ELECTROMAGNETIC THEORY EKT 241/4: ELECTROMAGNETIC THEORY PREPARED BY: NORDIANA MOHAMAD SAAID...

EKT 241/4: ELECTROMAGNETIC ELECTROMAGNETIC THEORYTHEORY

PREPARED BY: NORDIANA MOHAMAD [email protected]

CHAPTER 5 – TRANSMISSION LINES

UN

IVE

RS

ITI M

AL

AY

SIA

PE

RL

ISU

NIV

ER

SIT

I MA

LA

YS

IA P

ER

LIS

Chapter Outline

General ConsiderationsLumped-Element ModelTransmission-Line EquationsWave Propagation on a Transmission LineThe Lossless Transmission Line Input Impedance of the Lossless LineSpecial Cases of the Lossless LinePower Flow on a Lossless Transmission LineThe Smith Chart Impedance MatchingTransients on Transmission Lines

General Considerations

• Transmission line – a two-port network connecting a generator circuit to a load.

The role of wavelength

• The impact of a transmission line on the current and voltage in the circuit depends on the length of line, l and the frequency, f of the signal provided by generator.

• At low frequency, the impact is negligible• At high frequency, the impact is very significant

Propagation modes

• Transmission lines may be classified into two types:a) Transverse electromagnetic (TEM) transmission lines – waves propagating along these lines having electric and magnetic field that are entirely transverse to the direction of propagation b) Higher order transmission lines – waves propagating along these lines have at least one significant field component in the direction of propagation

Propagation modes

Lumped- element model


• A transmission line is represented by a parallel-wire configuration regardless of the specific shape of the line, i.e coaxial line, two-wire line or any TEM line.

• Lumped element circuit model consists of four basic elements called ‘the transmission line parameters’ : R’ , L’ , G’ , C’ .


• Lumped-element transmission line parameters:– R’ : combined resistance of both conductors

per unit length, in Ω/m– L’ : the combined inductance of both

conductors per unit length, in H/m– G’ : the conductance of the insulation medium

per unit length, in S/m– C’ : the capacitance of the two conductors per

unit length, in F/m


Note: µ, σ, ε pertain to the insulating material between conductors


• All TEM transmission lines share the following relation:

''CL

'

'

C

G

Note: µ, σ, ε pertain to the insulating material between conductors

Transmission line equations

• Complex propagation constant, γ

• α – the real part of γ - attenuation constant, unit: Np/m

• β – the imaginary part of γ - phase constant, unit: rad/m

j

CjG'LjR'

''

Transmission line equations

• The characteristic impedance of the line, Z0 :

• Phase velocity of propagating waves:

where f = frequency (Hz) λ = wavelength (m)

''

''0 CjG

LjRZ

fu p

Example 1

An air line is a transmission line for which air is the dielectric material present between the two conductors, which renders G’ = 0.

In addition, the conductors are made of a material with high conductivity so that R’ ≈0.

For an air line with characteristic impedance of 50Ω and phase constant of 20 rad/m at 700MHz, find the inductance per meter and the capacitance per meter of the line.

Solution to Example 1

• The following quantities are given:

• With R’ = G’ = 0,

• The ratio is given by

• We get L’ from Z0

Hz 107 MHz 700 rad/m, 20 ,50 80 fZ

'

'

'

' and ''''Im 0 C

L

Cj

LjZCLCjLj

pF/m 9.90501072

20'

80

Z

C

nH/m 227109.9050''' 1220 LCLZ

Lossless transmission line

• Lossless transmission line - Very small values of R’ and G’.

• We set R’=0 and G’=0, hence:

line) (lossless ''

line) (lossless 0

CL

line) (lossless '

'

0,G' and 0R' since

''

''

0

0

C

LZ

CjG

LjRZ

Lossless transmission line

• Using the relation properties between μ, σ, ε :

• Wavelength, λ

Where εr = relative permittivity of the insulating material between conductors

(m/s) 1

(rad/m)

pu

rr

p

f

c

f

u

01

Voltage reflection coefficient

• The load impedance, ZL

Where;

= total voltage at the load

V0- = amplitude of reflected voltage wave

V0+ = amplitude of the incident voltage wave

= total current at the load

Z0 = characteristic impedance of the line

00

~VVVL

LV~

0

0

0

0~

Z

V

Z

VI L

LI~

L

LL

I

VZ ~

~


• Hence, load impedance, ZL:

• Solving in terms of V0- :

0

00

00 ZVV

VVZ L

00

00 V

ZZ

ZZV

L

L


• Voltage reflection coefficient, Γ – the ratio of the amplitude of the reflected voltage wave, V0

- to the amplitude of the incident voltage wave, V0

+ at the load.

• Hence,

less)(dimension 1

1

0

0

0

0

0

0

ZZ

ZZ

ZZ

ZZ

V

V

L

L

L

L


• Z0 for lossless line is a real number while ZL in general is a complex number. Hence,

Where |Γ| = magnitude of Γ

θr = phase angle of Γ

• A load is matched to the line if ZL = Z0 because there will be no reflection by the load (Γ = 0 and V0

−= 0.

rje


• When the load is an open circuit, (ZL=∞), Γ = 1 and V0

- = V0+.

