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Chapter 2. Transmission Line Theory

Sept. 29th, 2008

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2.1 Transmission Lines

• A transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over the length of the line.

Lumped Element Model for a Transmission Line

• Transmission lines usually consist of 2 parallel conductors.

• A short segment Δz of transmission line can be modeled as a lumped-element circuit.

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Figure 2.1 Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit.

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• R = series resistance per unit length for both conductors

• L = series inductance per unit length for both conductors

• G = shunt conductance per unit length

• C = shunt capacitance per unit length

• Applying KVL and KCL,( , )

( , ) ( , ) ( , ) 0 (2.1 )i z t

v z t R zi z t L z v z z t at

( , )

( , ) ( , ) ( , ) 0 (2.1 )v z z t

i z t G zv z z t C z i z z t bt

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• Dividing (2.1) by Δz and Δz 0,

Time-domain form of the transmission line, or telegrapher, equation.

• For the sinusoidal steady-state condition with cosine-based phasors,

( , ) ( , )( , ) (2.2 )

v z t i z tRi z t L a

z t

( , ) ( , )

( , ) (2.2 )i z t v z t

Gv z t C bz t

( )( ) ( ) (2.3 )

dV zR j L I z a

dz

( )( ) ( ) (2.3 )

dI zG j C V z b

dz

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Wave Propagation on a Transmission Line

• By eliminating either I(z) or V(z):

where the complex propagation constant. (α = attenuation constant, β = phase constant)

• Traveling wave solutions to (2.4):

22

2

( )( ) (2.4 )

d V zV z a

dz

22

2

( )( ) (2.4 )

d I zI z b

dz

( )( )j R j L G j C

0 0 0 0( ) + , ( ) (2.6)z z z zV z V e V e I z I e I e

Wave propagation in +z directo

n

Wave propagation in -z direct

on

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• Applying (2.3a) to the voltage of (2.6),

• If a characteristic impedance, Z0, is defined as

• (2.6) can be rewritten

0 0( ) +z zI z V e V eR j L

0 , (2.7)R j L R j L

ZG j C

0 0

00 0

V VZ

I I

0 0

0 0

( ) (2.8)z zV VI z e e

Z Z

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• Converting the phasor voltage of (2.6) to the time domain:

• The wavelength of the traveling waves:

• The phase velocity of the wave is defined as the speed at which a constant phase point travels down the line,

+0 0( , ) cos( ) cos( ) (2.9)z zv z t V t z e V t z e

2= (2.10)

= = = (2.11)p

dzv f

dt

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Lossless Transmission Lines

• R = G = 0 gives or

• The general solutions for voltage and current on a lossless transmission line:

j j LC

, 0 (2.12)LC

0 = (2.13)L

ZC

0 0

00

0

( ) + ,

( ) (2.14)

j z j z

j z j z

V z V e V e

II z e I e

Z

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• The wavelength on the line:

• The phase velocity on the line:

2 2= = (2.15)

LC

1= = (2.16)pv

LC

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2.2 Field Analysis of Transmission Lines

• Transmission Line Parameters

Figure 2.2 (p. 53)Field lines on an arbitrary TEM transmission line.

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• The time-average stored magnetic energy for 1 m section of line:

• The circuit theory gives

• Similarly,

4m SW H H ds

2

0| | / 4mW L I

20| | S

L H H dsI

20, | | / 4

4e eSW E E ds W C V

20| | S

C E E dsV

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• Power loss per unit length due to the finite conductivity (from (1.130))

• Circuit theory (H || S)

• Time-average power dissipated per unit length in a lossy dielectric (from (1.92))

1 22s

c C C

RP H H dl

20| | / 2cP R I

1 22

0| |s

C C

RR H H dl

I

2d SP E E ds

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• Circuit theory

• Ex 2.1 Transmission line parameters of a coaxial line

• Table 2.1

20| | / 2dP G V

1 22

0| | C CG E E ds

V

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The Telegrapher Equations Derived form Field Analysis of a Coaxial Line• Eq. (2.3) can also be obtained from ME.

• A TEM wave on the coaxial line: Ez = Hz = 0.

