1. 6. 8. 20031 V–2 AC Circuits

2. 6. 8. 20032 Main Topics Power in AC Circuits. R, L and C in AC Circuits. Impedance. Description using Phasors. Generalized Ohm’s Law. Serial RC, RL and RLC AC Circuits. Parallel RC, RL and RLC AC Circuits. The Concept of the Resonance.

3. 6. 8. 20033 The Power The power at any instant is a product of the voltage and current: P(t) = V(t) I(t) = V 0 sin(  t)I 0 sin(  t +  ) The mean value of power depends on the phase shift between the voltage and the current:mean = V rms I rms cos  The quality cos  is called the power factor.

4. 6. 8. 20034 AC Circuit with R Only If a current I(t) = I 0 sin  t flows through a resistor R Ohm’s law is valid at any instant. The voltage on the resistor will be in-phase:resistor V(t) = RI 0 sin  t = V 0 sin  t V 0 = RI 0 = V rms I rms = RI rms 2 = V rms 2 /R We define the impedance of the resistor : X R = R

5. 6. 8. 20035 AC Circuit with L Only I If a current I(t) = I 0 sin  t supplied by some AC power-source flows through an inductance L Kirchhoff’s law is valid in any instant: inductance V(t) – LdI(t)/dt =0 This gives us the voltage on the inductor: V(t) =  LI 0 cos  t = V 0 sin(  t+  /2) V 0 =  LI 0

6. 6. 8. 20036 AC Circuit with L Only II There is a phase-shift between the voltage and the current on the inductor. The current is delayed by  =  /2 behind the voltage. The mean power now will be zero: = V rms I rms cos  = 0 We define the impedance of the inductance: X L =  L  V 0 = I 0 X L

7. 6. 8. 20037 AC Circuit with L Only III Since the impedance, in this case the inductive reactance, is a ratio of the peak (and also rms) values of the voltage over current we can regard it as a generalization or the resistance. Note the dependence on  ! The higher is the frequency the higher is the impedance.

8. 6. 8. 20038 AC Circuit with C Only I If a current I(t) = I 0 sin  t supplied by some AC power-source flows through an capacitor C Kirchhoff’s law is valid in any instant: capacitor V(t) – Q(t)/C =0 This is an integral equation for voltage:integral V(t) = –I 0 /  C cos  t = V 0 sin(  t –  /2) V 0 = I 0 /  C

9. 6. 8. 20039 AC Circuit with C Only II There is a phase-shift between the voltage and the current on the inductor. The voltage is delayed by  =  /2 behind the current. The mean power now will be again zero: = V rms I rms cos  = 0 We define the impedance of the capacitor: X C = 1/  C  V 0 = I 0 X C

10. 6. 8. 200310 AC Circuit with C Only III Since the impedance, in this case the capacitive reactance, is a ratio of the peak (and rms) values of the voltage over current we can regard it again as a generalization or the resistance. Note the dependence on  ! Here, the higher is the frequency the lower is the impedance.

11. 6. 8. 200311 A Loudspeaker Cross-over The different frequency behavior of the impedances of an inductor and a capacitor can be used in filters and for instance to simply separate sounds in a loud-speaker.filtersloud-speaker high-frequency speaker ‘a tweeter’ is connected is series with an capacitor. low-frequency speaker ‘a woofer’ is connected is series with an inductance.

12. 6. 8. 200312 General AC Circuits I If there are more R, C, L elements in an AC circuit we can always, in principle, build appropriate differential or integral equations and solve them. The only problem is that these equations would be very complicated even in very simple situations. There are, fortunately, several ways how to get around this more elegantly.

13. 6. 8. 200313 General AC Circuits II AC circuits are a two-dimensional problem. If we supply any AC circuit by a voltage V 0 sin  t, the time dependence of all the voltages and currents in the circuit will also oscillate with the same  t but possibly different phase. So it is necessary and sufficient to describe any quantity by two parameters its phase and magnitude.

14. 6. 8. 200314 General AC Circuits III There are two mathematical tools commonly used: Two-dimensional vectors, so called, phasors in a coordinate system which rotates with  t so all the phasors, which also rotate, are still Complex numbers in Gauss plane. This is preferred since more operations (e.g. division, roots) are defined for complex numbers.

15. 6. 8. 200315 General AC Circuits IV The description by both ways is similar: The magnitude of particular quality (voltage or current) is described by a magnitude of a phasor (vector) or an absolute value of a complex number and the phase is described by the angle with the positive x-axis or a real axis.

