Two Port Networks MCQ Quiz - Objective Question with Answer for Two Port Networks - Download Free PDF

Explanation:

Reciprocity in Two-Port Networks

Definition: A two-port network is said to be reciprocal if the transfer of signal or energy between port 1 and port 2 is the same in either direction under the same conditions. Reciprocity is a property often associated with linear passive networks and is crucial in the design of electrical systems such as filters, amplifiers, and communication networks. Mathematically, this property ensures that the forward and reverse transmission characteristics are identical under certain conditions.

Conditions for Reciprocity: Reciprocity of a two-port network can be analyzed using different network parameters such as impedance (Z), admittance (Y), hybrid (h), and transmission (ABCD) parameters. The condition for reciprocity varies depending on the type of network parameters being used. Here’s a breakdown:

• Impedance Parameters (Z): For the network to be reciprocal using Z-parameters, the condition is Z₁₂ = Z₂₁.
• Admittance Parameters (Y): For the network to be reciprocal using Y-parameters, the condition is Y₁₂ = Y₂₁.
• Hybrid Parameters (h): For the network to be reciprocal using h-parameters, the condition is h₁₂ = -h₂₁.
• Transmission Parameters (ABCD): For the network to be reciprocal using ABCD parameters, the condition is AD - BC = 1.

Correct Option Analysis:

The correct option is:

Option 4: AD - BC = 1

This condition pertains to the transmission (ABCD) parameters of a two-port network. The ABCD parameters, also known as transmission parameters, describe the input and output voltages and currents of the two-port network. They are expressed as:

V₁ = AV₂ + BI₂

I₁ = CV₂ + DI₂

Here, A, B, C, and D are the transmission parameters. For a network to be reciprocal, the determinant of the ABCD parameter matrix must equal 1, i.e.,:

AD - BC = 1

This means that the energy transfer between port 1 and port 2 is identical in both directions, confirming reciprocity. This condition is widely used in the analysis of transmission lines, filters, and communication systems to ensure symmetry in signal transmission.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Z₁₁ = Z₂₂

This option is incorrect. The condition Z₁₁ = Z₂₂ does not imply reciprocity in a two-port network. Instead, it suggests symmetry in the impedance parameters. Reciprocity in Z-parameters is defined by the condition Z₁₂ = Z₂₁.

Option 2: Y₂₁ = Y₂₁

This option is incorrect and is a typographical error. The correct condition for reciprocity in terms of Y-parameters is Y₁₂ = Y₂₁. This ensures that the admittance between the two ports is identical in both directions.

Option 3: h₂₁ = h₂₂

This option is incorrect. The condition h₂₁ = h₂₂ does not imply reciprocity in a two-port network. Reciprocity in terms of h-parameters is defined by the condition h₁₂ = -h₂₁, which involves a specific relationship between the forward and reverse hybrid parameters.

Option 5: No option is mentioned for Option 5, so it is irrelevant to the problem.

Conclusion:

Understanding the concept of reciprocity in two-port networks is vital for analyzing and designing electrical systems. The correct condition for reciprocity depends on the type of network parameters being used. In this case, the transmission (ABCD) parameters define reciprocity with the condition AD - BC = 1. This ensures symmetry in energy transfer between ports, making the network reciprocal. By analyzing the incorrect options, we see that they either misrepresent the reciprocity condition or describe different properties of the network parameters.

Explanation:

Two-Port Symmetric Bilateral Network

Definition: A two-port symmetric bilateral network is a type of electrical network that has two pairs of terminals, referred to as "ports." The network is termed symmetric when certain parameters are equal, and it is bilateral when the network behaves identically when the input and output ports are interchanged. These networks are often analyzed using transmission parameters (A, B, C, D), where the relationships between input and output voltages and currents are described as:

Equations:

    \( V_1 = A \cdot V_2 + B \cdot I_2 \)

    \( I_1 = C \cdot V_2 + D \cdot I_2 \)

For symmetric networks, the parameters satisfy the following conditions:

• \( A = D \) (Symmetry condition)
• \( A \cdot D - B \cdot C = 1 \) (Reciprocity condition)

Given:

• \( A = 3 \, \Omega \)
• \( B = 1 \, \Omega \)

We need to determine the value of the parameter \( C \).

