Bθ 2π r[ ] = µ0 I or Bθ =µ 0

2πI

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.



emf =r E ⋅ d ˆ l

L

∫ = −∂∂t

r B ⋅ ˆ n dA

S

∫∫ = −∂∂t

δ l Bθ dra

b

∫

= − µ0

2πδ l dr

ra

b

∫

∂I

∂t= − µ0

2πδ l ln b a( ){ }∂I

∂t

[ V-7 ]

Therefore, the inductance per unit line length is

ls =µ0

2πln 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

=2π

µ0 ln b a( )

ln b a( )2πε 0

=1

ε0 µ0

= phase velocity [ V-9a ]

Zc = ls cp =µ0 ln b a( )

2π

ln b a( )2πε 0

=ln b a( )

2π

µ 0

ε0

= 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,ω( ) = Zc ω( ) 1 + ΓV z,ω( )1 - ΓV z,ω( )

[ VI-1a ]

or



z =

1 + Γ1 - Γ

where z = Z z,ω( ) Zc ω( ) [ VI-1b ]

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

zr + j zi = 1 + Γr + j Γi

1 - Γr − j Γi

= 1 + Γr + j Γi

1 - Γr − jΓi

1 - Γr + j Γi

1 - Γr + jΓi

=1 - Γr

2 −Γ i2 + j 2Γi

1 - Γr( )2 + Γi( )2

[ VI-2 ]

Equating real and imaginary components on either side of the equation

zr =

1 - Γr2 −Γ i

2

1 - Γr( )2+ Γi( )2 [ VI-3a ]

zi =

2Γi

1 - Γr( )2+ Γi( )2 [ VI-3b ]

we obtain

1 − 2Γr +Γ r

2 +Γi2 =

1

zr

1 - Γr2 −Γi

2{ } [ VI-4a ]

1 − 2Γr +Γ r

2 +Γi2 =

1

zi

2Γi{ } [ VI-4b ]

or with further messaging

Γr

2 zr +1

zr

− 2Γr + Γi2 zr +1

zr

=1− zr

z r

[ VI-5a ]

Γr

2 − 2Γr +1 +Γi2 −

2Γi

zi

= 0 [ VI-5b ]



Completing the square in both cases

Γr

2 −2zr

zr +1+

zr

zr +1

2

+Γi2 =

zr

zr + 1

2

+1 − zr

zr +1=

1

zr +1

2

[ VI-6a ]

Γr

2 − 2Γr +1{ } + Γi2 −

2Γi

zi

+1

zi

2

=1

zi

2

[ VI-6b ]

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

circles -- viz.



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

Γr −

zr

zr + 1

2

+ Γi2 =

1

zr +1

2

[ VI-7a ] Γr −1{ }2

+ Γi −1

zi

2

=1

zi

2

[ VI-7b ]

radius =1

1 + z r

; center=zr

1 + zr

,0

radius =

1

zi

; center = 1,1

zi

1Γr =

Γr

Γi

zr = 1z =r 0

z <r 1

z >r 1

1Γr =

Γr

Γi

These isoresistance and isoreactance curves are the basis for the famous

and very useful Smith charts.1

1 P. H. Smith, Electronics 12 , 29 (1939); 17 , 130 (1944)

Documents

T RANSMISSION L INE T HEORY I. The Transmission Line Modelpeople.seas.harvard.edu/~jones/es151/lectures/transmission_lines_1.… · T RANSMISSION L INE T HEORY P AGE 2 R. V. Jones,