Circuit Theory: Impedance and Admittance Matrices

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Questions and Answers

How can the transfer impedance Zij be determined?

By measuring current at port j when all other ports are connected.

By measuring voltage at port i when all other ports are short-circuited.

By driving port j with current while open-circuiting all other ports. (correct)

By driving port i with a current and short-circuiting all other ports.

What does Zii represent in an N-port network?

The voltage at port i when all ports are short-circuited.

The short-circuit current at port i.

The input impedance seen looking into port i. (correct)

The total power dissipated at port i.

What condition allows the impedance matrix Z to be symmetric?

When the network contains active devices.

When the network is reciprocal. (correct)

When the network is composed of nonreciprocal media.

When the network is lossless.

What is the characteristic of the elements Zij or Yij in a lossless network?

They are purely imaginary. Signup and view all the answers

What does the admittance matrix describe in an N-port network?

The relationship between the total current and voltage across ports. Signup and view all the answers

In the context of measuring Yi j, what is done at other ports when measuring the short-circuit current?

All other ports are short-circuited. Signup and view all the answers

How many independent quantities or degrees of freedom does an N-port network have?

2N² Signup and view all the answers

What is the implication of a network being classified as 'reciprocal'?

It lacks any active devices or nonreciprocal media. Signup and view all the answers

What does the reciprocity theorem represent in the context of the network described?

It shows the relationship between two independent sources' fields at the network ports. Signup and view all the answers

When the boundary walls of the network and transmission lines are perfect conductors, which condition holds true?

The tangential electric field is zero on the walls. Signup and view all the answers

Which of the following conditions applies when evaluating the fields at the terminal planes?

The boundaries can be taken arbitrarily far away to render some fields negligible. Signup and view all the answers

What is represented by the expression $Ē 1b = V1b ē1$?

The electric field at terminal plane t1 due to source b. Signup and view all the answers

What do the variables $V1a$, $I1b$, $V2a$, and $I2b$ represent in the context of the given equations?

The voltages and currents at different terminals due to sources. Signup and view all the answers

In the integral of the reciprocity theorem, what do $d s̄$ terms signify?

The infinitesimal surface area elements along the network boundaries. Signup and view all the answers

Which statement best describes the result of substituting fields into the reciprocity theorem equation?

It leads to a zero-sum for the evaluated vectors indicating equilibrium. Signup and view all the answers

What do the scattering matrix parameters relate to in high-frequency networks?

The voltage waves incident on the ports Signup and view all the answers

What are the cross-sectional areas $S1$ and $S2$ referred to in the context of the given equations?

The specific areas at the terminal planes of the respective ports. Signup and view all the answers

How can scattering parameters be measured in a network?

With a vector network analyzer Signup and view all the answers

In the context of the scattering matrix, what does the term 'incident wave' refer to?

The amplitude of the voltage wave entering the port Signup and view all the answers

What is one of the primary challenges in defining voltages and currents for non-TEM lines?

Inability to measure phase shifts directly Signup and view all the answers

What is the main purpose of the scattering matrix in an N-port network?

To provide a complete description of the network as seen at its ports Signup and view all the answers

What is the relationship between voltage waves in the scattering matrix representation?

It links incident and reflected voltage waves Signup and view all the answers

Which of the following statements about the I1 and I2 relationship is correct?

Their difference is always zero Signup and view all the answers

What abstraction is often used when discussing high-frequency networks?

Equivalent voltage and current matrices Signup and view all the answers

What is the condition required for the admittance matrix Y in a reciprocal lossless N-port junction?

Y12 = Y21 Signup and view all the answers

Which of the following must be true for a network defined by the equation V1a I1b − V1b I1a + V2a I2b − V2b I2a = 0?

V1 and V2 must be independent Signup and view all the answers

What mathematical property does a symmetric admittance matrix imply about its inverse?

The inverse is also symmetric Signup and view all the answers

In a lossless network, what does Re{Pavg} equal?

0 Signup and view all the answers

When substituting values into the admittance matrix, which equation represents the relationship between currents and voltages effectively?

I1 = Y11 V1 + Y22 V2 Signup and view all the answers

What must be true regarding the elements of the impedance and admittance matrices for a reciprocal lossless junction?

They must be purely imaginary. Signup and view all the answers

Considering the equation (V1a V2b − V1b V2a )(Y12 − Y21) = 0, which condition must hold for any choice of sources?

Y12 must equal Y21 Signup and view all the answers

What does the term 1/2 [V]t [I]* represent in the context of power in a network?

Average power delivered Signup and view all the answers

What is the return loss when port 2 is terminated with a matched load?

16.5 dB Signup and view all the answers

How is the reflection coefficient seen at port 1 affected when port 2 is terminated with a short circuit?

It decreases to -0.452 Signup and view all the answers

What condition must be met for the reflection coefficient looking into port n to equal Snn?

All other ports must be matched Signup and view all the answers

What does the calculation S11 + |S21|^2 = 0.745 imply about the network's lossiness?

The network exhibits some loss Signup and view all the answers

What is the result of the return loss calculated as RL = −20 log |β| when β is -0.452?

6.9 dB Signup and view all the answers

Study Notes

Impedance and Admittance Matrices

The impedance matrix relates the total port voltages and currents.
The admittance matrix is the inverse of the impedance matrix.
Each element of the impedance or admittance matrix can be determined by driving one port with a current or voltage, open-circuiting/short-circuiting the other ports, and measuring the voltage/current at the desired port.

Reciprocal Networks

A reciprocal network contains no active devices, ferrites, or plasmas.
The reciprocity theorem states that the integral of the dot product of Ēa × H̄b and Ēb × H̄a over a closed surface S enclosing the network is equal.
If the boundary walls are perfect conductors or far enough away, the only non-zero contribution to the integrals come from the cross-sectional areas of the ports.
Applying the reciprocity theorem to a two-port network results in the admittance matrix being symmetric, i.e., Yi j = Y ji.
Since the impedance matrix is the inverse of the admittance matrix, it is also symmetric for reciprocal networks.

Lossless Networks

In a lossless network, the real power delivered to the network is zero.
The elements of the impedance and admittance matrices are purely imaginary for lossless networks.
Since the real power delivered to the network is the sum of the imaginary parts of the impedance matrix multiplied by the currents and their conjugates, the elements of the matrices must be purely imaginary for the sum to be zero.

The Scattering Matrix

The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports.
The scattering matrix provides a complete description of the network as seen at its ports.
It is defined as the matrix equation relating the reflected voltage waves at each port to the incident voltage waves at all ports.
The scattering parameters can be calculated using network analysis techniques or measured directly with a vector network analyzer.
The scattering parameters are used to characterize networks at microwave frequencies, where it's difficult to directly measure voltage and current.
The reflection coefficient looking into a port is not equal to the corresponding diagonal scattering parameter unless all other ports are matched.
The scattering matrix for a lossless network has a unitarity property, meaning the sum of the squared magnitudes of each element in a row is equal to 1.

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Description

This quiz covers the fundamental concepts of impedance and admittance matrices in circuit theory, including their definitions and applications within reciprocal networks. Explore how these matrices are interrelated and the significance of the reciprocity theorem in analyzing two-port networks.