1 EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 1: TRANSMISSION LINE THEORY

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EKT 441

MICROWAVE COMMUNICATIONS

CHAPTER 1:

TRANSMISSION LINE THEORY

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OUR MENU (PART 1)

Introduction to Microwaves Transmission Line Equations The Lossless Line Terminated Transmission Lines

Reflection Coefficient VSWR Return Loss

Transmission Lines Impedance Equations Special Cases of Terminated Transmission Lines

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SPECTRUM & WAVELENGTHS

Waves in the electromagnetic spectrum vary in size from very long radio waves the size of buildings, to very short gamma-rays smaller than the size of the nucleus of an atom.

Wavelength of a wave is the distance we have to move along the transmission line for the sinusoidal voltage to repeat its pattern

4

INTRODUCTION

Microwave refers to alternating current signals with frequencies between 300 MHz and 300 GHz.

Figure 1 shows the location of the microwave frequency

Long wave radio

AM broad

Casting radio

Short

wave

radio

VHF

TV

FM

broad

casting

radio

MicrowavesFar

infraredinfrared Visible

light

3 x105 3 x 106 3 x107 3x 108 3x109 3x1010 3x1011 3x1012 3x 1013 3x1014

103 102 101 1 10-1 10-2 10-3 10-4 10-5 10-6

Typical frequencies

AM broadcast band 535-1605 kHz VHF TV (5-6) 76-88 MHz

Shortwave radio 3-30 MHz UHF TV (7-13) 174-216 MHz

FM broadcast band 88-108 MHz UHF TV (14-83) 470-890 MHz

VHF TV (2-4) 54-72 MHz Microwave ovens 2.45 GHz

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MICROWAVE BAND DESIGNATION

Frequency (GHz)

Wavelength (cm) IEEE band

1 - 2 30 - 15 L

2 - 4 15 - 7.5 S

4 - 8 7.5 - 3.75 C

8 - 12 3.75 - 2.5 X

12 - 18 2.5 - 1.67 Ku

18 - 27 1.67 - 1.11 K

27 - 40 1.11 - 0.75 Ka

40 - 300 0.75 - 0.1 mm

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APPLICATION OF MICROWAVE ENGINEERING

Communication systems UHF TV Microwave Relay Satellite Communication Mobile Radio Telemetry

Radar system Search & rescue Airport Traffic Control Navigation Tracking Fire control Velocity Measurement

Microwave Heating Industrial Heating Home microwave ovens

Environmental remote sensing

Medical system Test equipment

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TYPICAL Rx ARCHITECTURE

Typical receiver (from RF & Microwave Wireless Systems, Wiley)

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TYPICAL Rx ARCHITECTURE

When signal arrives at Rx, normally it is amplified by a Low Noise Amplifier (LNA)

Mixer then produce a down-converted signal at freq of fIF+fm OR fIF-fm; fIF<fm

Signal is then filtered to remove undesired harmonics & spurious products from mixing process

Signal is then amplified by an intermediate freq (IF) amplifier

Output signal of amplifier goes to detector for the recovery of the original message

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TYPICAL Tx ARCHITECTURE

Typical transmitter architecture (from RF & Microwave Wireless System, Wiley)

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TYPICAL Tx ARCHITECTURE

Input baseband signals (video, data, or voice) is assumed to be bandlimited to a freq fm

Signal is filtered to remove any components beyond passband

Message signal is then mixed with a local oscillator (LO) to produce modulated carrier (up-conversion) (fLO+fm OR fLO-fm), fm<fLO

Modulated carrier can be amplified & transmitted by the antenna

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TRANSMISSION LINES

Low frequencies wavelengths >> wire length current (I) travels down wires easily for efficient power transmission measured voltage and current not dependent on position along wire

High frequencies wavelength » or << length of transmission medium need transmission lines for efficient power transmission matching to characteristic impedance (Zo) is very important for low reflection and maximum power transfer measured envelope voltage dependent on position along line

I+ -

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TRANSMISSION LINE EQUATIONS

Complex amplitude of a wave may be defined in 3 ways: Voltage amplitude Current amplitude Normalized amplitude whose squared modulus equals

the power conveyed by the wave Wave amplitude is represented by a complex phasor:

length is proportional to the size of the wave phase angle tells us the relative phase with respect to

the origin or zero of the time variable

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Transmission line is often schematically represented as a two-wire line.

i(z,t)

V(z,t)

Δz

z

Figure 1: Voltage and current definitions.

