Read the following statements regarding two port networks. (A) The condition of reciprocity and symmetry for a two-port network with 'g' parameter representation can be deduced from the interrelationship of 'h' and g parameters. (B) The condition of reciprocity for the h parameter representation leads to the condition of reciprocity for 'g' parameters representation as g 12 = −g 21 . (C) The condition of symmetry in h-parameters representation never leads to the condition of symmetry in 'g' parameters representation. (D) The symmetry condition in 'h' and 'g parameter representation can be deduced by putting h 11 = g 11 . Choose the correct answer from the options given below:
(A) and (B) only
Understanding Two-Port Network Parameters
Two-port networks are fundamental building blocks in electrical engineering, representing circuits or systems with two pairs of terminals (ports) through which electrical signals can enter or leave. These networks can be characterized by various sets of parameters, such as impedance (Z), admittance (Y), hybrid (h), inverse hybrid (g), and transmission (ABCD) parameters.
Each set of parameters relates the port voltages (\(V_1, V_2\)) and currents (\(I_1, I_2\)) in a specific manner. Understanding the relationships between these parameters and their conditions for network properties like reciprocity and symmetry is crucial for circuit analysis and design.
H-Parameters (Hybrid Parameters)
The hybrid parameters, or h-parameters, define the port variables as follows:
\[ \begin{pmatrix} V_1 \\ I_2 \end{pmatrix} = \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix} \begin{pmatrix} I_1 \\ V_2 \end{pmatrix} \]This expands to the equations:
- \(V_1 = h_{11}I_1 + h_{12}V_2\)
- \(I_2 = h_{21}I_1 + h_{22}V_2\)
For a two-port network described by h-parameters:
- The condition for reciprocity is \(h_{12} = -h_{21}\).
- The condition for symmetry is \(h_{11}h_{22} - h_{12}h_{21} = 1\) (or the determinant of the h-parameter matrix, \(\Delta_h\), equals 1).
G-Parameters (Inverse Hybrid Parameters)
The inverse hybrid parameters, or g-parameters, define the port variables as follows:
\[ \begin{pmatrix} I_1 \\ V_2 \end{pmatrix} = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} \begin{pmatrix} V_1 \\ I_2 \end{pmatrix} \]This expands to the equations:
- \(I_1 = g_{11}V_1 + g_{12}I_2\)
- \(V_2 = g_{21}V_1 + g_{22}I_2\)
For a two-port network described by g-parameters:
- The condition for reciprocity is \(g_{12} = -g_{21}\).
- The condition for symmetry is \(g_{11}g_{22} - g_{12}g_{21} = 1\) (or the determinant of the g-parameter matrix, \(\Delta_g\), equals 1).
Relationship Between H and G Parameters
The h and g parameters are inverses of each other in a specific way. The conversion formulas between h and g parameters can be derived by rearranging the defining equations. If the h-parameter matrix is \(\mathbf{H}\) and the g-parameter matrix is \(\mathbf{G}\), the relationships between parameters are:
| G Parameters in terms of H Parameters | H Parameters in terms of G Parameters |
|---|---|
| \(g_{11} = \frac{h_{22}}{\Delta_h}\) | \(h_{11} = \frac{g_{22}}{\Delta_g}\) |
| \(g_{12} = \frac{-h_{12}}{\Delta_h}\) | \(h_{12} = \frac{-g_{12}}{\Delta_g}\) |
| \(g_{21} = \frac{-h_{21}}{\Delta_h}\) | \(h_{21} = \frac{-g_{21}}{\Delta_g}\) |
| \(g_{22} = \frac{h_{11}}{\Delta_h}\) | \(h_{22} = \frac{g_{11}}{\Delta_g}\) |
where \(\Delta_h = h_{11}h_{22} - h_{12}h_{21}\) and \(\Delta_g = g_{11}g_{22} - g_{12}g_{21}\).
Note that \(\Delta_g = \frac{1}{\Delta_h}\) (provided \(\Delta_h \neq 0\)).
Analyzing the Given Statements on Two-Port Networks
Let's evaluate each statement based on the definitions and relationships discussed:
(A) The condition of reciprocity and symmetry for a two-port network with 'g' parameter representation can be deduced from the interrelationship of 'h' and g parameters.
The reciprocity condition for g-parameters is \(g_{12} = -g_{21}\). Using the interrelationships: \(-\frac{h_{12}}{\Delta_h} = -(-\frac{h_{21}}{\Delta_h})\) This simplifies to \(h_{12} = -h_{21}\) (assuming \(\Delta_h \neq 0\)), which is the reciprocity condition for h-parameters. Thus, if we know the interrelationship and the h-parameter reciprocity condition, we can deduce the g-parameter reciprocity condition. The symmetry condition for g-parameters is \(\Delta_g = 1\). Using the interrelationship \(\Delta_g = \frac{1}{\Delta_h}\), the condition \(\Delta_g = 1\) is equivalent to \(\frac{1}{\Delta_h} = 1\), which means \(\Delta_h = 1\). This is the symmetry condition for h-parameters. So, the symmetry condition for g-parameters can also be deduced from the interrelationship and the h-parameter symmetry condition. Therefore, statement (A) is correct.
(B) The condition of reciprocity for the h parameter representation leads to the condition of reciprocity for 'g' parameters representation as \(g_{12} = -g_{21}\).
