rf&me lecture 4, 5_microwave transmission lines

7/28/2019 Rf&Me Lecture 4, 5_microwave Transmission Lines

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Microwave Transmission Lines

Engr. Ghulam Shabbir



Transmission Lines:

Fundamental Concepts



Transmission Line Theory

Introduction:

In an electronic system, the delivery of powerrequires the connection of two wires between the

source and the load. At low frequencies, power isconsidered to be delivered to the load through the wire.

In the microwave frequency region, power isconsidered to be in electric and magnetic fields that are

guided from place to place by some physical structure.Any physical structure that will guide an electromagneticwave, place to place is called a Transmission Line.



• A lumped circuit is one where all the terminalvoltages and currents are functions of timeonly. Lumped circuit elements includeresistors, capacitors, inductors, independentand dependent sources.

• An distributed circuit is one where the

terminal voltages and currents are functionsof position as well as time. Transmission linesare distributed circuit elements.

Circuit Theory:

Lumped vs. Distributed Systems




• A lumped system is one in which the dependent variables of

interest are a function of time alone. In general, this will

mean solving a set of ordinary differential equations (ODEs )

• A distributed system is one in which all dependent variables

are functions of time and one or more spatial variables. In

this case, we will be solving partial differentialequations (PDEs )

• For example, consider the following two systems:

http://www.dsprelated.com/dspbooks/pasp/Lumped_Models.html

http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html


http://mathworld.wolfram.com/PartialDifferentialEquation.html











• The first system is a distributed system, consisting of an

infinitely thin string, supported at both ends; the dependentvariable, the vertical position of the string is indexed

continuously in both space and time.

• The second system, a series of ``beads'' connected by

massless string segments, constrained to move vertically,can be thought of as a lumped system, perhaps an

approximation to the continuous string.

• For electrical systems, consider the difference between a

lumped RLC network and a transmission line.

• The importance of lumped approximations to distributed

systems will become obvious later, especially

for waveguide -based physical modeling , because it enables

one to cut computational costs by solving ODEs at a few

points, rather than a full PDE (generally much more costly)




http://en.wikipedia.org/wiki/Waveguide

http://en.wikipedia.org/wiki/Model_(physical)




http://en.wikipedia.org/wiki/Waveguide






Circuit Theory: Capacitors Circuit

i(t) +

-

v(t)

The

rest

of

the

circuit

dt

t dvC t i

)()(

t

dx xiC

t v )(1

)(

t

t

dx xiC

t vt v

0

)(1)()( 0

)(2

1

)(

2

t Cvt wC Energy stored in the

capacitors



Circuit Theory: Inductors Circuits

dt

t di Lt v

)()( i(t)

+

-

v(t)

The

rest

of

the

circuit

L

t

dx xv L

t i )(1

)(

t

t

dx xv L

t it i

0

)(1

)()( 0

)(2

1

)(2

t Lit w L Energy stored in the

inductor



Propagation Velocity

• Physical example:

Wave propagates in z direction

• Circuit:L = [nH/cm]

C = [pF/cm]

t

I Ldz dz

z

V

• Total voltage change across Ldz

(use ):dt

dI LV

• Total current change across Cdz (use ):

dt dV C I

t V Cdz dz

z I

[1]

[2]

• Simplify [1] & [2] to get the

Telegraphist’s Equations

[3a]

t

I L z

V

t

V C

z

I

[3b]

I

V

Ldz

Cdz

dz

V+ dzdV

dz

I+ dzdI

dz



Propagation Velocity (2)

• Phase velocity definition:

LC v

1 [7]

• Equation in terms of current:

2

2

22

2

2

2 1

t

I

t

I LC

z

I

[8]

•

Equate [4] & [5]: [6]2

2

22

2

2

2

1t V

t V LC

z V

• Differentiate [3b] by z: [5] z t

I L

z

V

2

2

2

• Differentiate [3a] by t : [4]2

22

t

V C

t z

I

• Equation [6] is a form of the wave equation.

• The solution to [6] contains forward and backward traveling

wave components, which travel with a phase velocity.



