w2&w3. ch2.0 transmission line theory v0.2

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Ch2: Transmission Line and Matching Circuits

Dr. Rashid A. Saeed

ECE 4337: RF Circuit and DevicesDepartment of Electrical and Computer Engineering,

IIUM

Agenda

• Transmission Line– Wave Propagation on a transmission line– Lossless transmission lines– Special Cases of Lossless Terminated Lines

• Impedance Matching– The Quarter-wave Transformer– Matching using L-sections– Single-stub tuning

• Microwave Network Analysis – Impedance and admittance matrices– The scattering matrix– The transmission (ABCD) matrix

2

z

5

z

Let’s V=Voejwt , I = Ioejwt

Therefore

Vjdt

dV Ijdt

dI

then

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Wave Propagation on a transmission line

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to relate the voltage and current on the line as

Then (2.6b) can be rewritten in the following form:

the wavelength on the line is

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• The above solution was for a general transmission line, including loss effects, In many practical cases, the loss of the line is very small and so can be neglected,

• The characteristic impedance of (2.7) reduces to

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Assume that an incident wave of the form is generated from a source at

assume incident and reflected waves are:

Amplitude of incident wave

Amplitude of reflected wave

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Solving ZL equation for

z=0 at the load ZL so

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The incident wave and reflected wave is called standing waves

To obtain the load impedance ZL must be equal to the characteristic impedance Z0 of the transmission line

Matched load impedanceMatched load impedance

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ZL =Z0

The Power flow along the lineThe Power flow along the line

The middle two terms in the brackets are of the form A - A* = 2jlm(A) and so are purely imaginary. This simplifies the result to

the average power flow is constant at any point on the line,if =0, maximum power is delivered to the load, while no power is delivered for lf =1.

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Return Loss (RL)Return Loss (RL)

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

This "loss" is called return loss (RL),

0dB return loss means all incident power is reflected.

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Standing Wave RatioStanding Wave Ratio

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At a distance from the load, the input impedance seen looking toward the load is

from

We can write

from

We can write

and

The transmission line impedance equation

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Solution

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Special Cases of Lossless Terminated Lines1. A transmission line terminated in a

short circuit.

At z=0 (at the load) V=0 at the load (as expected, for a short circuit), while the current is a maximum there

the reflection coefficient for a short circuit load is

= -1

from We can write

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2. The open-circuited line

from We can write

the reflection coefficient for an open circuit load is

= 1

which shows that now I =0 at the load, as expected for an open circuit, while the voltage is a maximum, The input impedance is

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3. terminated transmission lines with some special lengths.

meaning that a half-wavelength line (or any multiple of ) does not alter or transform the load impedance, regardless of the characteristic impedance.

Such a line is known as a quarter-wave transformer because it has the effect of transforming the load impedance, in an inverse manner, depending on the characteristic impedance of the line.

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Agenda




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Impedance Matching Techniques

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• Impedance matching techniques includes:1. The quarter-wave transformer2. Lumped element matching network or some time called L-

sections3. Single stub tuning

• Factors that may be important in the selection of a particular matching network include the following:

• Complexity- A simpler matching network is usually cheaper, more reliable, and less lossy than a more complex design.

• Bandwidth- In many applications, however, it is desirable to match a load over a band of frequencies. However, this is done by increasing complexity.

• Implementation- For example, tuning stubs are much easier to implement in waveguide than L-section or quarter-wave transformers, while L-section is easy to be implemented in RFIC.

• Adjustability- In some applications the matching network may require adjustment to match a variable load impedance.

by applying that to Zin equation we can get

which yields

• The above condition applies only when the length of the matching section is /4 or an odd multiple (2n + 1) of /4 long, so that a perfect match may be achieved at one frequency, but mismatch will occur at other frequencies

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1. The Quarter-Wave Transformer

Frequency response of a quarter-wave transformer

• Consider a load resistance RL=100, to be matched to a 50 line with a quarter-wave transformer.

1. Find the characteristic impedance of the matching section and 2. plot the magnitude of the reflection coefficient versus normalized frequency, f/f0,

where f0 is the frequency at which the line is /4 long.