The transmission line always have at least two conductors.Figure 1 can be modeled as a lumped-element circuit, as shown in Figure 2.

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The parameters are expressed in their respective name per unit length.

RΔz LΔz

GΔz CΔz

i(z,t) i(z + Δz,t)

Δz

v(z + Δz,t)

R = series resistant per unit length, for both conductors, in Ω/mL = series inductance per unit length, for both conductors, in H/mG = shunt conductance per unit length, in S/mC = shunt capacitance per unit length, in F/m

Figure 2: Lumped-element equivalent circuit

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The series L represents the total self-inductance of the two conductors.

The shunt capacitance C is due to close proximity of the two conductors.

The series resistance R represents the resistance due to the finite conductivity of the conductors.

The shunt conductance G is due to dielectric loss in the material between the conductors.

NOTE: R and G, represent loss.

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By using the Kirchoff’s voltage law, the wave equation for V(z) and I(z) can be written as:

022

2

zVdz

zVd 022

2

zIdz

zId

CjGLjRj where

γ is the complex propagation constant, which is function of frequency.

α is the attenuation constant in nepers per unit length, β is the phase constant in radians per unit length.

[1] [2]

[3]

17


The traveling wave solution to the equation [2] and [3] before can be found as:

zz

zz

eIeIzI

eVeVzV

00

00[4]

[5]

The characteristic impedance, Z0 can be defined as:

CjG

LjRLjRZ

0[6]

Note: characteristic impedance (Zo) is the ratio of voltage to current in a forward travelling wave, assuming there is no backward wave

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Zo determines relationship between voltage and current waves Zo is a function of physical dimensions and r Zo is usually a real impedance (e.g. 50 or 75 ohms)

characteristic impedancefor coaxial airlines (ohms)

10 20 30 40 50 60 70 80 90 100

1.0

0.8

0.7

0.6

0.5

0.9

1.5

1.4

1.3

1.2

1.1

norm

aliz

ed v

alue

s 50 ohm standard

attenuation is lowest at 77 ohms

power handling capacity peaks at 30 ohms

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Voltage waveform can be expressed in time domain as:

zz eztVeztVtzv coscos, 00

The factors V0+ and V0

- represent the complex quantities. The Φ± is the phase angle of V0

±. The quantity βz is called the electrical length of line and is measured in radians.

Then, the wavelength of the line is:

2

[7]

[8]

and the phase velocity is:

fvp

[9]

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EXAMPLE 1.1

A transmission line has the following parameters:

R = 2 Ω/m G = 0.5 mS/m f = 1 GHz

L = 8 nH/m C = 0.23 pF

Calculate:

1. The characteristic impedance.

2. The propagation constant.

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THE LOSSLESS LINE

The general transmission line are including loss effect, while the propagation constant and characteristic impedance are complex.

On a lossless transmission line the modulus or size of the wave complex amplitude is independent of position along the line; the wave is neither growing not attenuating with distance and time

In many practical cases, the loss of the line is very small and so can be neglected. R = G = 0

So, the propagation constant is:

LCjj [10]

LC 0

[10a]

[10b]

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THE LOSSLESS LINE

For the lossless case, the attenuation constant α is zero.

Thus, the characteristic impedance of [6] reduces to:

C

LZ 0

[11]

The wavelength is:

LC

22

and the phase velocity is:

LCvp

1

[11a]

[11b]

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EXAMPLE 1.2

A transmission line has the following per unit length parameters: R = 5 Ω/m, G = 0.01 S/m, L = 0.2 μH/m and C = 300 pF. Calculate the characteristic impedance and propagation constant of this line at 500 MHz. Recalculate these quantities in the absence of loss (R=G=0)

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TERMINATED TRANSMISSION LINES

RF

Incident

Reflected

Transmitted

LightwaveDUT

• Network analysis is concerned with the accurate measurement of the ratios of the reflected signal to the incident signal, and the transmitted signal to the incident signal.

Waves travelling from generator to load have complex amplitudes usually written V+ (voltage) I+ (current) or a (normalised power amplitude).

Waves travelling from load to generator have complex amplitudes usually written V- (voltage) I- (current) or b (normalised power amplitude).

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TERMINATED LOSSLESS TRANSMISSION LINE

Most of practical problems involving transmission lines relate to what happens when the line is terminated

Figure 3 shows a lossless transmission line terminated with an arbitrary load impedance ZL

This will cause the wave reflection on transmission lines.