Step 1: Use the Reciprocity Condition

From the reciprocity condition, we know:

    \( A \cdot D - B \cdot C = 1 \)

Since the network is symmetric, \( A = D \). Substituting \( A = 3 \, \Omega \) and \( B = 1 \, \Omega \), we get:

    \( 3 \cdot 3 - 1 \cdot C = 1 \)

    \( 9 - C = 1 \)

Step 2: Solve for \( C \)

Rearranging the equation:

    \( C = 9 - 1 \)

    \( C = 8 \, \text{s} \)

Thus, the value of the parameter \( C \) is \( 8 \, \text{s} \).

Step 3: Evaluate the Correct Option

From the options provided, the correct answer corresponds to:

• Option 1: \( 8 \, \text{s} \)

Important Information:

To analyze why the other options are incorrect, let us revisit the reciprocity condition:

    \( A \cdot D - B \cdot C = 1 \)

For a symmetric network:

• \( A = D = 3 \, \Omega \)
• \( B = 1 \, \Omega \)

Substituting these values into the equation, we calculated \( C = 8 \, \text{s} \). Any other value for \( C \) would violate the reciprocity condition. Let’s examine the incorrect options:

Option 2: \( C = 6 \, \text{s} \)

If \( C = 6 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 6 = 9 - 6 = 3 \)

Here, the result is \( 3 \), which does not satisfy the reciprocity condition (\( A \cdot D - B \cdot C = 1 \)). Thus, this option is incorrect.

Option 3: \( C = 4 \, \text{s} \)

If \( C = 4 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 4 = 9 - 4 = 5 \)

Here, the result is \( 5 \), which does not satisfy the reciprocity condition. Thus, this option is incorrect.

Option 4: \( C = 16 \, \text{s} \)

If \( C = 16 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 16 = 9 - 16 = -7 \)

Here, the result is \( -7 \), which does not satisfy the reciprocity condition. Thus, this option is incorrect.

Option 5: \( C = 1 \, \text{s} \)

If \( C = 1 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 1 = 9 - 1 = 8 \)

While the arithmetic is correct, the parameter \( C \) is defined as a reciprocal value in seconds (\( \text{s} \)), and the units here do not align with the requirements of the problem. Thus, this option is also incorrect.

Conclusion:

The correct value of \( C \) in the given two-port symmetric bilateral network is \( 8 \, \text{s} \), satisfying the reciprocity condition. This corresponds to Option 1. The other options fail to meet the necessary conditions or have incorrect units, as demonstrated in the analysis above.

The correct option is 3

Concept:

Two-port network parameters such as Z, Y, h, and T (ABCD) describe the relationship between voltages and currents at the input and output ports of a network. When two such networks are cascaded, their parameters combine in specific ways depending on the type of parameter used.

Calculation:

When two two-port networks A and B are cascaded:

• Impedance [Z] parameters cannot be directly multiplied. They are added when connected in series, not cascaded.
• Admittance [Y] parameters are added for parallel connection, not cascade.
• Hybrid [h] parameters cannot be used directly for cascading either.
• Transmission (ABCD) [T] parameters are specifically designed for cascade connection. When two networks are cascaded, the overall transmission matrix is the product of individual matrices:

[T] = [TA] × [TB] 

Explanation:

The h-parameters represent a two-port network with the following equations:

V1 = h11 * I1 + h12 * V2 I2 = h21 * I1 + h22 * V2

where:

• V1 is the input voltage
• I1 is the input current
• V2 is the output voltage
• I2 is the output current
• h11, h12, h21, and h22 are the h-parameters

The condition for reciprocity in a two-port network represented by h-parameters is:

h12 = -h21

Therefore, the correct answer is h12 = -h21.