Equivalent Circuit of a Real Transmission Line



Characteristic Impedance (Lossless)

Ldz

Cdx

Z1 Z2 Z3

Ldz

Cdz

Ldz

Cdz

dz dz

V1

V3

V2 to

a

f ed

cb

dz

dz= segment length

C = capacitance per segment

L = inductance per segment




•

The input impedance (Z1) is the impedance of the firstinductor (Ldz) in series with the parallel combination of the

impedance of the capacitor (Cdz) and Z2.

[.9] Cdz j Z Cdz j Z Ldz j Z

/1

/1

2

21

0/1/1/1 2221 Cdz j Z Cdz j Z Ldz jCdz j Z Z




• Assuming a uniform line, the input impedance should be

the same when looking into node pairs a-d , b-e, c - f , and so

forth. So, Z2 = Z1 = Z0.

0/1/1/1 0000 Cdz j Z Cdz j Z lLdz jCdz j Z Z [10]

Cdz j

Ldz jdz LZ j Z

Cdz j

Z

Cdz

Ldz dz LZ j

Cdz j

Z Z

0

2

00

002

0 0

• Allow dz to become very small, causing the frequency

dependent term to drop out:

00

2

0 C

Ldz LZ j Z [11]

02

0 C

L Z [12]

• Solve for Z 0 called as Characteristic Impedance:

C L Z 0

[13]



Visualizing Transmission Line Behavior

• Water flow

– Potential = Wave height [m]

– Flow = Flow rate [liter/sec]

I

I

V

+++++++

- - - - - - -

• Transmission Line

• Potential = Voltage [V]

• Flow = Current [A] = [C/sec]

• Just as the wave front of the water flows in the pipe,

the voltage propagates in the transmission line.

The same holds true for current.

•

Voltage and current propagate as waves in thetransmission line.

h flow



Visualizing Transmission Line Behavior

•

Extending the analogy – The diameter of the pipe relates the flow rate and height of the

water. This is analogous to electrical impedance.

– Ohm’s law and the characteristic impedance define the relationship

between current and potential in the transmission line.

• Effects of impedance discontinuities

– What happens when the water encounters a ledge or a barrier?

– What happens to the current and voltage waves when the

impedance of the transmission line changes?

– The answer to this question is a key to understanding transmissionline behavior.

– It is useful to try to visualize current/voltage wave propagation on a

transmission line system in the same way that we can, for water

flow in a pipe.



General Transmission Line Model (No Coupling)

•

Transmission line parameters are distributed (e.g.capacitance per unit length).

• A transmission line can be modeled using a network of

resistances, inductances, and capacitances, where the

distributed parameters are broken into small discrete

elements.

R L

G C

R L

G C

R L

G C



General Transmission Line Model #2

Parameter Symbol Units

Conductor Resistance R W•cm-1

Self Inductance L nH•cm-1

Total Capacitance C pF•cm-1

Dielectric Conductance G W-1•cm -1

Parameters

Characteristic ImpedanceC jG

L j R Z

0 [14]

Propagation Constant jC jG L j R [15]

= attenuation constant = rate of exponential attenuation = phase constant = amount of phase shift per unit length

p

Phase Velocity [16]

In general, and are frequency dependent.



Frequency DependenceFrom [14] and [15] note that:

• Z 0 and depend on the frequency content of the signal.• Frequency dependence causes attenuation and edge rate

degradation.

Attenuation

Edge rate degradation

Output signal from lossy transmission line

Signal at driven end of transmission line

Output signal from

lossless transmission line

• R and G are sometimes

negligible, particularly

at low frequencies• Simplifies to the

lossless case:

• no attenuation

&

• no dispersion



Lossless Transmission Lines

Quasi-TEM Assumption

• The electric and magnetic fields are perpendicular to the

propagation velocity in the transverse planes.

x

z y

H E



Lossless Line Parameters

• Lossless line characteristics are frequency independent.

• As noted before, Z 0 defines the relationship between

voltage and current for the traveling waves. The units are

ohms [].• defines the propagation velocity of the waves. The units

are cm/ns.

– Sometimes, we use the propagation delay, (units are

ns/cm).

C

L Z

0

LC v

1

Characteristic Impedance


[17]

[18]

Lossless transmission lines are characterized by the

following two parameters:



Lossless Line Equivalent Circuit

• The transmission line equivalent circuit shown on the left

is often represented by the coaxial cable symbol.