Figure 3: A transmission line terminated in an arbitrary load ZL

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Assume that an incident wave of the form V0+e-jβz is

generated from the source at z < 0. The ratio of voltage to current for such a traveling

wave is Z0, the characteristic impedance [6].

If the line is terminated with an arbitrary load ZL= Z0 , the ratio of voltage to current at the load must be ZL.

The reflected wave must be excited with the appropriate amplitude to satisfy this condition.

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The total voltage on the line is the sum of incident and reflected waves:

zjzj eVeVzV 00

The total current on the line is describe by:

zjzj eZ

Ve

Z

VzI

0

0

0

0

[12]

[13]

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

0

00

00

0

0Z

VV

VV

I

VZL

[14]

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Solving for V0+ from [14] gives:

00

00 V

ZZ

ZZV

L

L [15]

The amplitude of the reflected wave normalized to the amplitude of the incident wave is defined as the voltage reflection coefficient, Γ:

0

0

0

0

ZZ

ZZ

V

V

L

L

[16]

The total voltage and current waves on the line can then be written as:

zjzj eeVzV 0

zjzj eeZ

VzI

0

0

[17]

[18]

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The time average power flow along the line at the point z:

2

0

2

01

2

1

Z

VPav [19]

• [19] shows that the average power flow is constant at any point of the line.

• The total power delivered to the load (Pav) is equal to the incident

power minus the reflected power

• If |Γ|=0, maximum power is delivered to the load. (ideal case)

• If |Γ|=1, there is no power delivered to the load. (worst case)• So reflection coefficient will only have values between 0 < |Γ| < 1

0

2

0

2Z

V

0

22

0

2Z

V

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STANDING WAVE RATIO (SWR)

When the load is mismatched, the presence of a reflected wave leads to the standing waves where the magnitude of the voltage on the line is not constant.

ljzj eVeVzV 20

20 11 [21]

ljeV 20 1

The maximum value occurs when the phase term ej(θ-2βl) =1. 10max VV

The minimum value occurs when the phase term ej(θ-2βl) = -1.

10min VV

[22]

[23]

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STANDING WAVE RATIO (SWR)

As |Γ| increases, the ratio of Vmax to Vmin increases, so the measure of the mismatch of a line is called standing wave ratio (SWR) can be define as:

1

1

min

max

V

VSWR [24]

• This quantity is also known as the voltage standing wave ratio, and sometimes identified as VSWR.

• SWR is a real number such that 1 ≤ SWR ≤

• SWR=1 implies a matched load

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RETURN LOSS

When the load is mismatched, not all the of the available power from the generator is delivered to the load.

This “loss” is called return loss (RL), and is defined (in dB) as:

log20RL [20]

• If there is a matched load |Γ|=0, the return loss is dB (no reflected power).

• If the total reflection |Γ|=1, the return loss is 0 dB (all incident power is reflected).•So return loss will have only values between 0 < RL <

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SUMMARY

Three parameters to measure the ‘goodness’ or ‘perfectness’ of the termination of a transmission line are:

1. Reflection coefficient (Γ)

2. Standing Wave Ratio (SWR)

3. Return loss (RL)

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EXAMPLE 1.3

Calculate the SWR, reflection coefficient magnitude, |Γ| and return loss values to complete the entries in the following table:

SWR |Γ| RL (dB)

1.00 0.00

1.01

0.01

30.0

2.50

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EXAMPLE 1.3

The formulas that should be used in this calculation are as follow:

[20]

[24]

mod from [20]

mod from [24]

log20RL

1

1SWR

)20/(10 RL

1

1

SWR

SWR

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EXAMPLE 1.3

Calculate the SWR, reflection coefficient magnitude, |Γ| and return loss values to complete the entries in the following table:

SWR |Γ| RL (dB)

1.00 0.00

1.01 0.005 46.0

1.02 0.01 40.0

1.07 0.0316 30.0

2.50 0.429 7.4

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Since we know that total voltage on the line is

zjzj eVeVzV 00

And the reflection coefficient along the line is defined as Γ(z):

zjzj

zj

eV

V

eV

eV

zincidentV

zreflectedVz

2

0

0

0

0

)(

)(

[12]

Defining or ΓL as the reflection coefficient at the load;

)0(0

0

V

VL

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TRANSMISSION LINE IMPEDANCE EQUATION