SolutionFrom The reflection coefficient magnitude is given as

• where is a function of

frequency from the term l, which can be written in terms of f/f0, as

This method of impedance matching is limited to real load impedances

2. L-section Impedance Matching• This technique is used extensively in lower frequency circuit design• It has advantages over quarter-wavelength that the load impedance need not to be real.• Has two possible configuration,

– If the normalized load impedance is inside the circle on the smith chart, then this circuit should be used:

– If the normalized load impedance is outside the circle on the smith chart, then this circuit should be used:

– The 1+jx circle is the resistance circle on the impedance Smith chart for which r=1.

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inside outside

Example : L-section impedance matching

Design an L-section matching network to match a series RC load with an impedance ZL=200 - j100 , to a 100 line, at a frequency of 500 MHz.

Solutions

• The normalized load impedance is zL=2 - j1, which is inside the 1+jx circle

• so we will use this matching circuit

• Since the first element from the load is a shunt susceptance helpful to convert to admittance Smith chart

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zL=2 - j1

• This matching circuit consists of a shunt capacitor and a series inductor.

• For a frequency of f= 500 MHz, the capacitor has a value of

• and the inductor has a value of

There is no difference in the BW

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3.Single Stub Tuning

• A matching technique that uses a single open-circuited or short-circuited length of transmission line (a "stub"),

• Stub is connected either in parallel or in series with the transmission feed line at a certain distance from the load

Series Stub

Shunt Stub

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• Two degree of freedom d and Y• For the shunt-stub case, the basic idea is to select d

so that the admittance, Y, seen looking into the line at distance d from the load is of the form Y0+jB.– Then the stub susceptance is chosen as -jB , resulting in a

matched condition.

• For the series stub case, the distance d is selected so that the impedance, Z, seen looking into the line at a distance d from the load, is of the form Z0+jX. – Then the stub reactance is chosen as -jX, resulting in a

matched condition.

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Example: single Stub shunt tuning

1. normalized load impedance zL =1.2 - jI .6,2. Construct the appropriate SWR circle, and 3. convert to the load admittance, yL,4. SWR circle intersects the 1 +jb circle at two points, denoted as

y1 and y2

5. Thus the distance d, from the load to the stub, is given by either of these two intersections. Reading the WTG scale, we obtain

6. At the two intersection points, the normalized admittances are

7. The stub length

solution

Agenda




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Microwave Network Analysis

• Two representations can give complete description for any microwave network:1. Impedance and admittance matrices (Z, Y)2.The scattering matrix (Sxy)

• The scattering matrix representation is used for high frequency applications,– Where it is difficult to measure total voltages

and currents, but easier to measure the incident and reflected voltages.

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1.Impedance and admittance matrices

• considering an arbitrary N-port microwave network,

Now at the nth terminal plane (tn), the total voltage and current is given by

When z=0

• The impedance matrix [Z] of the microwave network then relates these voltages and currents:

• or in matrix form as• Similarly, we can define an admittance matrix [Y] as

• or in matrix form as• Inverse

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Example

• Evaluation of impedance parameters:Find the Z parameters of the two-port T-

network shown in the Figure below.

Solution:Need to calculate Z11, Z12, Z21, Z22

A two-port T-network

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• Z11 can be found as the input impedance of port 1 when port 2 is open-circuited:

• The transfer impedance Z12 can be found measuring the open-circuit voltage at port 1 when a current I2 is applied at port 2. By voltage division,

• You can verify that Z21= Z12, indicating that the circuit is reciprocal.

• Finally, Z22 is found as

A two-port T-network

Where

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Thévenin's theorem -

2.The Scattering matrix

• For some components and circuits, the scattering parameters can be calculated using network analysis techniques. – Otherwise, the scattering parameters can be measured

directly with a vector network analyzer (VNA)

• The scattering matrix, or [S] matrix, is defined in relation to the incident Vn

+ and reflected Vn

-

voltage waves as

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•Thus, Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, and

•Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads

• A specific element of the [S] matrix can be determined as

• Sij is found by driving port j with an incident wave of voltage Vn

+ and measuring the reflected wave amplitude, Vn

- , coming out of port i.

• The incident waves on all ports except the jth port are set to zero, – which means that all ports should be terminated in

matched loads to avoid reflections.

