1. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr1 Signals and Systems Introduction EEE393 Basic Electrical Engineering K.A.Peker Bilkent University

2. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 2 From Circuits to Signals and Systems

3. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 3 From Circuits to Signals and Systems Output(t) = System{ Input(t) } y(t) = H{ x(t) } + V in I out + V in I in + V out - + V out - H{} x(t)y(t)

4. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 4 Some System Properties Linear Systems:  y 1 (t) = H{ x 1 (t) }  y 2 (t) = H{ x 2 (t) } Time-Invariant Systems:  y(t) = H{ x(t) }  y(t - t 0 ) = H{ x(t - t 0 ) } same behavior at all times H { a · x 1 (t) + b · x 2 (t) } = a · y 1 (t) + b · y 2 (t)

5. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 5 Linear Time Invariant Circuits R, L, C  linear, time invariant elements  V = R·i  linear, time invariant  C and L are also linear, time invariant (diff. is linear): R-L-C circuits are also linear, time invariant systems

6. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 6 ODE Linear time-invariant (LTI) systems are usually modeled using constant coefficient, linear, ordinary differential (or integral) equations  Time-invariant  Linear system (no square etc. of derivatives) + v(t) R L i(t) C + v(t) R L i(t) + v(t) Ri(t) C

7. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 7 Same math: LTI systems from other fields M F F friction =D·v M F F spring = - k·x k M F k D  auto suspension system

8. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 8 Satellite control, riding a bicycle, process control, nuclear system control, … LTI systems from other fields (cont’d) A 1 : area 1 R2R2 R1R1 A 2 : area 2 h1h1 h2h2 u(t) input Output (OR “state variables”) R h

9. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 9 LTI systems from other fields (cont’d) (Arms race)  X: arms production of country 1  Y: arms production of country 2 competition current strength

10. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 10 LTI systems from other fields (cont’d) (Oligopoly/duopoly) – economics, game theory  Two companies in a market, producing same product If change in production is proportional to marginal change in profit

11. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 11 So… RLC circuits are …linear time-invariant systems …described by linear ordinary differential equations (ODE) …which are used to model many other systems in different sciences and engineering fields In fact, they’re mathematically same as many other physical/social/economic etc. systems (analogous) Hence, the math and the techniques we learn here apply to many other problems in other domains

12. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 12 Response of LTI Systems - Response to complex exponentials - AC circuit analysis

13. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 13 How do we handle these diff. equations? There are various ways to approach this problem  We will start with Steady state response … to complex exponentials There are two components of the solution (Homogenous equation) x t : Complementary solution, Natural response (systems own nature), Transient response (usually dies away) x s : Particular solution, Steady state response

14. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 14 The complex exponential Everything starts with the simple but important fact that; Corollaries:  If we assume complex exponential input, then all currents/voltages in an RLC circuit are also complex exponentials  Differential equations become algebraic (polynomial) equations Derivative of a complex exponential is itself, multiplied by a constant Complex exponentials are eigen functions of LTI systems Linear systems – vector spaces – basis expansion Laplace domain – exp(st), Fourier domain – exp(jw)

15. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 15 The complex exponential (cont’d) From diff. equations to polynomial equations Note: Complex exp. input  Complex exp. output (works both ways)

16. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 16 Complex exponentials with (s=j  ) Let’s assume our inputs are in the form: Why? Because …   and, cos(  t+  ) is AC circuits (I.e. very important practical case) Or, more generally (linearity, superposition) Special case of for Phasor Can obtain AC response by summing complex exp. responses (shortcut: use Re{.} operation)

17. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 17 Element characteristics for e j  t input R: C: L: Phasor Only amplitude change (attenuated) v=R·i still holds. Same signal, multiplied by j  C Like v=R·i. Ratio is a constant, independent of time. Can solve like a DC (constant) circuit.

18. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 18 Impedance For complex exponential v(t), i(t), with frequency   Capacitor characteristics C:  Inductor characteristics L:  and are just like resistance values (constant ratio), but are complex values  We call this ratio of complex v to i, ( ), “Impedance” (Z)

19. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 19 Impedance (cont’d) R: Resistive part X: Reactance |Z|: Magnitude (how it attenuates the input – change on input amplitude)  z : Phase (change on input phase) NOTE: A LTI system changes only TWO things in a complex exponential (or a sinusoid): - Amplitude (multiplied by magnitude of transfer function, |H(  )|) - Phase (shifted by phase of transfer function,  H ) Multiply magnitudes, add phase angles Impedance can be purely real (Resistance), purely imaginary (Capacitance or Inductance), or a mix (mixed circuits).

20. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 20 A closer look at C, L impedances C: L: Negative Reactance: “Capacitive reactance” Positive Reactance: “Inductive reactance” Magnitude inversely proportional to . Pass high frequencies, stop low (in: v, output: i) Open circuit for DC, short circuit for high freq. Magnitude proportional to . Pass low frequencies, stop high (in: v, output: i) Short circuit for DC, open circuit for high freq. Phase shifted by –90 0 (voltage 90 0 behind current) Phase shifted by 90 0 (voltage 90 0 ahead of current) v = Z·i

21. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 21 AC circuit analysis For a given input AC voltage (or current):  Find corresponding complex signal V (or I), such that:  Replace all C with and all L with,  Solve like a DC circuit (reductions, equivalent Z, loop currents, node voltages, superposition, etc),  Find complex output I (or V)  Convert back to real world by Phasor

22. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 22 AC circuit analysis - phasors A small simplification: Phasor carries the amplitude and phase information LTI system (i.e. RLC circuit) changes only the amplitude and phase Phasor For single  AC analysis, we can use only the phasors of voltages and currents Replace voltage (or current) sources with corresponding phasors When multiplying/dividing with Z, multiply/divide magnitudes and amplitudes, add/subtract phase angles Convert output phasor back to a sinusoid

23. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 23 Simple AC circuits Series RC: Series RL: + v(t) Ri(t) C + v(t) Ri(t) L Transfer Function

24. Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker kpeker@bilkent.edu.tr 24 Simple AC circuits Series RLC: Parallel RLC: + v(t) R L i(t) C + v(t) RL i(t) C C L Transfer Function