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In electrical engineering, analyzing complex circuits can often be simplified by breaking them down into smaller, manageable parts. One such powerful tool is the two port network. A two port network is an electrical network or device with two pairs of terminals (ports) through which electrical signals enter and exit. This abstraction allows engineers to model and analyze components like amplifiers, filters, and transmission lines efficiently.
Two port networks are widely used in communication systems, power electronics, and signal processing. They help in understanding how input signals are transformed into output signals, characterizing devices by parameters, and simplifying the analysis of interconnected systems.
A two port network is a black box with two distinct pairs of terminals called ports. Each port consists of two terminals where voltages and currents are defined. The first port is called the input port and the second the output port.
For a two port network, the following variables are defined:
Currents \( I_1 \) and \( I_2 \) are conventionally taken as entering the network at their respective ports.
Port conditions ensure that each port behaves like a two-terminal device, meaning the current entering one terminal of a port must leave through the other terminal. This condition is crucial for defining parameters consistently.
Port conditions guarantee that the network behaves predictably, allowing us to define parameters relating voltages and currents at the ports without ambiguity. Violating these conditions would make parameter definitions invalid and analysis inconsistent.
The impedance parameters, or Z-parameters, relate the voltages at the ports to the currents entering the ports using linear equations. They are defined as:
\[ \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \]
Here, each \( Z_{ij} \) is an impedance parameter measured in ohms (Ω).
Measurement of Z-parameters: To find these parameters practically, the output port is kept open (no current flows out), and input currents or voltages are applied to measure corresponding voltages and currents.
Z-parameters are intuitive because they relate voltages directly to currents, similar to Ohm's law. They are especially useful when dealing with series-connected networks or when open-circuit conditions are easy to realize.
Admittance parameters or Y-parameters relate the currents at the ports to the voltages at the ports:
\[ \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} \]
Each \( Y_{ij} \) is an admittance parameter measured in Siemens (S), which is the reciprocal of ohms.
Measurement of Y-parameters: To measure Y-parameters, the output port is short-circuited (voltage across it is zero), and currents and voltages are measured accordingly.
Y-parameters are convenient when dealing with parallel-connected networks or admittance-based circuit analysis. Short circuit conditions are often easier to realize in practical measurements, making Y-parameters popular in certain applications.
Hybrid parameters or h-parameters mix voltage and current variables to relate input voltage and output current to input current and output voltage:
\[ \begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} \]
Each parameter has a specific physical meaning:
h-parameters are especially useful in modeling transistor amplifiers, where input and output variables are naturally mixed.
Because h-parameters mix voltage and current variables, they fit well with transistor input-output relations, making them ideal for amplifier analysis and design.
Transmission parameters or ABCD-parameters relate input voltage and current to output voltage and current as:
\[ \begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix} \]
These parameters are useful for analyzing cascaded networks because the overall ABCD matrix of cascaded two port networks is the product of their individual ABCD matrices.
ABCD-parameters simplify the analysis of cascaded networks, such as transmission lines and multi-stage amplifiers, by allowing straightforward multiplication of parameter matrices to find overall behavior.
Since different parameter sets describe the same two port network, it is often necessary to convert between them. The most common conversions are:
These conversions require careful matrix algebra and attention to units.
Step 1: To find \( Z_{11} \), set output port open-circuited (\( I_2 = 0 \)) and apply a test current \( I_1 \) at input port. Calculate \( V_1 \) and find \( Z_{11} = \frac{V_1}{I_1} \).
Step 2: For \( Z_{12} \), set \( I_1 = 0 \) (input port open) and apply \( I_2 \) at output port. Measure \( V_1 \) and calculate \( Z_{12} = \frac{V_1}{I_2} \).
Step 3: Similarly, find \( Z_{21} = \frac{V_2}{I_1} \) with \( I_2=0 \), and \( Z_{22} = \frac{V_2}{I_2} \) with \( I_1=0 \).
Step 4: Using circuit analysis (Ohm's law and series-parallel resistor combinations), calculate each parameter:
Answer:
\[ \mathbf{Z} = \begin{bmatrix} 22 & 12 \\ 12 & 27.5 \end{bmatrix} \Omega \]
Step 1: Recall that \( \mathbf{Y} = \mathbf{Z}^{-1} \), so we need to find the inverse of the matrix \( \mathbf{Z} \).