• When the load is a short circuit (ZL=0), Γ = -1 and V0

- = V0+.

Example 2

• A 100-Ω transmission line is connected to a load consisting of a 50-Ω resistor in series with a 10pF capacitor. Find the reflection coefficient at the load for a 100-MHz signal.


• The following quantities are given

• The load impedance is

• Voltage reflection coefficient is

Hz10MHz100 ,100 F,10 ,50 80

11LL fZCR

1595010102

150

/

118

LLL

jj

CjRZ

7.6076.0159.15.0

159.15.0

1/

1/

0L

0L

j

j

ZZ

ZZ

Standing Waves

• Interference of the reflected wave and the incident wave along a transmission line creates a standing wave.

• Constructive interference gives maximum value for standing wave pattern, while destructive interference gives minimum value.

• The repetition period is λ for incident and reflected wave individually.

• But, the repetition period for standing wave pattern is λ/2.

Standing Waves

• For a matched line, ZL = Z0, Γ = 0 and

= |V0+| for all values of z. zV

~

Standing Waves

• For a short-circuited load, (ZL=0), Γ = -1.

Standing Waves

• For an open-circuited load, (ZL=∞), Γ = 1.

The wave is shifted by λ/4 from short-circuit case.

Standing Waves

• First voltage maximum occurs at:

• First voltage minimum occurs at:

• For next voltage maximum:

0 where24

rmax n

nl

4/ if 4/

4/ if 4/

maxmax

maxmaxmin

ll

lll

0 if ...... 2, 1, 0, n

0 if ...... 3, 2, 1, n

0 where24

r

r

rmax

nn

l

Where θr = phase angle of Γ

Voltage standing wave ratio

• VSWR is the ratio of the maximum voltage amplitude to the minimum voltage amplitude:

• VSWR provides a measure of mismatch between the load and the transmission line.

• For a matched load with Γ = 0, VSWR = 1 and for a line with |Γ| - 1, VSWR = ∞.

less)(dimension 1

1~

~

min

max

V

VVSWR

Example 3

A 50- transmission line is terminated in a load with ZL = (100 + j50)Ω . Find the voltage reflection coefficient and the voltage standing-wave ratio (VSWR).


• We have,

• VSWR is given by:

6.26

0L

0L 45.05050100

5050100

1/

1/ jej

j

ZZ

ZZ

6.245.01

45.01

1

1

VSWR

Input impedance of a lossless line

• The input impedance, Zin is the ratio of the total voltage (incident and reflected voltages) to the total current at any point z on the line.

• or

ljZZ

ljZZZ

ljZlZ

ljZlZZlZ

tan

tan

sincos

sincos

L0

0L0

L0

0L0in

zj

zj

in

e

eZ

zI

zVzZ

2

2

0 1

1

)(~

)(~

)(

Special cases of the lossless line

• For a line terminated in a short-circuit, ZL = 0:

• For a line terminated in an open circuit, ZL = ∞:

ljZlI

lVZ tan~

~

0

sc

scscin

ljZ

lI

lVZ cot~ 0

oc

ococin

Application of short-circuit and open-circuit measurements

• The measurements of short-circuit input impedance, and open-circuit input impedance, can be used to measure the characteristic impedance of the line:

• and

scinZ

ocinZ

ocin

scin ZZZo

ocin

scintan

Z

Zl

Length of line

• If the transmission line has length , where n is an integer,

• Hence, the input impedance becomes:

2/nl

0tan

2//2tantan

n

nl

2/for ZZ Lin nl

Quarter wave transformer

• If the transmission line is a quarter wavelength,

with ,

where , we have

, then the input impedance becomes:

2/4/ nl integer positiveany or 0n

24

2

l

2/4/for Z

ZZ

L

20

in nl

Example 4

A 50-Ω lossless transmission line is to be matched to a resistive load impedance with ZL=100Ω via a quarter-wave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer.


• To eliminate reflections at terminal AA’, the input impedance Zin looking into the quarter-wave line should be equal to Z01 (the characteristic impedance of the feedline). Thus, Zin = 50Ω .

• Since the lines are lossless, all the incident power will end up getting transferred into the load ZL.

7.701005002

L

202

in

Z

Z

ZZ

Matched transmission line

• For a matched lossless transmission line, ZL=Z0:

1) The input impedance Zin=Z0 for all locations z on the line,

2) Γ =0, and

3) all the incident power is delivered to the load, regardless of the length of the line, l.

Power flow on a lossless transmission line

• Two ways to determine the average power of an incident wave and the reflected wave;– Time-domain approach– Phasor domain approach

• Average power for incident wave;

• Average power for reflected wave:

(W) 2 0

2

0iav Z

VP

iav

2

0

2

02rav 2

PZ

VP

Power flow on a lossless transmission line

• The net average power delivered to the load:

(W) 12

2

0

2

0rav

iavav

Z

VPPP

Smith Chart

• Smith chart is used to analyze & design transmission line circuits.

