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A transfer function is a mathematical representation used in control theory and signal processing to describe the relationship between the input and output of a system, typically a linear time-invariant system. It is a key concept in fields like electrical engineering, mechanical engineering, and control systems engineering. The transfer function is particularly useful for analyzing and designing systems in the frequency domain.
In its simplest form, a transfer function relates the Laplace transform of the output to the Laplace transform of the input in a linear time-invariant system. The Laplace transform is used to analyze systems in the frequency domain, making it easier to understand how a system responds to different frequencies of input signals.
The general form of a transfer function is represented as:
H(s) = Y(s) / X(s)
Where:
H(s) is the transfer function of the system.
Y(s) is the Laplace transform of the system's output.
X(s) is the Laplace transform of the system's input.
s is a complex variable representing complex frequency.
The transfer function describes the system's behavior, showing how different frequencies of input signals are attenuated or amplified by the system and how phase shifts occur at various frequencies. By analyzing the transfer function, engineers can determine the system's stability, frequency response, and overall performance characteristics.
Voltage transfer function can be expressed as the ratio of two cofactors of the admittance matrix:
where a is the input node, b is the output node, Δab is the cofactor of the admittance matrix with removed row a and removed column b, Δaa is the cofactor of the admittance matrix with removed row and column a.
Example:
Circuit Description:
The circuit being discussed has three nodes. In circuit analysis, a node is a point in the circuit where two or more components are connected. Nodes are important for formulating the system of equations that describe the circuit behavior.
Creation of a 3x3 Matrix:
Since there are three nodes in this circuit, an empty 3x3 matrix will be created. This matrix is known as the nodal admittance matrix (Y-matrix). The elements of this matrix will be populated with the conductance (or admittance) values corresponding to the components in the circuit.
Node Connections and Matrix Stamps:
The elements in the matrix are filled using their stamps. A "stamp" in circuit analysis is a numerical value that represents a component's contribution to the matrix. The term "stamps" often refers to the values derived from circuit elements based on the nodal analysis technique.
Resistor R1 is connected between nodes 1 and 2. The stamp for a resistor between two nodes in the admittance matrix is typically 1/R1(where R is the resistance). So, the value1/R1 would be placed in the matrix at the locations corresponding to nodes 1 and 2, and also at nodes 2 and 1 (since the matrix is symmetric).
Resistor R2 and capacitor C1 are connected between nodes 0 and 1. Similar to R1, the stamps for R2 and C1 would be placed in the matrix at the appropriate positions for nodes 0 and 1, and vice versa.
In the Python code the inclusion of stamps is done using the addRES and addCAP functions.
Base Node and Removal:
The base node in this example is labeled with "0." The base node is often chosen for reference in nodal analysis. After including the elements in the matrix, the base node is no longer needed for analysis, so it is removed. This is a common practice in nodal analysis. Removal of the base node is done in remCR function is the Python code
Finding the transfer function
Since the Y-matrix is now known the two cofactors can be found. the input node “a” is number 2, and the output node “b” is number 1. To find the numerator of the transfer function it is necessary to remove row 2 and column 1 and find the determinant of the resulting matrix. For the denominator the procedure is similar, row 2 and column 2 are removed and the determinant is calculated. Final step is the find the ratio between the two cofactors.
Finding the cofactors of the Y-matrix and their ratio is done in the Av function in the Python code.
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