Circuit Theory/Circuit Functions

Circuit Functions
This appendix page will list the various values of the variable H that have been used throughout the circuit theory textbooks. These values of H are all equivalent, but are represented in different domains. All of the H functions are a ratio of the circuit input over the circuit output.

The "Impulse Response"
The impulse response is the time-domain relationship between the circuit input and the circuit output, denoted with the following notation:


 * $$h(t)$$

The impulse response is, strictly speaking, the output that the circuit will produce when an ideal impulse function is the input. The impulse response can be used to determine the output from the input through the convolution operation:


 * $$y(t) = h(t) * x(t)$$

The "Network Function"
The network function is the phasor-domain representation of the impulse response. The network function is denoted as such:


 * $$\mathbb{H}(\omega)$$

The network function is related to the input and output of the circuit through the following relationships:


 * $$\mathbb{Y}(\omega) = \mathbb{H}(\omega) \mathbb{X}(\omega)$$

Similarly, the network function can be received by dividing the output by the input, in the phasor domain.

The "Transfer Function"
The transfer function is the laplace-transformed representation of the impulse response. It is denoted with the following notation:


 * $$H(s)$$

The transfer function can be obtained by one of two methods:


 * 1) Transform the impulse response.
 * 2) Transform the circuit, and solve.

The Transfer function is related to the input and output as follows:


 * $$Y(s) = H(s) X(s)$$

The "Frequency Response"
The Frequency Response is the fourier-domain representation of the impulse response. It is denoted as such:


 * $$H(j \omega)$$

The frequency response can be obtained in one of three ways:


 * 1) Transform the impulse response
 * 2) Transform the circuit and solve
 * 3) Substitute $$s = j \omega$$ into the transfer function

The frequency response has the following relationship to the circuit input and output:


 * $$Y(j \omega) = H(j \omega) X(j \omega)$$

The frequency response is particularly useful when discussing a sinusoidal input, or when constructing a bode diagram.