• Due to the azimuthal symmetry, no φ-variation

ə/əφ = 0

• The fields inside the coaxial line will satisfy ME.

where

E j H

H j E

j

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1ˆ ˆˆ ˆˆ ( ) ( )

1ˆ ˆˆ ˆˆ ( ) ( )

E Ez E j H H

z z

H Hz H j E E

z z

Since the z-components must vanish,

( ) ( ),

f z g zE H

From the B.C., Eφ = 0 at ρ = a, b Eφ = 0 everywhere

0H

,E H

j H j Ez z

( )h zE

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( ) ( )( ), ( ),

h z g zj g z j h z

z z

The voltage between 2 conductors

( ) ( , ) ( ) ( ) lnb b

a a

d bV z E z d h z h z

a

The total current on the inner conductor at ρ = a2

0( ) ( , ) 2 ( )I z H a z ad g z

( ) ln / ( ) 2 ( )( ), ( )

2 ln /

V z b a I z V zj I z j j

z z b a

( ) ( )( ), ( ) ( )

V z I zj LI z G j C V z

z z

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Propagation Constant, Impedance, and Power Flow for the Lossless Coaxial Line• From Eq. (2.24)

• For lossless media,

• The wave impedance

• The characteristic impedance of the coaxial line

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20

EE

z

2 2

LC

/w

EZ

H

00

0

ln / ln / ln /

2 2 2

E b aV b a b aZ

I H

Ex 2.1

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• Power flow ( in the z direction) on the coaxial line may be computed from the Poynting vector as

• The flow of power in a transmission line takes place entirely via the E & H fields between the 2 conductors; power is not transmitted through the conductors themselves.

20 0

0 020

1 1 1

2 2 2 ln / 2

b

s a

V IP E H ds d d V I

b a

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2.3 The Terminated Lossless Transmission Lines

The total voltage and current on the line

0 0

0 0

0 0

( ) + ,

( ) (2.34)

j z j z

j z j z

V z V e V e

V VI z e e

Z Z

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• The total voltage and current at the load are related by the load impedance, so at z = 0

• The voltage reflection coefficient:

• The total voltage and current on the line:

0 00

0 0

(0)= =

(0)L

V VVZ Z

I V V

00 0

0

L

L

Z ZV V

Z Z

0 0

0 0

(2.35)L

L

V Z Z

V Z Z

0

0

0

( ) + ,

( ) (2.36)

j z j z

j z j z

V z V e e

VI z e e

Z

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• It is seen that the voltage and current on the line consist of a superposition of an incident and reflected wave. standing waves

• When Γ= 0 matched.

• For the time-average power flow along the line at the point z:

2

20 2 2

0

2

20

0

1 1Re ( ) ( ) Re 1

2 2

11

2

j z j zavg

VP V z I z e e

Z

V

Z

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• When the load is mismatched, not all of the available power from the generator is delivered to the load. This “loss” is return loss (RL):

RL = -20 log|Γ| dB

• If the load is matched to the line, Γ= 0 and |V(z)| = |V0

+| (constant) “flat”.

• When the load is mismatched, 2 2 ( 2 )

0 0 0( ) 1 1 1 (2.39)j z j l j lV z V e V e V e

max 0 min 01 , 1 (2.40)V V V V

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• A measure of the mismatch of a line, called the voltage standing wave ratio (VSWR)

(1< VSWR<∞)

• From (2.39), the distance between 2 successive voltage maxima (or minima) is l = 2π/2β = λ/2 (2βl = 2π), while the distance between a maximum and a minimum is l = π/2β = λ/4.

• From (2.34) with z = -l,

1

1SWR

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0

( ) (0) (2.42)j l

j lj l

V el e

V e

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• At a distance l = -z,

Transmission line impedance equation

20

0 0 20

0 00

0 0

00

0

00

0

( ) 1 (2.43)

( ) 1

( ) ( )

( ) ( )

cos sin

cos sin

tan

tan

j l j l j l

in j l j l j l

j l j lL L

j l j lL L

L

L

L

L

VV l e e eZ Z Z

I l V e e e

Z Z e Z Z eZ

Z Z e Z Z e

Z l jZ lZ

Z l jZ l

Z jZ lZ

Z jZ l

(2.44)

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Special Cases of Terminated Transmission Lines

• Short-circuited line

ZL = 0 Γ= -1

0 0

0 0

0 0

( ) 2 sin ,

( ) 2 cos

j z j z

j z j z

V z V e e jV z

V VI z e e z

Z Z

0 tan (2.45)inZ jZ l

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Figure 2.6 (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along a short-circuited transmission line.

