1. Lect15EEE 2021 Systems Concepts Dr. Holbert March 19, 2008

2. Lect15EEE 2022 Introduction Several important topics today, including: –Transfer function –Impulse response –Step response –Linearity and time invariance

3. Lect15EEE 2023 Transfer Function The transfer function, H(s), is the ratio of some output variable (y) to some input variable (x) The transfer function is portrayed in block diagram form as H(s) ↔ h(t) X(s) ↔ x(t)Y(s) ↔ y(t) InputOutput System

4. Lect15EEE 2024 Common Transfer Functions The transfer function, H(s), is bolded because it is a complex quantity (and it’s a function of frequency, s = jω) Since the transfer function, H(s), is the ratio of some output variable to some input variable, we may define any number of transfer functions –ratio of output voltage to input voltage (i.e., voltage gain) –ratio of output current to input current (i.e., current gain) –ratio of output voltage to input current (i.e., transimpedance) –ratio of output current to input voltage (i.e., transadmittance)

5. Lect15EEE 2025 Finding a Transfer Function Laplace transform the circuit (elements) –When finding H(s), all initial conditions are zero (makes transformation step easy) Use appropriate circuit analysis methods to form a ratio of the desired output to the input (which is typically an independent source); for example:

6. Lect15EEE 2026 Transfer Function Example C +–+– v in (t) R v out (t) + – 1/sC +–+– V in (s) R V out (s) + – Time DomainFrequency Domain Using voltage division, we find the transfer function

7. Lect15EEE 2027 Transfer Function Use We can use the transfer function to find the system output to an arbitrary input using simple multiplication in the s domain Y(s) = H(s) X(s) In the time domain, such an operation would require use of the convolution integral:

8. Lect15EEE 2028 Impulse Response Let the system input be the impulse function: x(t) = δ(t); recall that X(s) = L [δ(t)] = 1 Therefore:Y(s) = H(s) X(s) = H(s) The impulse response, designated h(t), is the inverse Laplace transform of transfer function y(t) = h(t) = L -1 [H(s)] With knowledge of the transfer function or impulse response, we can find the response of a circuit to any input

9. Lect15EEE 2029 (Unit) Step Response Now, let the system input be the unit step function:x(t) = u(t) We recall that X(s) = 1/s Therefore: Using inverse Laplace transform skills, and a specific H(s), we can find the step response, y(t)

10. Lect15EEE 20210 Step Response from Convolution We could also use the convolution integral in combination with the impulse response, h(t), to find the system response to any other input Either form of the convolution integral above can be used, but generally one expression leads to a simpler, or more interpretable, result We shall use the first formulation here

11. Lect15EEE 20211 Impulse – Step Response Relation The step input function is The convolution integral becomes We observe that the step response is the time integral of the impulse response

12. Lect15EEE 20212 (Unit) Ramp Response Besides the impulse and step responses, another common benchmark is the ramp response of a system (because some physical inputs are difficult to create as impulse and step functions over small t) The unit ramp function ist·u(t) which has a Laplace transform of1/s 2 The ramp response is the time integral of the unit step response

13. Lect15EEE 20213 Pole-Zero Plot For a pole-zero plot place "X" for poles and "0" for zeros using real-imaginary axes Poles directly indicate the system transient response features Poles in the right half plane signify an unstable system Consider the following transfer function Re Im

14. Lect15EEE 20214 Linearity Linearity is a property of superposition αx 1 (t) + βx 2 (t) → αy 1 (t) + βy 2 (t) A system with a constant (additive) term is nonlinear; this aspect results from another property of linear systems, that is, a zero input to a linear system results in an output of zero Circuits that have non-zero initial conditions are nonlinear An RLC circuit initially at rest is a linear system

15. Lect15EEE 20215 Time-Invariant Systems In broad terms, a system that does not change with time is a time-invariant system; that is, the rule used to compute the system output does not depend on the time at which the input is applied The coefficients to any algebraic or differential equations must be constant for the system to be time-invariant An RLC circuit initially at rest is a time-invariant system

16. Lect15EEE 20216 Class Examples Drill Problems P7-1, P7-2, P7-4