Substituting ΓL into eq [14 and 15], the impedance along the line is given as:

ljlj

ljlj

ee

eeZ

zI

zVzZ

0)(

At x=0, Z(x) = ZL. Therefore;

L

LL ZZ

1

10

0

00 ZZ

ZZZe

L

LjLL

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TRANSMISSION LINE IMPEDANCE EQUATION

At a distance l = -z from the load, the input impedance seen looking towards the load is:

0

0

0 ZeeV

eeV

lI

lVZ

ljlj

ljlj

in

02

2

1

1Z

e

elj

lj

[25a]

When Γ in [16] is used: lj

Llj

L

ljL

ljL

in eZZeZZ

eZZeZZZZ

00

000

ljZlZ

ljZlZZ

L

L

sincos

sincos

0

00

ljZZ

ljZZZ

L

L

tan

tan

0

00

[26a]

[26b]

[26c]

[25b]

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EXAMPLE 1.4

A source with 50 source impedance drives a

50 transmission line that is 1/8 of wavelength

long, terminated in a load ZL = 50 – j25 .

Calculate:

(i) The reflection coefficient, ГL

(ii) VSWR

(iii) The input impedance seen by the source.

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SOLUTION TO EXAMPLE 1.4

It can be shown as:

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(i) The reflection coefficient,

076

0

0

242.0502550

502550 j

L

LL

ej

j

ZZ

ZZ

(ii) VSWR

64.11

1

L

LVSWR

SOLUTION TO EXAMPLE 1.4 (Cont’d)

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(iii) The input impedance seen by the source, Zin

8.38.30

255050

50255050

tan

tan

0

00

j

j

jj

jZZ

jZZZZ

L

Lin

48

2 1

4tan

Need to calculate

Therefore,

SOLUTION TO EXAMPLE 1.4 (Cont’d)

49

SPECIAL CASE OF LOSSLESS TRANSMISSION LINES

For the transmission line shown in Figure 4, a line is terminated with a short circuit, ZL=0.

From [16] it can be seen that the reflection coefficient Γ= -1. Then, from [24], the standing wave ratio is infinite.

VL=0

V(z),I(z)

Z0, β

z0-l

IL=0

ZL=0

Figure 4: A transmission line terminated with short circuit

0

0

0

0

ZZ

ZZ

V

V

L

L

1

1SWR

50


Referred to Figure 4, equation [17] and [18] the voltage and current on the line are:

zjVeeVzV zjzj sin2 00 [27]

zZ

Vee

Z

VzI zjzj cos

2

0

0

0

0

[28]

From [26c], the ratio V(-l) / I(-l), the input impedance is:

ljZZ in tan0 [29]

When l = 0 we have Zin=0, but for l = λ/4 we have Zin = ∞ (open circuit)

Equation [29] also shows that the impedance is periodic in l.

51


V(z)/2jV0+

z-λ/4

-λ/2

-3λ/4

-λ

1

-1(a)

I(z)Z0/2V0+

z-λ/4

-λ/2

-3λ/4

-λ

1

-1(b)

Xin/Z0

z-λ/4

-λ/2

-3λ/4

-λ

1

-1(c)

Figure 5: (a) Voltage (b) Current (c) impedance (Rin=0 or ∞) variation along a short circuited transmission line

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For the open circuit as shown in Figure 6, ZL=∞

The reflection coefficient is Γ=1.

The standing wave is infinite.

VL=0

V(z),I(z)

Z0, β

z0-l

IL=0

ZL=∞

Figure 6: A transmission line terminated in an open circuit.

0

0

0

0

ZZ

ZZ

V

V

L

L

1

1SWR

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For an open circuit I = 0, while the voltage is a maximum. The input impedance is:

ljZZ in cot0 [30]

When the transmission line are terminated with some special lengths such as l = λ/2,

Lin ZZ [31]

For l = λ/4 + nλ/2, and n = 1, 2, 3, … The input impedance [26c] is given by:

Lin Z

ZZ

20 [32]

Note: also known as quarter wave transformer.

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(a)

(b)

(c)

V(z)/2V0+

z-λ/4

-λ/2

-3λ/4

-λ

1

-1

I(z)Z0/-2jV0+

z-λ/4

-λ/2

-3λ/4

-λ

1

-1

Xin/Z0

z-λ/4

-λ/2

-3λ/4

-λ

1

-1

Figure 7: (a) Voltage (b) Current (c) impedance (R = 0 or ∞) variation along an open circuit transmission line.

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1 EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 1: TRANSMISSION LINE THEORY