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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 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 Y ( s) H ( s) X ( s) Sometimes transfer function is denoted also by T(s) 3 Definition of transfer function (cont.) Voltage transfer function U 2 ( s) H u (s) U1 (s) Current transfer function 4 I 2 (s) H i ( s) I1 ( s ) Definition of transfer function (cont.) Voltage-to-current transfer function H ui ( s) U 2 ( s) I1 ( s ) Current-to-voltage transfer function H iu ( s ) I 2 (s) U1 ( s) Special case of transfer function is the input impedance U1 ( s ) Z we ( s) I1 ( s ) 5 Transfer function of RLC circuits Each RLC element has its operator description Element Operator description Resistance R ZR R Inductance L Z L sL Mutual inductance M Z M sM Capacitance C ZC 1 sC General form of transfer function 6 L(s) bm s m bm1s m1 ... b1s b0 H ( s) n M ( s) s an1s n1 ... a1s a0 Impulse and step responses Impulse response is the time response of the circuit for Dirac impulse excitation at zero initial conditions H ( s) Y ( s) Y ( s) 1 1 Y ( s) H ( s) y(t ) L Y ( s) L H ( s) h(t ) X ( s) 1 Step response is the time response of the circuit for unity Heaviside excitation at zero initial conditions Y ( s) Y ( s) 1 H ( s) Y ( s) H ( s) X ( s) 1 / s s 7 1 y (t ) L Y ( s ) L H ( s ) s 1 1 Example Transfer function of the circuit is given in the form H ( s) 1 s 1s 5 Impulse response 1 1 st 1 st 1 t 1 5t h(t ) L lim s 1 e lim s 5 e e e s5 s 1 4 4 s 1s 5 1 Step response 8 1 1 st y (t ) L1 lim e s 0 s 1s 5 s s 1s 5 1 1 lim s 1 e st lim s 5 e st 0,2 0,25e t 0,05e 5t ss 5 ss 1 Example (cont.) Impulse response 9 Step response 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. 10 Dependence of stability on the placement of poles Impulse response of 2nd order transfer function 11 12 Frequency 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. H ( j ) H ( s) s j • Magnitude characteristics (magnitude of spectral function) H ( j ) • Phase characteristics (phase of spectral function) ( ) arg( H ( j ) • Logarithmic magnitude characteristics 20 log 10 H ( j) Example Transfer function is given in the form 0.003s 4 0.082 s 2 0.287 H ( s) 4 s 0,945s 3 1,487 s 2 0,778s 0,322 Magnitude characteristics 0.003 4 0.082 2 0.287 H ( j ) 4 1,487 2 0,322 j 0,945 3 0,778 13 Linear and logarithmic form of magnitude characteristics First order transfer functions 1) Integrator k H (s) s Frequency characteristics k k j 90 H ( j ) e j Magnitude and phase characteristics H ( j ) k ( ) 90 14 , First order transfer functions (cont.) 2) Differentiator H ( s ) ks Frequency characteristics H ( j) kj ke Magnitude and phase characteristics H ( j ) k, ( ) 90 15 j 90 First order transfer functions (cont.) 3) Phase shifter sa H (s) sa Frequency characteristics j a 2 a 2 e j j 2 H ( j ) 1 e , ( ) arctg j 2 2 j a a a e Magnitude and phase characteristics H ( j ) 1, ( ) 2 arctg a 16 Frequency characteristics of nth order transfer function General form bm s m bm1s m1 ... b1s b0 H ( s) an s n an 1s n 1 ... a1s a0 Frequency characteristics b j bm1 j ... b1 j b0 H ( j ) m A( ) jB( ) n n 1 an j an 1 j ... a1 j a0 m 1 m Magnitude and phase characteristics B( ) H ( j ) A ( ) B ( ) , ( ) arctg A( ) 2 17 2 Example Determine the voltage transfer function of the circuit. Assume: R=1, L=2H, C=1F Solution: Operator form of the circuit 18 Example (cont.) Current I(s) I (s) U1 ( s ) sC 2 U1 ( s ) R sL 1 / sC s LC sRC 1 Output voltage 1 1 U 2 ( s) I (s) 2 U1 ( s ) sC s LC sRC 1 Voltage transfer function 19 1 U 2 (s) LC H u ( s) , U1 ( s) s 2 s R 1 L LC 0,5 H u ( s) 2 s 0,5s 0,5