CIRCUITS and SYSTEMS – part II Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 12 Transfer function concept

3 Definition of transfer function Transfer function is defined as the ratio of Laplace transform of output signal Y(s) and input signal X(s) at zero initial conditions Sometimes transfer function is denoted also by T(s)

4 Definition of transfer function (cont.) Voltage transfer function Current transfer function

5 Definition of transfer function (cont.) Voltage-to-current transfer function Current-to-voltage transfer function Special case of transfer function is the input impedance

6 Transfer function of RLC circuits Each RLC element has its operator description General form of transfer function

7 Impulse and step responses Impulse response is the time response of the circuit for Dirac impulse excitation at zero initial conditions Step response is the time response of the circuit for unity Heaviside excitation at zero initial conditions

8 Example Impulse response Step response Transfer function of the circuit is given in the form

9 Example (cont.) Impulse response Step response

10 Stability of linear circuits Stability BIBO (Bounded Input – Bounded Output): the circuit is stable if at bounded input excitation the output signal is also bounded at any time t. Dependence of stability on the placement of poles

11 Impulse response of 2nd order transfer function

12 Frequency characteristics Magnitude characteristics (magnitude of spectral function) Phase characteristics (phase of spectral function) Logarithmic magnitude characteristics Spectral transfer function is the frequency characteristics of the circuit. It represents the dependence of output signal on the frequncy at the sinusoidal input signal of unity magnitude and changing frequency.

13 Example Transfer function is given in the form Magnitude characteristics Linear and logarithmic form of magnitude characteristics

14 First order transfer functions 1) Integrator Frequency characteristics Magnitude and phase characteristics

15 First order transfer functions (cont.) 2) Differentiator Frequency characteristics Magnitude and phase characteristics

16 First order transfer functions (cont.) 3) Phase shifter Frequency characteristics Magnitude and phase characteristics

17 Frequency characteristics of nth order transfer function General form Frequency characteristics Magnitude and phase characteristics

18 Example Determine the voltage transfer function of the circuit. Assume: R=1 , L=2H, C=1F Solution: Operator form of the circuit

19 Example (cont.) Current I(s) Voltage transfer function Output voltage