L

C

L

C

L

C

Z , v , length Z 0 , , length

L= Self Inductance per segment

C = Total Capacitance per segment

C

L Z

0

LC v

1

Characteristic Impedance


Length of segment = dz



Homogeneous Media

• A homogeneous dielectric medium is uniform in all

directions.

– All field lines are contained within the dielectric.

• For a transmission line in a homogeneous medium, the

propagation velocity depends only on material

properties:

r r r

nscmc LC

v

/3011 0 [19]

0 r Dielectric Permittivity

cm F x 140 10854.8 Permittivity of free space

cm H x

8

0 10257.1 Magnetic Permeability

0 Permeability of free space

er is the relative permittivity or dielectric constant.

Note: only r is

required to

calculate .



Non-Homogeneous Media

• A non-homogenous medium contains multiple materials

with different dielectric constants.

• For a non-homogeneous medium, field lines cut across

the boundaries between dielectric materials.

• In this case the propagation velocity depends on the

dielectric constants and the proportions of the materials.Equation [19] does not hold:

11

LC

v

• In practice, an effective dielectric constant, er,eff is often

used, which represents an average dielectric constant.



Phase Constant (Lossless Case)• Recall the basic voltage divider circuit:

R1

R2 V 1

+

V 2

-

I

• We want to find the ratio of the input voltage, V 1, to the

output voltage, V 2, to the output voltage, V 2..

• Now, we apply it to our transmission line equivalent

circuit.

0211

IR IRV 21

1

R R

V I

21

2122

R R

RV IRV

2

1

2

21

2

1 1 R

R

R

R R

V

V



Phase Constant (Lossless Case)• The analogous transmission line circuit looks like this:

• The phase shift is the ratio of V 1 to V 2:

• Substitute the expressions for Z C , Z L, and Z 0 :

00

0

00

0

2

1 11111

Z Z Z

Z Z

Z Z Z

Z Z

Z

Z Z

Z Z Z

V

V

C

L

C

C L

C

L

C

C L

L Z R 1

0

002

Z Z Z Z Z Z R

C

C C

Cdz j Z C

1

Ldz j Z L

C L Z 0

LC dz j LC dz j Ldz j

Cdz jCdz j Ldz j

Z Z Z

V

V

C

L

222222

02

1 1111

1

LC dz j LC dz V

V

22

2

1

1

Ldz

CdzV1

+

V2

-

Z0

I

CharacteristicImpedance



Phase Constant (Lossless Case) #3

• The amplitude of the phase constant is:

• The phase angle, denoted as tanl , is:

• Now, we make the assumption that dz is small enough that

the applied frequency, , is much smaller than the

resonant frequency, , of each subsection, so that:

LC dz LC dz V

V 22222

2

1

1

LC dz

LC dz l 22

1

tan

LC dz 1

122 LC dz

• The phase angle becomes: LC dz l tan

• Since , tanl is, very small. Therefore:

LC dz l l tan

122

LC dz

Ph C t t (L l C ) #4



Phase Constant (Lossless Case) #4

• The phase shift per unit length is:

p

• l represents the amount by which the input voltage, V 1,

leads the output voltage, V 2.

• We can simplify the amplitude ratio by using the condition

of small l :

• So, there is no decrease in the amplitude of the voltage

along the line, for the lossless case. Only a shift in phase.• From our definition of phase velocity in equation [16] we

get

LC dz

l

1122222

2

1 LC dz LC dz V

V

C

L p



Transmission-Lines Equations



Transmission-Lines Equations

A typical engineering problem involves the transmission of a

signal from a generator to a load. A transmission line is thepart of the circuit that provides the direct link between

generator and load.

Transmission lines can be realized in a number of ways.

Common examples are the parallel-wire line and the coaxialcable.

For simplicity, we use in most diagrams the parallel-wire

line to represent circuit connections, but the theory applies

to all types of transmission lines.



Examples of Transmission-Lines



Microwave Transmission Lines and Solutions

Conventional two-conductor transmission lines are

commonly used for transmitting microwave energy. If a line is properly matched to its characteristic impedance

at each terminal, its efficiency can reach a maximum.

In ordinary circuit theory, it is assumed that all impedance

elements are lumped constants. This is not true for long transmission line over a wide range

of frequencies. Frequencies of operation are so high that

inductances of short lengths of conductors and capacitances

between short conductors and their surroundings cannot be

neglected.