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Reciprocal and Lossless network

• the [S] matrix for a reciprocal network is symmetric,

• If the network is lossless, then no real power can be delivered to the network.– if the characteristic impedances of all the ports are

identical and assumed to be unity, the average power delivered to the network is (all the incident power in port 1 will be transferred to port 2)

Purely imaginary

A- A*

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Reflected power to port

2

Incident power in port

1

– so

– Using – Then, we can write– so that, for nonzero

– the [S] matrix for a lossless network is unitary matrix.

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Example: Application of scattering parameters

• A two-port network is known to have the following scattering matrix:

1. Determine if the network is reciprocal, and lossless.

2. If port 2 is terminated with a matched load, what is the return loss seen at port 1?

3. If port 2 is terminated with a short circuit, what is the return loss seen at port 1?

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DUT

1. Determine if the network is reciprocal, and lossless.

• Since [S] is not symmetric, the network is not reciprocal.• To be lossless, the S parameters must satisfy:

and

• Taking the first column gives

• So the network is not lossless.

2. If port 2 is terminated with a matched load, what is the return loss seen at port 1?

• In this case the reflection coefficient seen at port 1 is

• So the return loss is

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the [S] matrix for a lossless network is unitary matrix.

3. If port 2 is terminated with a short circuit, what is the return loss seen at port 1?

• for a short circuit at port 2, so• From the definition of the scattering matrix, we can write

•

• The second equation gives•

• Dividing the first equation by

• So the return loss is

22 VV

2

1

2221

1211

2

1

V

V

SS

SS

V

VS21

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1V

22

21

1

2

1 S

S

V

V

4.16 A four-port network has the scattering matrix shown below.(a) Is this network lossless?

(b) Is this network reciprocal?

(c) What is the return loss at port 1 when all other ports are terminated with matched loads?

(d) What is the insertion loss and phase delay between ports 2 and 4, when all other ports are terminated with matched loads?

(e) What is the reflection coef0cient seen at port 1 if a short circuit is placed at the terminal plane of port 3, and all other ports are terminated with matched loads?

yes

Phase delay 600

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Conclusions of S parameters• An important point to understand about S

parameters is that the reflection coefficient looking into port n is not equal to Snn, unless all other ports are matched.– Similarly, the transmission coefficient from port

m to port n is not equal to Snm, unless all other ports are matched.

• Changing the terminations or excitations of a network does not change its S parameters, – but may change the reflection coefficient seen at

a given port, or the transmission coefficient between two ports.

Frequency GHzFrequency GHz

Frequency GHz Frequency GHz

using a NVA

Output RL

Input RL

S-P

aram

eter

(dB

)S

-Par

amet

er (

dB)

S-P

aram

eter

(dB

)S

-Par

amet

er (

dB)

Forward Response or gain of the

system

reverse Response or gain of the

system

The Transmission (ABCD) Matrix

• The Z , Y, and S parameter representations can be used to characterize a microwave network with an arbitrary number of ports, – but in practice many microwave networks consist of a

cascade connection of two or more two-port networks.

• In this case it is convenient to define a 2 x 2 transmission, or ABCD matrix, for each two-port network.

• The ABCD matrix is defined for a two-port network in terms of the total voltages and currents

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A cascade connection of two-port networks.

Relationship between ABCD and impedance Z

Direction of I2 is opposite then substitute by –I2

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V

2221212

2121111

IZIZV

IZIZV

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• If the network reciprocal , then Z12=Z21 , so we can write

2221212

2121111

IZIZV

IZIZV

ABCD representation

Z Impedance representation

1 BCAD

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The ABCD parameters of some useful two-port Circuits

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BACKUP

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Passive and Active RF Devices

• What is the difference between passive and active devices? – Active devices are capable of changing their operational

performance, may deliver power to the circuit, and can perform interesting mathematical functions.

– While a Passive devices that does not require a source of energy for its operation.

– Active devices generate Non-linear network, while passive devices generate linear network.

Passive RF Devices Active RF Devices

•RFID Tag•Directional coupler

•Power divider •Antenna

•Filter•Waveguide

all passive devices use Impedance, Inductance and capacitance

•RFID Interrogator •Mixer

•Amplifier•Oscillators and Frequency Synthesizers All these active devices use Diodes and

Transistors

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Wikipedia

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Demonstration

• Try Falstad's demos• http://www.falstad.com/circuit/

index.html

w2&w3. ch2.0 transmission line theory v0.2

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