The reciprocity condition for h-parameters is \(h_{12} = -h_{21}\). Using the interrelationships: \(g_{12} = -\frac{h_{12}}{\Delta_h}\) and \(g_{21} = -\frac{h_{21}}{\Delta_h}\). If \(h_{12} = -h_{21}\), then \(g_{12} = -\frac{-h_{21}}{\Delta_h} = \frac{h_{21}}{\Delta_h}\). Also, \(-g_{21} = -(-\frac{h_{21}}{\Delta_h}) = \frac{h_{21}}{\Delta_h}\). Since \(g_{12} = \frac{h_{21}}{\Delta_h}\) and \(-g_{21} = \frac{h_{21}}{\Delta_h}\), it follows that \(g_{12} = -g_{21}\) (assuming \(\Delta_h \neq 0\)). Thus, the reciprocity condition in h-parameters (\(h_{12} = -h_{21}\)) indeed leads to the reciprocity condition in g-parameters (\(g_{12} = -g_{21}\)). Therefore, statement (B) is correct.
(C) The condition of symmetry in h-parameters representation never leads to the condition of symmetry in 'g' parameters representation.
The symmetry condition in h-parameters is \(\Delta_h = 1\). The relationship between the determinants is \(\Delta_g = \frac{1}{\Delta_h}\). If \(\Delta_h = 1\), then \(\Delta_g = \frac{1}{1} = 1\). The condition of symmetry in g-parameters is \(\Delta_g = 1\). So, the symmetry condition in h-parameters (\(\Delta_h = 1\)) actually leads to the symmetry condition in g-parameters (\(\Delta_g = 1\)). Therefore, statement (C) is incorrect.
(D) The symmetry condition in 'h' and 'g parameter representation can be deduced by putting \(h_{11} = g_{11}\).
The symmetry condition for h-parameters is \(\Delta_h = 1\). The symmetry condition for g-parameters is \(\Delta_g = 1\). The interrelationship states \(g_{11} = \frac{h_{22}}{\Delta_h}\). Setting \(h_{11} = g_{11}\) means \(h_{11} = \frac{h_{22}}{\Delta_h}\). This is one equation relating \(h_{11}\), \(h_{22}\), and \(\Delta_h\). This single equation does not imply that \(\Delta_h = 1\) or \(\Delta_g = 1\). For instance, if \(\Delta_h = 2\), the condition becomes \(h_{11} = h_{22}/2\). This is not equivalent to \(\Delta_h = 1\). Therefore, statement (D) is incorrect.
Conclusion
Based on the analysis of each statement using the definitions and interrelationships of h and g parameters for two-port networks, statements (A) and (B) are correct.
Revision Table: Key Two-Port Network Conditions
| Parameter Type | Defining Equations | Reciprocity Condition | Symmetry Condition | Determinant |
|---|---|---|---|---|
| Hybrid (h) | \(V_1 = h_{11}I_1 + h_{12}V_2\) \(I_2 = h_{21}I_1 + h_{22}V_2\) |
\(h_{12} = -h_{21}\) | \(\Delta_h = 1\) | \(\Delta_h = h_{11}h_{22} - h_{12}h_{21}\) |
| Inverse Hybrid (g) | \(I_1 = g_{11}V_1 + g_{12}I_2\) \(V_2 = g_{21}V_1 + g_{22}I_2\) |
\(g_{12} = -g_{21}\) | \(\Delta_g = 1\) | \(\Delta_g = g_{11}g_{22} - g_{12}g_{21}\) |
Additional Information: Two-Port Network Parameters
Besides h and g parameters, other common representations for two-port networks include:
- Z-parameters (Impedance Parameters): Relate voltages to currents (\(V_1, V_2\) in terms of \(I_1, I_2\)). Reciprocity: \(z_{12} = z_{21}\). Symmetry: \(\Delta_z = 1\).
- Y-parameters (Admittance Parameters): Relate currents to voltages (\(I_1, I_2\) in terms of \(V_1, V_2\)). Reciprocity: \(y_{12} = y_{21}\). Symmetry: \(\Delta_y = 1\).
- ABCD Parameters (Transmission Parameters): Relate port 1 variables to port 2 variables (\(V_1, I_1\) in terms of \(V_2, -I_2\)). Primarily used for cascaded networks. Reciprocity: \(AD - BC = 1\). Symmetry: \(A = D\).
Conversions between any two sets of parameters are possible, provided the required determinant is non-zero. For instance, Z parameters are defined only when the network is open-circuited at both ports, and Y parameters when short-circuited at both ports. H and G parameters are suitable for networks that are neither open-circuited nor short-circuited at both ports simultaneously.
Important Questions from Two Port Networks
A Two - port network is reciprocal if and only if:
A passive 2-port network is in a steady-state. Compared to its input, the steady state output can never offer ________.
The condition for symmetric property of ABCD parameter of two port network is:
- When port 1 of two-port circuit is short-circuited, I 1 = 4I 2 & V 2 = 0.25 I 2 . Which of the following is true?
A two-port network has scattering parameters given \(\left[ s \right] = \left[ {\begin{array}{*{20}{c}} {{s_{11}}}&{{s_{12}}}\\ {{s_{211}}}&{{s_{22}}} \end{array}} \right]\). If the port 2 of the two-port is short-circuited, the s11 parameter for the resultant one-port network is