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• Open-circuited line

ZL = ∞ Γ= 1

0 0

0 0

0 0

( ) 2 cos ,

2( ) sin (2.46)

j z j z

j z j z

V z V e e V z

V jVI z e e z

Z Z

0 cot inZ jZ l

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Figure 2.8 (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along an open-circuited transmission line.

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• Terminated transmission lines with special lengths.

• If l = λ/2, Zin = ZL.

• If the line is a quarter-wavelength long, or, l = λ/4+ nλ/2 (n = 1,2,3…), Zin = Z0

2/ZL. quarter-wave transformer

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Figure 2.9 (p. 63)Reflection and transmission at the junction of two transmission lines with different characteristic impedances.

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2.4 The Smith Chart

• A graphical aid that is very useful for solving transmission line problems.

Derivation of the Smith Chart

• Essentially a polar plot of the Γ(= |Γ|ejθ).

• This can be used to convert from Γto normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart.

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Figure 2.10 (p. 65)The Smith chart.

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• If a lossless line of Z0 is terminated with ZL, zL = ZL/Z0 (normalized load impedance),

• Let Γ= Γr +jΓi, and zL = rL + jxL.

1

1jL

L

ze

z

1

1

j

L j

ez

e

(1 )

(1 )r i

L Lr i

jr jx

j

2 2

2 2

1

(1 )r i

Lr i

r

2 2

2

(1 )i

Lr i

x

2 2

2

2 22

1,

1 1

1 11

Lr i

L L

r iL L

r

r r

x x

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• The Smith chart can also be used to graphically solve the transmission line impedance equation of (2.44).

• If we have plotted |Γ|ejθ at the load, Zin seen looking into a length l of transmission line terminates with zL can be found by rotating the point clockwise an amount of 2βl around the center of the chart.

2

0 2

1 (2.57)

1

j l

in j l

eZ Z

e

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• Smith chart has scales around its periphery calibrated in electrical lengths, toward and away from the “generator”.

• The scales over a range of 0 to 0.5 λ.

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Ex 2.2 ZL = 40+j70, l = 0.3λ, find Γl, Γin and Zin

Figure 2.11 (p. 67)Smith chart for Example 2.2.

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The Combined Impedance-Admittance Smith Chart

• Since a complete revolution around the Smith chart corresponds to a line length of λ/2, a λ/4 transformation is equivalent to rotating the chart by 180°.

• Imaging a give impedance (or admittance) point across the center of the chart to obtain the corresponding admittance (or impedance) point.

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Ex 2.3 ZL = 100+j50, YL, Yin ? when l = 0.15λ

Figure 2.12 (p. 69)ZY Smith chart with solution for Example 2.3.

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The Slotted Line

• A transmission line allowing the sampling of E field amplitude of a standing wave on a terminated line.

• With this device the SWR and the distance of the first voltage minimum from the load can be measured, from this data ZL can be determined.

• ZL is complex 2 distinct quantities must be measured.

• Replaced by vector network analyzer.

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Figure 2.13 (p. 70)An X-band waveguide slotted line.

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1| |

1

SWR

SWR

• Assume for a certain terminated line, we have measured the SWR on the line and lmin , the distance from the load to the first voltage minimum on the line.

• Minimum occurs when

• The phase of Γ =

• Load impedance

( 2 ) 1j le

min2 l

0

1

1LZ Z

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Ex 2.4

• With a short circuit load, voltage minima at z = 0.2, 2.2, 4.2 cm

• With unknown load, voltage minima at z = 0.72, 2.72, 4.72 cm

• λ = 4 cm,

• If the load is at 4.2 cm, lmin = 4.2 – 2.72 = 1.48 cm = 0.37 λ

?, ?, ?LZ

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Figure 2.14 (p. 71)Voltage standing wave patterns for Example 2.4. (a) Standing wave for short-circuit load. (b) Standing wave for unknown load.