Solution:

To find the Z parameters (Z₁₁, Z₁₂, Z₂₁, Z₂₂) for the given network, we need to understand the two-port network analysis. The Z-parameters or impedance parameters are defined by the following set of equations:

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

Where:

• V₁ is the input voltage
• V₂ is the output voltage
• I₁ is the input current
• I₂ is the output current

To determine the Z-parameters, we need to perform the following steps:

Step 1: Calculate Z₁₁

Z₁₁ is found by setting I₂ = 0 (open-circuit output). Under this condition, the input impedance is:

Z₁₁ = V₁ / I₁ (with I₂ = 0)

Step 2: Calculate Z₁₂

Z₁₂ is found by setting I₂ = 0 (open-circuit output). Under this condition, the reverse transfer impedance is:

Z₁₂ = V₁ / I₂ (with I₁ = 0)

Step 3: Calculate Z₂₁

Z₂₁ is found by setting I₁ = 0 (open-circuit input). Under this condition, the forward transfer impedance is:

Z₂₁ = V₂ / I₁ (with I₂ = 0)

Step 4: Calculate Z₂₂

Z₂₂ is found by setting I₁ = 0 (open-circuit input). Under this condition, the output impedance is:

Z₂₂ = V₂ / I₂ (with I₁ = 0)

Let's now solve these for the given options:

Correct Option Analysis:

The correct option is:

Option 3: 30Ω, 20Ω, 20Ω, 30Ω

We will validate this by calculating each of the Z parameters for the given network:

Z₁₁:

Given that I₂ = 0, V₁ = Z₁₁I₁. If Z₁₁ = 30Ω, then:

V₁ = 30Ω × I₁

Z₁₂:

Given that I₂ = 0 and assuming Z₁₂ = 20Ω, then:

V₁ = 20Ω × I₂

Z₂₁:

Given that I₁ = 0 and assuming Z₂₁ = 20Ω, then:

V₂ = 20Ω × I₁

Z₂₂:

Given that I₁ = 0, V₂ = Z₂₂I₂. If Z₂₂ = 30Ω, then:

V₂ = 30Ω × I₂

Thus, the Z-parameters are correctly given by option 3: Z₁₁ = 30Ω, Z₁₂ = 20Ω, Z₂₁ = 20Ω, Z₂₂ = 30Ω.

Let's analyze the other options to understand why they are incorrect:

Option 1: 40Ω, 20Ω, 20Ω, 40Ω

While Z₁₂ and Z₂₁ are the same as in the correct answer, Z₁₁ and Z₂₂ are different. For the given network, these values do not match the correct Z-parameter values.

Option 2: 40Ω, 30Ω, 30Ω, 40Ω

All the Z-parameter values here differ from the correct option. These values do not satisfy the given network's conditions.

Option 4: 30Ω, 30Ω, 30Ω, 30Ω

In this case, Z₁₂ and Z₂₁ are incorrectly given as 30Ω instead of 20Ω. This does not align with the correct Z-parameter values.

Conclusion:

By understanding the principles of Z-parameters and carefully analyzing the network, we have determined that the correct Z-parameters for the given network are Z₁₁ = 30Ω, Z₁₂ = 20Ω, Z₂₁ = 20Ω, and Z₂₂ = 30Ω, which corresponds to option 3.

