1 Lecture #18 EGR 272 – Circuit Theory II Transfer Functions (Network Functions) A transfer function, H(s), can be used to describe a system or circuit in the s-domain in terms of its input and output as illustrated below. Read: Chapter 13 in Electric Circuits, 6 th Edition by Nilsson

2 Lecture #18 EGR 272 – Circuit Theory II H(s) is defined more specifically as: Notes: Transfer functions are always defined with zero initial condition (or zero initial stored energy). Therefore, V C (0) = I L (0) = 0, so the voltage sources in the models for capacitors and inductors disappear. Y(s) and X(s) typically represent voltages or currents. The input and the output must be designated by the user (there might be many possible transfer functions for a given circuit or system). However, most circuits or systems have well-defined inputs and outputs. H(s) completely characterizes the circuit or system. Once H(s) is known, you can calculate circuit outputs for various inputs without ever seeing the circuit or system again.

3 Lecture #18 EGR 272 – Circuit Theory II Example: Find H(s) for the circuit shown below if H(s) is defined as:

4 Lecture #18 EGR 272 – Circuit Theory II So we can determine the output of a circuit by taking the inverse Laplace transform of the product of the transfer function and the Laplace transform of the input to the circuit. Finding the output, y(t) using H(s) and the input, x(t) Example: Find the output, y(t) for the circuit below if the input is x(t) = 25u(t)V. x(t) + y(t) - + -

5 Lecture #18 EGR 272 – Circuit Theory II Example: Find the output, V o, for the circuit below if the input is V i = 10 u(t)V using the transfer function H(s) found in a previous example.

6 Lecture #18 EGR 272 – Circuit Theory II Unit Step Response and Impulse Response Although H(s) can be used to find the output for any given input, there are two special cases that are often of interest: 1) Impulse response - the output when the input is  (t) 2) Unit step response - the output when the input is u(t) Impulse response – the output to a circuit when the input x(t) =  (t) If x(t) =  (t), then X(s) = 1, so Y(s) = H(s)  1 = H(s), so y(t) = L –1 {H(s)} = h(t). Unit step response – the output to a circuit when the input x(t) = u(t)

7 Lecture #18 EGR 272 – Circuit Theory II Example: For the circuit shown: A) Find the transfer function H(s) = I(s)/V(s)

8 Lecture #18 EGR 272 – Circuit Theory II B) Use the transfer function to determine the output if v(t) = 10e -2t V C) Use the transfer function to determine the impulse response D) Use the transfer function to determine the unit step response