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Figure 2.15 (p. 72)Smith chart for Example 2.4.

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2.5 The Quarterwave Transformer

Impedance Viewpoint

• For βl = (2π/λ)(λ/4) = π/2

• In order for Γ = 0, Zin = Z0

11

1

tan

tanL

inL

R jZ lZ Z

Z jR l

21

inL

ZZ

R

1 0 LZ Z R

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Figure 2.16 (p. 73)The quarter-wave matching transformer.

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Ex 2.5 Frequency Response of a Quarter-Wave Transformer• RL = 100, Z0 = 50

1 0 70.71LZ Z R

0

0

| | in

in

Z Z

Z Z

0 0

0

2

4 2 2

fl

f

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Figure 2.17 (p. 74)Reflection coefficient versus normalized frequency for the quarter-wave transformer of Example 2.5.

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The Multiple Reflection Viewpoint

Figure 2.18 (p. 75)Multiple reflection analysis of the quarter-wave transformer.

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1 0 0 1 11 2 1 3

1 0 0 1 1

, , L

L

Z Z Z Z R Z

Z Z Z Z R Z

011 2

1 0 1 0

22,

ZZT T

Z Z Z Z

2 2 2

1 1 2 3 1 2 3 1 2 3

1 1 2 3 2 30

1 2 3 1 1 2 3 1 2 31

2 3 2 3

( )

1 1

n

n

TT TT TT

TT

TT TT

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• Numerator2

1 1 2 3 1 2 3 1 3 1 1 2

1 0 1 1 1 01 3

1 0 1

21 0

1 0 1

( )

( )( ) ( )( )

( )( )

2( )

( )( )

L L

L

L

L

TT TT

Z Z R Z R Z Z Z

Z Z R Z

Z Z R

Z Z R Z

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2.6 Generator and Load Mismatches

• Because both the generator and load are mismatched, multiple reflections can occur on the line.

• In the steady state, the net result is a single wave traveling toward the load, and a single reflected wave traveling toward the generator.

• In Fig. 2.19, where z = -l,2

00 02

0

1 tan (2.67)

1 tan

j ll l

in j ll l

e Z jZ lZ Z Z

e Z jZ l

0

0

(2.68)ll

l

Z Z

Z Z

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Figure 2.19 (p. 77)Transmission line circuit for mismatched load and generator.

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• The voltage on the line:

• Power delivered to the load:

0

0

( ) ( )

1 (2.70)

j l j ling l

in g

ing j l j l

in g l

ZV l V V e e

Z Z

ZV V

Z Z e e

0

0 20

(2.71)1

j l

g j lg l g

Z eV V

Z Z e

By (2.67) &

0

0

gg

g

Z Z

Z Z

2

2 2

22 2

1 1 1 1 1Re Re | | Re | |

2 2 2

1| | (2.39)

2 ( ) ( )

inl in in in g

in in g in

ing

in g in g

ZP V I V V

Z Z Z Z

RV

R R X X

1

1l

l

SWR

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• Case 1: the load is matched to the line, Zl = Z0, Γl = 0, SWR = 1, Zin = Z0,

• Case 2: the generator is matched to the input impedance of a mismatched line, Zin = Zg

• If Zg is fixed, to maximize Pl,

2 02 2

0

1| | (2.40)

2 ( )l gg g

ZP V

Z R X

2

2 2

1| | (2.41)

2 4g

l g

g g

RP V

R X

22 2 2 2

2 ( )10 0

( ) ( ) ( ) ( )

in in gl

in in g in g in g in g

R R RP

R R R X X R R X X

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or

or

• Therefore, Rin = Rg and Xin = -Xg, or Zin = Zg*

• Under these conditions

• Finally, note that neither matching for zero reflection (Zl = Z0), nor conjugate matching (Zin = Zg

*), necessary yields a system with the best efficiency.

2 2 2( ) 0g in in gR R X X

22 2

2 ( )0 0

( ) ( )

in in gl

in in g in g

X X XP

X R R X X

( ) 0in in gX X X

21 1| | (2.44)

2 4l gg

P VR

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2.7 Lossy Transmission Lines