• Impedances on Smith chart are represented by normalized value, zL for example:

• the normalized load impedance, zL is dimensionless.

0Z

Zz L

L

Smith Chart

• Reflection coefficient, Γ :

• Since , Γ becomes:

• Re-arrange in terms of zL:

• Normalized load admittance:

1/

1/

0

0

ZZ

ZZ

L

L

0Z

Zz L

L 1

1

L

L

z

z

LL jxrz

1

1L

1

11

LL z

y

Smith Chart

• The complex Γ plane.

Smith Chart

• The families of circle for rL and xL.

Smith Chart

• Plotting normalized impedance, zL = 2-j1

Input impedance

• The input impedance, Zin:

• Γ is the voltage reflection coefficient at the load.

• We shift the phase angle of Γ by 2βl, to get ΓL. This will zL to zin. The |Γ| is the same, but the phase is changed by 2βl.

• On the Smith chart, this means rotating in a clockwise direction (WTG).

lj

lj

in e

eZZ

2

2

0 1

1

Input impedance

• Since β = 2π/λ, shifting by 2 βl is equal to phase change of 2π.

• Equating:

• Hence, for one complete rotation corresponds to l = λ/2.

• The objective of shifting Γ to ΓL is to find Zin at an any distance l on the transmission line.

2

222 ll

Example 5

• A 50-Ω transmission line is terminated with ZL=(100-j50)Ω. Find Zin at a distance l =0.1λ from the load.


at B, zin = 0.6 –j0.66

VSWR, Voltage Maxima and Voltage Minima

VSWR, Voltage Maxima and Voltage Minima

• Point A is the normalized load impedance with zL=2+j1.

• VSWR = 2.6 (at Pmax).

• The distance between the load and the first voltage maximum is lmax=(0.25-0.213)λ.

• The distance between the load and the first voltage minimum is lmin=(0.037+0.25)λ.

Impedance to admittance transformations

yL=0.25 - j0.6

zL=0.6 + j1.4

Example 6

• Given that the voltage standing-wave ratio, VSWR = 3. On a 50-Ω line, the first voltage minimum occurs at 5 cm from the load, and that the distance between successive minima is 20 cm, find the load impedance.


• The distance between successive minima is equal to λ/2. Hence, λ = 40 cm.

• First voltage minimum (in wavelength unit) is at

on the WTL scale from point B.

• Intersect the line with constant SWR circle = 3.• The normalized load impedance at point C is:

• De-normalize (multiplying by Z0) to get ZL:

125.040

5min l

8.06.0L jz

40308.06.050L jjZ

Impedance Matching

• Transmission line is matched to the load when Z0 = ZL.

• This is usually not possible since ZL is used to serve other application.

• Alternatively, we can place an impedance-matching network between load and transmission line.

Single- stub matching

• Matching network consists of two sections of transmission lines.

• First section of length d, while the second section of length l in paralllel with the first section, hence it is called stub.

• The second section is terminated with either short-circuit or open circuit.


• The distance d is chosen so as to transform the load admittance, YL=1/ZL into an admittance of the form Yd = Y0+jB when looking towards the load at MM’.

• The length l of the stub is chosen so that its input admittance, YS at MM’ is equal to –jB.

• Hence, the parallel sum of the two admittances at MM’ yields Y0, which is the characteristic admittance of the line.

Example 7

50-Ω transmission line is connected to an antenna with load impedance ZL = (25 − j50)Ω. Find the position and length of the short-circuited stub required to match the line.


• The normalized load impedance is

(located at A).

• Value of yL at B is which locates at position 0.115λ on the WTG scale.

• Draw constant SWR circle that goes through points A and B.

• There are two possible matching points, C and D where the constant SWR circle intersects with circle rL=1 (now gL =1 circle).

jj

Z

Zz

5.0

50

5025

0

LL

8.04.0L jy


First matching points, C.• At C, is at 0.178λ on WTG scale.• Distance B and C is• Normalized input admittance

at the juncture is:

E is the admittance of short-circuit stub, yL=-j∞.

Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale gives:

58.11d jy 063.0155.0178.0 d

58.1

58.1101

s

s

sin

jy

jyj

yyy d

09.025.034.01 l


Second matching point, D.• At point D,• Distance B and C is• Normalized input admittance at G. • Rotating from point E to point G, we get

58.11 jyd 207.0115.0322.02 d

58.1s jy

41.016.025.02 l

ELECTROMAGNETIC THEORY EKT 241/4: ELECTROMAGNETIC THEORY PREPARED BY: NORDIANA MOHAMAD SAAID...

Documents

Transcript of ELECTROMAGNETIC THEORY EKT 241/4: ELECTROMAGNETIC THEORY PREPARED BY: NORDIANA MOHAMAD SAAID...