Concept:

Admittance Matrix:

• It is also known as a short circuit matrix or Y matrix.
• Y matrix is represented as:

\(\begin{bmatrix} Y_{11} & Y_{12}\\ Y_{21}& Y_{22} \end{bmatrix}\)= \(\begin{bmatrix} Y_{A}+ Y_{C}& -Y_{C}\\ -Y_{C}& Y_{B}+ Y_{C} \end{bmatrix}\)

• The condition of symmetry and reciprocity in Y parameters are given by:

Symmetry: \(Y_{11}= Y_{22}\)

Reciprocity: \(Y_{12}= Y_{21}\)

Explanation:

Given, that the Y matrix is \(\left[ {\begin{array}{*{20}{c}} 0\\ {\frac{1}{2}} \end{array}\begin{array}{*{20}{c}} { - \frac{1}{2}}\\ 0 \end{array}} \right]\)

Here, \(Y_{12}\neq Y_{21}\)

Hence Y matrix is not reciprocal.

The shunt admittance dissipates energy, hence it is a passive element.

Therefore, option 1 is correct.

A two-port network is said to be symmetrical if the input and output ports can be interchanged without altering the port voltages and currents.

A network is said to be reciprocal if the ratio of the response to the excitation is invariant to an interchange of the positions of the excitation and response of the network.

Conditions of reciprocity and symmetry in terms of different two-port parameters are:

The application of different two port network parameters are shown below.

h₁₂ = V₁/V₂ → dimensionless

h₂₁ = I₂/I₁ → dimensionless

h₁₁ = V₁/I₁ → ohms

B = V₁/I₂ → ohms

C = I₁/V₂ → mho

AD → dimensionless

Concept:

Z parameter:

We will get the following set of two equations by considering the variables V₁ & V2 as dependent and I₁ & I₂ as an independent. The coefficients of independent variables, I₁ and I₂ are called Z parameters.

V₁ = Z₁₁I₁ +Z₁₂I₂

V₂ = Z₂₁I₁+ Z₂₂I₂

Calculation:

Apply KVL on loop 1:

⇒ V₁ = 5I₁ + 20(I₁ + I₂)

⇒ V1 = 25I1 + 20I2 ------------ (1)

Apply KVL on loop 2:

⇒  V₂ = 10I2 + 20(I1 + I2)

⇒ V2 = 20I1 + 30I2 -------------(2)

Compare equation 1 and 2 with Z parameter equations:

 Z11 = 25Ω, Z22 = 30Ω

Alternate Method For a T equivalent network as shown below 

Z₁₁ = (Z₁ + Z₃) = (5+20) = 25Ω 

Z₂₂ = (Z₂ + Z₃) = (10 + 20) = 30Ω 

Z₁₂ = Z₂₁ = Z₃ = 20Ω 

Concept:

The Z-parameter for a T-equivalent two-port network is,

\(\left[ Z \right] = \left[ {\begin{array}{*{20}{c}} {{Z_{11}}}&{{Z_{12}}}\\ {{Z_{21}}}&{{Z_{22}}} \end{array}} \right]\)

Where

Z₁₁ is open circuit impedance

Z₁₂ = Z₂₁ = Transfer impedance

Z₂₂ = Open circuit output impedance

Calculation:

By using delta to star conversion, the given circuit can be reduced.

\({R_1} = \frac{{4 \times 2}}{{4 + 2 + 2}} = \frac{8}{8} = 1\;{\rm{\Omega }}\)

\({R_2} = \frac{{2 \times 2}}{{4 + 2 + 2}} = \frac{4}{8} = \frac{1}{2}\;{\rm{\Omega }}\)

\({R_3} = \frac{{4 \times 2}}{{4 + 2 + 2}} = \frac{8}{8} = 1\;{\rm{\Omega }}\)

The modified circuit is,

Now, by comparing the above circuit with T-equivalent network of Z-parameter matrix,

\(\left[ Z \right] = \left[ {\begin{array}{*{20}{c}} 4&3\\ 3&{3.5} \end{array}} \right]\)

A two-port network is said to be symmetrical if the input and output ports can be interchanged without altering the port voltages and currents.

A network is said to be reciprocal if the ratio of the response to the excitation is invariant to an interchange of the positions of the excitation and response of the network.

Conditions of reciprocity and symmetry in terms of different two-port parameters are:

Concept:

Y parameters

These are also called the admittance parameters.

