BRANCH :- E & TC ENGINEERING

SEMESTER :- 3^RD

SUBJECT :- CIRCUIT THEORY

CHAPTER :- 6 – Two Port Network Analysis

TOPIC :-MULTIPLE NETWORK

AY-2021-2022 (Winter-2021)

FACULTY :- Er. Biswajit Mandal(Sr. Lecturer Electrical Engg.)

Presentation On:-

Two Port Netwaork

Multiport Networks

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Note:

Equal and opposite currents are assumed on the two wires of a port.

A general circuit can be represented by a multi-port network, where the “ports” are defined as access terminals at which we can define voltages and currents.

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Examples:

1) One-port network

2) Two-port network

3) N-port network

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Multiport Networks (cont.)

Note: Passive sign convention is used at the ports.

To represent multi-port networks we use:

• Z (impedance) parameters
• Y (admittance) parameters
• ABCD parameters

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• S (scattering) parameters

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Not easily measurable at high frequency

Measurable at high frequency

Multiport Networks (cont.)

N-port network

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Consider a 2-port linear network:

In terms of Z-parameters, we have (from superposition):

Impedance (Z) matrix

Two-Port Networks

Therefore

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Elements of Z-Matrix: Z-Parameters

Port 2 open circuited

Port 1 open circuited

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Z-Parameters (cont.)

We inject a current into port j and measure the voltage (with an ideal voltmeter) at port i. All ports are open-circuited except j.

N-port network

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Summary of Z Parameters

N-port network

Summary of Z Parameters

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Admittance (Y) Parameters

Consider a 2-port linear network:

Admittance matrix

or

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Y-Parameters (cont.)

We apply a voltage across port j and measure the current (with an ideal current meter) at port i. All ports are short-circuited except j.

N-port network

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N-port network

Summary of Y Parameters

Summary of Y Parameters

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Relation Between Z and Y Parameters

Relation between [Z] and [Y] matrices:

Hence:

Therefore

It follows that

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Reciprocal Networks

If a network does not contain non-reciprocal devices or materials*

(i.e. ferrites, or active devices), then the network is “reciprocal”, which means that the Z and Y matrices are symmetric.

* A reciprocal material is one that has symmetric permittivity and permeability matrices. A reciprocal device is one that is made from reciprocal materials.

Note:

The inverse of a symmetric matrix is symmetric.

Example of a nonreciprocal material: a biased ferrite

(This is very useful for making isolators and circulators.)

(proof omitted)

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Reciprocal Materials

Ferrite:

is not symmetric!

Reciprocal:

(not a reciprocal material)

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We can show that the equivalent circuits for reciprocal 2-port networks are:

T-equivalent

Pi-equivalent

Reciprocal Networks (cont.)

“Π network”

“T network”

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ABCD-Parameters

There are defined only for 2-port networks.

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Cascaded Networks

A nice property of the ABCD matrix is that it is easy to use with cascaded networks: you simply multiply the ABCD matrices together.

• At high frequencies, Z, Y, and ABCD parameters are difficult � (if not impossible) to measure.
• V and I are not always uniquely defined (e.g., microstrip, waveguides).
• Even if defined, V and I are extremely difficult to measure (particularly I).
• Required open and short-circuit conditions are often difficult to achieve.�
• Scattering (S) parameters are often the best� representation for multi-port networks at high frequency.

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Scattering Parameters

Note: We can always convert from S parameters to Z, Y, or ABCD parameters.

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Scattering Parameters (cont.)

Keysight (formerly Agilent) VNA shown performing a measurement.

A Vector Network Analyzer (VNA) is usually used to measure S parameters.

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For scattering parameters, we think in terms of incident and reflected waves on transmission lines connected to a device.

Scattering Parameters (cont.)

• The a coefficients represent incident waves.
• The b coefficients represent reflected waves.

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On each transmission line:

Scattering Parameters (cont.)

We define normalized voltage wave functions:

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Why are the wave functions (a and b) defined as they are?

Scattering Parameters (cont.)

• The normalization makes power calculation easy (see next slide).
• The [S] matrix is unitary (discussed later).

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Power Calculations

Recall:

Scattering Parameters (cont.)

(assuming lossless lines, so Z₀ is real)

Similarly,

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A One-Port Network

For a one-port network, S₁₁ is the same as Γ_L.

Recall:

Definition of S₁₁ for a one-port

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Scattering matrix

A Two-Port Network

From linearity:

or

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Output is matched

Input is matched

input reflection coef. � w/ output matched

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reverse transmission coef.� w/ input matched

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forward transmission coef.� w/ output matched

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output reflection coef. � w/ input matched

Output is matched

Input is matched

A Two-Port Network (cont.)

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A Two-Port Network (cont.)

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Illustration of a three-port network

To Illustrate:

Three-Port Network

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Illustration of S₂₁:

Three-Port Network (cont.)

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For a general multiport network:

All ports except port j are semi-infinite (or with matched load at ports) with no incident wave.

N-port network

We send in an incident wave on port j and measure the outgoing wave on port i, when all lines except j are semi-infinite (or terminated in a matched load), and thus there is an incident wave only on port j.

N-Port Network

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N-port network

Summary of S Parameters

Summary of S Parameters

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A microwave system may have waveguides entering a device. In this case, the transmission lines are TEN models for the waveguides.

Scattering Parameters with Waveguides

TEN

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(illustrated for a two-port)

Shift in Reference Planes

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For reciprocal networks (networks containing only reciprocal materials), the S-matrix is symmetric:

Properties of the S Matrix

Example of a nonreciprocal material: a biased ferrite

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• For lossless networks, the S-matrix is unitary^†.

Identity matrix

Properties of the S Matrix (cont.)

Hence,

Alternate notation:

(“Hermetian conjugate” or “Hermetian transpose”)

^†A proof is in the Pozar book.

(a “unitary” matrix)

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Properties of the S Matrix (cont.)

N-port network

Start with the first part of the unitary equation:

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Properties of the S Matrix (cont.)

Interpretation: The inner product of columns i and j is zero unless i = j.

S₁ vector

S₃ vector

The rows also form orthogonal sets (this follows from starting with the second part of the unitary equation).

Physical interpretation: All of the power outgoing on the ports is equal to all of the power incident on the ports.

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Comment on Normalization

Note: If all lines entering the network have the same characteristic impedance, then

“unnormalized” scattering parameters

“normalized” scattering parameters

In general (different characteristic impedances):

Note:

The unitary property of the scattering matrix requires normalized parameters.

We use normalized parameters in these notes.

In this case, there is no difference between normalized and unnormalized scattering parameters.

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Note on Z₀

Important Note:

The S parameters depend on Z₀.

(The Z and Y parameters do not.)

Example: The device is a section of transmission line.

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Scattering Parameters (cont.)

A general formula for converting from Z parameters to S parameters is:

(This assumes all transmission lines are identical with characteristic impedance Z₀.)

Note:

The derivation is in the Pozar book.

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Example

Find the S parameters for a series impedance Z.

Note that two different coordinate systems are being used here!

Thank You