These inductances and capacitances are distributed along

the length of a conductor, and their effects combine at each

point of the conductor.



Microwave Transmission Lines and Solutions

Since the wavelength is short in comparison to the physical

length of the line, distributed parameters cannot berepresented accurately by means of lumped-parameter

equivalent circuit.

Thus microwave transmission lines can be analyzed in terms

of voltage, current, and impedance only by distributedcircuit theory.

If the spacing between the lines is smaller than the

wavelength of the transmitted signal, the transmission line

must be analyzed as a waveguide.



Transmission-Lines Equations and Solutions

If you are only familiar with low frequency circuits, you are

used to treat all lines connecting the various circuitelements as perfect wires, with no voltage drop and no

impedance associated to them (lumped impedance

circuits).

This is a reasonable procedure as long as the length of thewires is much smaller than the wavelength of the signal. At

any given time, the measured voltage and current are the

same for each location on the same wire.




d l




i i i i d l i



Why is the study of transmission lines (TL) and

waveguide important in understanding

nanotechnology?

One of the practical applications of nanotechnology is the

generation and transmission of light.

The short wavelength of light makes almost every small

conducting tree appear as a TL or waveguide that can divert

light from its intended path.

Laser diodes, for example, are dependent on the creation of waveguides in the nm range for operation.



Transmission-Line Theory:

Two Basic Approaches to Analysis

Lumped – Element Modelo Uses discrete components and circuit theory

o Assumes dimensions smaller than the wavelength

Field Analysiso Uses distributed components and Maxwell’s Equations

o Considers dimensions vs. wavelength



Transmission-Line Theory:

Two Basic Approaches to Analysis

Lumped circuits: resistors, capacitors, inductors

neglect time delays (phase)

account for propagation and

time delays (phase change)

Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length of a

line is significant compared with a wavelength.



Lumped – Element Circuit Model




z

,i z t

+ + + + + + +

- - - - - - - - - - ,v z t x x xB

R z L z

G z C z

z

v( z + z , t )

+

-

v( z , t )

+

-

i( z , t ) i( z + z , t )

z

z

T i i Li



Transmission Line

2 conductors

4 per-unit-length parameters:

C = Shunt capacitance/unit length [F/m

]

L = Series inductance/unit length (both conductors) [H/m]

R = Series resistance/unit length (both conductors [/m]

G = Shunt conductance/unit length [ /m or S/m]

z

L l T i i Li



Loss-less Transmission Line




Loss less Transmission Line



Lossy Transmission Line




T i i Li E ti d S l ti



A transmission line can be analyzed either by the solution of

Maxwell’s field equations or by the methods of distributed

circuit theory.

The solution of Maxwell’s equations involves three space

variables in addition to the time variable. The distributed –circuit method, however, involves only one

space variable in addition to the time variable.

Transmission-Line Equations and Solutions:Transmission-Line Equations

Based on uniformly distributed-circuit theory, the schematiccircuit of a conventional two-conductor transmission line

with constant parameters R, L, G and C is shown as under:

T i i Li E ti d S l ti



Elementary section of a transmission line

The parameters are expressed in their respective names per

unit length, and the wave propagation is assumed in the

positive direction.

Transmission-Line Equations and Solutions:Transmission-Line Equations

Transmission-Line Equations and Solutions:



By Kirchhoff’s voltage law, the summation of the voltage

drops around the central loop is given by

Rearranging this equation, dividing it by ∆z, and then

omitting the argument (z, t), which is understood, we obtain

Using Kirchhoff’s current law, the summation of the currents

at point B in above Figure can be expressed as

Transmission-Line Equations




By rearranging the preceding equation, dividing it by ∆z,omitting ((z, t), and assuming ∆z equal to zero, we have

Transmission Line Equations and Solutions:Transmission-Line Equations

Then by differentiating Eq. (3-1-2) with respect to z and Eq.

(3-1-4) with respect to t and combining the results, the final

transmission-line equation in voltage form is found to be




Also, by differentiating Eq.(3-1-2) with respect to t and Eq.(3-

1-4) with respect to z and combining the results, the final

transmission-line equation in current form is


All these transmission-line equations are applicable to the

general transient solution.




The voltage and current on the line are the functions of bothposition z and time t.

The instantaneous line voltage and current can be expressed as


where Re stands for “real part of ” .