The Y parameters for the two-port network are shown as:

\(\left[ {\begin{array}{*{20}{c}} {{I_1}}\\ {{I_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{Y_{11}}}&{{Y_{12}}}\\ {{Y_{21}}}&{{Y_{22}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{V_2}} \end{array}} \right]\)

I1 = Y11V1 + Y12V2

I2 = Y21V1 + Y22V2

Calculation:

When V₂ = 0 or short circuit port 2

Then, I₁ = Y₁₁ V₁

I₂ = Y₂₁ V₁

V₁ and I₁ relation can be drawn from current division rule as

\(V_1 = 0.5\times ({\frac{0.5}{0.5+0.5}})I_1\)

V₁ = 0.25I₁

I₁ = 4V₁

Y11 = 4 S

similarly, V1 and I2 relation can be drawn as

V₁ = 0.5(-I₂)

I₂ = -2V₁

Y21 = -2 S

By applying the same procedure for port 1

When V1 = 0 or short circuit port 1

Then, I1 = Y12 V2

I2 = Y22 V2

V2 and I2 relation can be drawn from current division rule as

\(V_2 = 0.5\times ({\frac{0.5}{0.5+0.5}})I_2\)

V2 = 0.25I2

I2 = 4V2

Y22 = 4 S

similarly, V1 and I2 relation can be drawn as

V2 = 0.5(-I1)

I1 = -2V2

Y12 = -2 S

\( \left[ {\begin{array}{*{20}{c}} {{Y_{11}}}&{{Y_{12}}}\\ {{Y_{21}}}&{{Y_{22}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {4}&{-2}\\ {-2}&{4} \end{array}} \right]~S\)

Hence option 1 is correct

For the given π network 

Y parameter can be calculated by

\(\left[ {\begin{array}{*{20}{c}} {{I_1}}\\ {{I_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{Y_1} + {Y_2}}&{ - {Y_2}}\\ { - {Y_2}}&{{Y_2} + {Y_3}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{V_2}} \end{array}} \right]\\ \)

Substituting the value of Y₁, Y_2, and Y₃

\(= \left[ {\begin{array}{*{20}{c}} {\left( {2 +2} \right)}&{\left( { - 2} \right)}\\ {\left( { - 2} \right)}&{\left( {2 + 2} \right)} \end{array}} \right] \\\)

\(= \left[ {\begin{array}{*{20}{c}} {4}&{ - 2}\\ { - 2}&{4} \end{array}} \right]~S\)

Concept:

I₁ = Y₁₁ V₁ + Y₁₂ V₂

I₂ = Y₂₁ V₁ + Y₂₂ V₂

When port 1 is short cirucited, i.e. V₁ = 0,

I₁ = Y₁₂ V₂ and I₂ = Y₂₂ V₂

Calculation:

The given equations are:

I₁ = 4I₂ and V₂ = 0.25 I₂

⇒ I₂ = 4V₂ ⇒ Y₂₂ = 4

I₁ = 4 (4V₂) = 16 V₂

Concept:

Y parameter:

I₁ = V₁ Y₁₁ + V₂ Y₁₂

I₂ = V₁ Y₂₁ + V₂ Y₂₂

\({I_1} = \frac{{{V_1} - {V_2}}}{Z}\)

\({I_1} = \frac{{{V_1}}}{Z} - \frac{1}{Z}\;{V_2}\)     ...1)

\({I_2} = - \frac{1}{Z}\;{V_1} + \frac{1}{Z}\;{V_2}\)     ...2)

\(\left[ y \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{Z}}&{ - \frac{1}{Z}} \\ { - \frac{1}{Z}}&{\frac{1}{Z}} \end{array}} \right]\)

Calculation:

For the given question Z = 5 Ω

\(\therefore \left[ y \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{5}}&{ - \frac{1}{5}} \\ { - \frac{1}{5}}&{\frac{1}{5}} \end{array}} \right]\)