Step 2: Calculate the determinant of \( \mathbf{Z} \):
\[ \det(\mathbf{Z}) = (4)(3) - (2)(2) = 12 - 4 = 8 \]
Step 3: Find the inverse matrix:
\[ \mathbf{Z}^{-1} = \frac{1}{8} \begin{bmatrix} 3 & -2 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.375 & -0.25 \\ -0.25 & 0.5 \end{bmatrix} \, \text{S} \]
Answer:
\[ \mathbf{Y} = \begin{bmatrix} 0.375 & -0.25 \\ -0.25 & 0.5 \end{bmatrix} \, \text{Siemens} \]
Step 1: Use the cascading formula:
\[ [\mathbf{ABCD}]_{total} = [\mathbf{ABCD}]_1 \times [\mathbf{ABCD}]_2 \]
Step 2: Multiply the matrices:
\[ \begin{bmatrix} 1 & 50 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 2 & 0 \\ 0.02 & 0.5 \end{bmatrix} = \begin{bmatrix} (1)(2) + (50)(0.02) & (1)(0) + (50)(0.5) \\ (0)(2) + (1)(0.02) & (0)(0) + (1)(0.5) \end{bmatrix} \]
\[ = \begin{bmatrix} 2 + 1 & 0 + 25 \\ 0 + 0.02 & 0 + 0.5 \end{bmatrix} = \begin{bmatrix} 3 & 25 \\ 0.02 & 0.5 \end{bmatrix} \]
Answer: The overall ABCD matrix is:
\[ [\mathbf{ABCD}]_{total} = \begin{bmatrix} 3 & 25 \\ 0.02 & 0.5 \end{bmatrix} \]
Step 1: Given the h-parameter relations:
\[ V_1 = h_{11} I_1 + h_{12} V_2, \quad I_2 = h_{21} I_1 + h_{22} V_2 \]
Step 2: The load connected at output port is \( R_L = 1\,k\Omega \), so:
\[ V_2 = -I_2 R_L \]
Step 3: Substitute \( I_2 \) into the load equation:
\[ V_2 = -R_L (h_{21} I_1 + h_{22} V_2) \implies V_2 + R_L h_{22} V_2 = -R_L h_{21} I_1 \]
\[ V_2 (1 + R_L h_{22}) = -R_L h_{21} I_1 \implies V_2 = \frac{-R_L h_{21}}{1 + R_L h_{22}} I_1 \]
Step 4: Substitute \( V_2 \) back into the first equation:
\[ V_1 = h_{11} I_1 + h_{12} V_2 = h_{11} I_1 + h_{12} \left( \frac{-R_L h_{21}}{1 + R_L h_{22}} I_1 \right) = I_1 \left( h_{11} - \frac{h_{12} R_L h_{21}}{1 + R_L h_{22}} \right) \]
Step 5: Input impedance \( Z_{in} = \frac{V_1}{I_1} \):
\[ Z_{in} = h_{11} - \frac{h_{12} R_L h_{21}}{1 + R_L h_{22}} \]
Substitute values (convert units where needed):
Calculate denominator:
\[ 1 + R_L h_{22} = 1 + 1000 \times 0.00002 = 1 + 0.02 = 1.02 \]
Calculate numerator:
\[ h_{12} R_L h_{21} = 2 \times 1000 \times 50 = 100,000 \]
Therefore:
\[ Z_{in} = 1000 - \frac{100,000}{1.02} \approx 1000 - 98,039 = -97,039\,\Omega \]
Negative input impedance indicates active behavior (amplification).
Step 6: Voltage gain \( A_v = \frac{V_2}{V_1} \). From step 3:
\[ V_2 = \frac{-R_L h_{21}}{1 + R_L h_{22}} I_1 \]
From step 5:
\[ V_1 = Z_{in} I_1 \]
So:
\[ A_v = \frac{V_2}{V_1} = \frac{-R_L h_{21}}{(1 + R_L h_{22}) Z_{in}} = \frac{-1000 \times 50}{1.02 \times (-97039)} \approx 0.5 \]
Answer: Input impedance \( Z_{in} \approx -97\,k\Omega \) (active), voltage gain \( A_v \approx 0.5 \).
Step 1: Recall the Z-parameter equations:
\[ V_1 = Z_{11} I_1 + Z_{12} I_2, \quad V_2 = Z_{21} I_1 + Z_{22} I_2 \]
Step 2: For \( I_2 = 0 \), measurements give:
\[ V_1 = Z_{11} I_1 \implies Z_{11} = \frac{V_1}{I_1} = \frac{10}{2} = 5\,\Omega \]
\[ V_2 = Z_{21} I_1 \implies Z_{21} = \frac{V_2}{I_1} = \frac{5}{2} = 2.5\,\Omega \]
Step 3: For \( I_1 = 0 \), measurements give:
\[ V_1 = Z_{12} I_2 \implies Z_{12} = \frac{V_1}{I_2} = \frac{4}{1} = 4\,\Omega \]
\[ V_2 = Z_{22} I_2 \implies Z_{22} = \frac{V_2}{I_2} = \frac{8}{1} = 8\,\Omega \]
Answer:
\[ \mathbf{Z} = \begin{bmatrix} 5 & 4 \\ 2.5 & 8 \end{bmatrix} \Omega \]
When to use: When deciding which parameter set to use or how to measure parameters practically.
When to use: While converting between Z and Y parameters to avoid calculation mistakes.
When to use: When combining multiple two port networks in series.
When to use: To reduce complexity and save time during problem solving.
When to use: During quick revision or while attempting parameter measurement questions.
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