The factors V(z) and I(z) are complex quantities of thesinusoidal functions of position z on the line and are known as

phasors.





The phasors give the magnitudes and phases of the sinusoidal function at each position of z, and they can be expressed as

Where V+ and I+ indicate complex amplitudes in the positive zdirection, V- and I- signify complex amplitudes in the negative z

direction, α is the attenuation constant in nepers per unit

length, and β is the phase constant in radians per unit length.





If we substitute j ω for ∂/∂t in Eqs. (3-1-2), (3-1-4), (3-1-5), and(3-1-6) and divide each equation by e jωt, the transmission-line

equations in phasor form of the frequency domain become

In which the following substitutions have been made:

using the phasor representation





For a lossless line, R = G = 0, and the transmission-line

equations are expressed as





It is interested to note that Eqs. (3-1-14) and (3-1-15) for atransmission line are similar to Eqs (3-1-21) and (3-1-22) of

electric and magnetic waves, respectively.

The only difference is that the transmission-line equations are

one-dimensional.




Solutions of Transmission-Line Equations

The one possible solution for Eq. (3-1-14) is

The factors V+ and V- represent complex quantities.

The term involving e-jβz shows a wave travelling in the positive

z direction, and the term with the factor e jβz is a wave going in

the negative z direction.

The quantity β z is called the electrical length of the line and is

measured in radians.

Similarly, the one possible solution for Eq. (3-1-15) is





In Eq. (3-1-24) the characteristic impedance of the line is

defined as

The magnitude of both voltage and current waves on the line is

shown in Fig.

Fig. Magnitude of voltage and current travelling waves

At microwave frequencies it can be seen that





By using the binomial expansion, the propagation constant can

be expressed as

Therefore, the attenuation and phase constants are,

respectively, given by





Therefore, the attenuation and phase constants are,

respectively, given by

Similarly, the characteristic impedance is found to be





From eq. (3-1-29) the phase velocity is

The product of LC is independent of the size and separation of

the conductors and depends only on the permeability µ and

permittivity of ϵ of the insulating medium.

If a lossless transmission line used for microwave frequencies

has an air dielectric and contains no ferromagnetic materials,

free-space parameters can be assumed.

Thus the numerical value of 1/ for air-insulatedconductor is approximately equal to the velocity of light in

vacuum. That is,





When the dielectric of a lossy microwave transmission line is

not air, the phase velocity is smaller than velocity of light invacuum and is given by

In general, the relative phase velocity factor can be defined as

A low-loss transmission line filled only with dielectric medium,

such as a coaxial line with solid dielectric between conductors,

has a velocity factor on the order of about 0.65.

Example 3-1-1:Li Ch i i I d d P i C



Line Characteristic Impedance and Propagation Constant

Reflection Coefficient and

T i i C ffi i



Transmission CoefficientReflection Coefficient

In the analysis of the solutions of transmission-line equation,the traveling wave along the line contains two components:

o One traveling wave in the positive z direction

o other traveling the negative z direction.

+

V +( z )

-

I + ( z )

z

+

V -( z )

-

I - ( z )

z

0

( )

( )

V z Z

I z

0

0

( )

( )

z

z

V z V e

I z I e

00

0

V Z

I

( Z 0

is a number, not a

function of z .)

so

0

( )

( )

V z Z

I z

0

( )

( )

V z

Z I z

soNote: The reference directions for voltage and current are the same as for the forward wave.

General Case (Waves in Both Directions)



00

0 0

j z j j z

z z

z j z V e e

V z V e V

V e e e

e

e

0

0 cos

c

, R

os

ej t

z

z

V e t

v z t V z

z

V z

e

e t

Note:

wave in + z

direction wave in - z direction

General Case



A general superposition of forward and backward

traveling waves:

Most general case:

Note: The reference

directions for voltage

and current are the

same for forward and

backward waves.

+

V ( z )

-

I ( z )

z

1

2

12

0

0 0

0 0

0 0

z z

z z

V z V e V e

V V I z e e

Z

j R j L G j C

R j L Z

G j

Z

C

phase velocity v p

[m/s] pv

The Terminated Lossless Transmission Line




• In the analysis of the solutions of transmission-line equations,

the traveling wave along the line contains two components:• One traveling in the positive z direction and the other

traveling the negative z direction.