16. 6. 8. 200316 General AC Circuits V The complex number approach: Describe voltages V, currents I, impedances Z and admittances Y = 1/Z by complex numbers. Then a general complex Ohm’s law is valid: V = ZI or I=YV Serial combination: Z s = Z 1 + Z 2 + … Parallel combination Y p = Y 1 + Y 2 + … Kirchhoff’s laws are valid for complex I and V

17. 6. 8. 200317 General AC Circuits VI The table of complex impedances and admitancess of ideal elements R, L, C, j is the imaginary unit j 2 = -1: R: Z R = RY R = 1/R L:Z L = j  L Y L = -j/  L C:Z C = -j/  C Y C = j  C

18. 6. 8. 200318 RC in Series Let’s illustrate the complex number approach on a serial RC combination: Let I, common for both R and C, be real. Z = Z R + Z C = R – j/  C |Z| = (ZZ*) 1/2 = (R 2 + 1/  2 C 2 ) 1/2 tg  = –1/  RC < 0 … capacity like

19. 6. 8. 200319 RL in Series Let’s have a R and L in series: Let I, common for both R and L, be real. Z = Z R + Z C = R + j  L |Z| = (ZZ*) 1/2 = (R 2 +  2 L 2 ) 1/2 tg  =  L/R > 0 … inductance like

20. 6. 8. 200320 RC in Parallel Let’s have a R and L in parallel: Let V, common for both R and C, be real. Y = Y R + Y C = 1/R + j  C |Y| = (YY*) 1/2 = (1/R 2 +  2 C 2 ) 1/2 tg  = –[  C/R] < 0 … again capacity like

21. 6. 8. 200321 RLC in Series I Let’s have a R, L and C in series: Let again I, common for all R, L, C be real. Z = Z R + Z C + Z L = R + j(  L - 1/  C) |Z| = (R 2 + (  L - 1/  C) 2 ) 1/2 The circuit can be either inductance-like if:  L > 1/  C …  > 0 or capacitance-like:  L < 1/  C …  < 0

22. 6. 8. 200322 RLC in Series II New effect of resonance takes place when:resonance  L = 1/  C   2 = 1/LC Then the imaginary parts cancel and the whole circuit behaves as a pure resistance: Z, V have minimum, I maximum It can be reached by tuning L, C or f !

23. 6. 8. 200323 RLC in Parallel I Let’s have a R, L and C in parallel: Let now V, common for all R, L, C be real. Y = Y R + Y C + Y L = 1/R + j(  C - 1/  L) |Y| = (1/R 2 + (  C - 1/  L) 2 ) 1/2 The circuit can be either inductance-like if:  L > 1/  C …  > 0 or capacitance-like:  L < 1/  C …  < 0

24. 6. 8. 200324 RLC in Parallel II Again the effect of resonance takes place when the same condition is fulfilled:  L = 1/  C   2 = 1/LC Then the imaginary parts cancel and the whole circuit behaves as a pure resistance: Y, I have minimum, Z,V have maximum It can be reached by tuning L, C or f !

25. 6. 8. 200325 Resonance General description of the resonance: If we need to feed some system capable of oscillating on its frequency  0 then we do it most effectively if the frequency our source  matches  0 and moreover is in phase. Good mechanical example is a swing. The principle is used in e.g. in tuning circuits of receivers.

26. 6. 8. 200326 Impedance MatchingMatching From DC circuits we already know that if we need to transfer maximum power between two circuits it is necessary that the output resistance of the first one matches the input resistance of the next one. In AC circuits we have to match (complex) impedances the same way. Unmatched phase may lead to reflection!

27. 6. 8. 200327 Homework Chapter 31 – 1, 2, 3, 4, 7, 12, 13, 24, 25, 40.

28. 6. 8. 200328 Things to read and learn This lecture covers: The rest of Chapter 31 Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!

29. The Mean Power I We choose the representative time interval  = T:

30. The Mean Power II Since only the first integral in non-zero. ^

31. AC Circuit with C I From definition of the current I = dQ/dt and relation for a capacitor V c = Q(t)/C: The capacitor is an integrating device. ^

32. LC Circuit I We use definition of the current I = -dQ/dt and relation of the charge and voltage on a capacitor V c = Q(t)/C: We take into account that the capacitor is discharged by the current. This is homogeneous differential equation of the second order. We guess the solution.

33. LC Circuit II Now we get parameters by substituting into the equation: These are un-dumped oscillations.

34. LC Circuit III The current can be obtained from the definition I = - dQ/dt: Its behavior in time is harmonic. ^

35. LC Circuit IV The voltage on the capacitor V(t) = Q(t)/C: is also harmonic but note, there is a phase shift between the voltage and the current. ^