•

Figure shows a lossless transmission line terminated in anarbitrary load impedance Z ℓ .

• This problem will illustrate wave reflection on transmission

lines, a fundamental property of distributed systems.

• Assume that an incident wave of the form V +e ─ is

generated from a source at z < 0.

• The ratio of voltage to current for such a traveling wave is

Z 0 , the characteristic impedance of the line.

• However, when the line is terminated in an arbitrary load

Z ℓ ≠ Z 0 , the ratio of voltage to current at the load must beZℓ .





• Thus, a reflected wave must be excited with the appropriate

amplitude to satisfy this condition.• The total voltage and current on the line can then be written as

a sum of incident and reflected waves:

• Similarly, the total current on the line is described by

Reflection Coefficient andTransmission Coefficient




It is usually more convenient to start solving the

transmission-line problem from the receiving rather than the

sending end, since the voltage-to-current relationship at the

load point is fixed by the load impedance.

Fig: Transmission line terminated in a load impedance

If the load impedance is equal to the line characteristicimpedance, the reflected traveling wave does not exist.

Figure below shows a transmission line terminated in an

impedance Zℓ.





The incident voltage and current waves traveling along the

transmission line are given by

in which the current wave can be expressed in terms of the

voltage by

If the line has a length of ℓ, the voltage and current at thereceiving end become





The ratio of the voltage to the current at the receiving end is

the load impedance. That is,

• The amplitude of the reflected voltage wave normalized to the

amplitude of the incident voltage wave is defined as the voltage

reflection coefficient, which is designated by (gamma), as





If Eq. (3-2-6) is solved for the ratio of the reflected voltage at

the receiving end , which is V-eℓ ,

to the incident voltage at

the receiving end, which is V+e ─ℓ, the result is the reflection

coefficient at the receiving end:

From Fig., let = ℓ − . Then the reflection coefficient atsome point located a distance from the receiving end is


T i i C ffi i t




The reflection coefficient at that point can be expressed in

term of the reflection coefficient at the receiving end as

Here the phase of Γℓ is assumed to be θℓ .

, the reflection coefficient Γ at a distance of d from

the load end is .

Its magnitude will be with a phase


T i i C ffi i t




This is a very useful

equation for determining

the reflection coefficient

at any point along the

line.

For a lossy line, both the

magnitude and phase of the reflection coefficient

are changing in an

inward –spiral way as

shown:

Reflection coefficient for a lossy line


Transmission Coefficient




For a lossless line, = , the magnitude of the reflectioncoefficient remains constant, and only the phase of is

changing circularly towards the generator with an angle of

− as shown

Reflection coefficient for a lossless line


Transmission Coefficient




It is evident that ℓ will be zero and there will be no

reflection from the receiving end when the terminating

impedance is equal to the characteristic impedance of the

line.

Thus a terminating impedance that differs from thecharacteristic impedance will create a reflected wave

traveling towards the source from the termination.

The reflection, upon reaching the sending end, will itself be

reflected if the source impedance is different from the linecharacteristic impedance at the sending end.




Transmission CoefficientTransmission Coefficient A transmission line terminated in its characteristic

impedance Z0 is called a properly terminated line.

Otherwise it is called an improperly terminated line.

There is a reflection coefficient at any point along an

improperly terminated line.

According to the principle of conservation of energy, theincident power minus the reflected power must be equal to

the power transmitted to the load. This can be expressed as

The letter T represents the transmission coefficient, which is

defined as




Transmission CoefficientTransmission Coefficient The following figure shows the transmission of power along a

transmission line where P inc is the incident power, P ref the

reflected power, and P tr the transmitted power.

Let the travelling waves at the receiving end be




Transmission CoefficientTransmission Coefficient

Multiplication of Equation (3-2-16) by Zℓ and submission of the result in Equation (3-2-15) yields

which, in turn, on submission back into Equation (3-2-15),

results in

• The power carried by the two waves in the side of the

incident and reflected waves is




Transmission CoefficientTransmission Coefficient

The power carried to the load by the transmitted wave is

• By setting Pinc

= Ptr

and using Equation (3-2-17) and (3-2-

18), we have

•

This relation verifies the previous statement that thetransmitted power is equal to the difference of the

incident power and reflected power.

Example 3-2-1



Example 3 2 1

Solution

rf&me lecture 4, 5_microwave transmission lines

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