1. Capacitor • A capacitor is a device that stores charge • Capacitor is like reservoir that stores water • Capacitors differs in their capacity in storing charges Insulator (vacuum, ceramic, plastic, metal oxide) conductor

2. Charge up a capacitor • Initially VC = 0, I R VC=0 V electrons

3. Because of the insulator in the capacitor, electrons moving into the –ve side are trapped • the –ve side becomes –ve charged, +ve voltage side becomes +ve charged, • VC across capacitor charge stored

4. Capacitor • Water analogy

5. Definition of Capacitor • Tub stores water, capacitor stores charge • C = Tub size • Q = Volume of water • V = Height of water level • I = Rate of flow of water • If a capacitor contains Q amount of water (charges), and the height of water (V), then the size is given by

6. Capacitor • C = size of the capacitor Same charge (Q) but different V V V Large cap Small cap

7. Capacitors in parallel • What is the combined capacitance C? • Two tubs in parallel, can store more water, C should be larger C? V V = C2 C1

8. Capacitor in parallel (C = C1+C2 ) • Parallel circuit, same voltage across C1 and C2 • Q1 = C1V, Q2 = C2V • Total charge Q=Q1+Q2=(C1+C2)V • C = = C1+ C2 V C2 C1 Q Same V for C1 and C2 V V C2 C1

9. Capacitors in series • What is the combined capacitance C? C1 C? V V = C2

10. Capacitor in series • Same current • Therefore same amount of Q in both capacitors I C1 V1 I V Same Q V depends on smaller cap V2 C2

11. Capacitor in series

12. Resistor in series : R = R1 + R2 • Resistor in parallel : • Capacitor in series : • Capacitor in parallel : C = C1 + C2

13. Capacitor • Dump dQ amount of water to a tub of size C, so that the water level is changed by dV • But • Hence • i.e. the rate of change of the height of water is proportional to the current (inflow of water)

14. Constant I • Rate of increase of V = I/C, • If I is constant, voltage increases constantly

15. DC and AC circuits • DC (direct current) circuits • voltage is constant over time • e.g. battery or lab power supply • AC (alternate current) circuits • voltage is varying over time • Extremely important! Signal is represented as voltage variation • e.g. your voice, modem signal, data signal

16. DC circuit • Initially VC =VC(0), find VC (t) I R VC V0

17. RC circuit

18. The general formula for charging/discharging a capacitor Destination Distance =Destination-Initial

19. Example - charging a capacitor • At t=0, VC=0. What is VC(t)? • Destination=V0 • Distance=V0-0=V0 • VC(t) = V0-V0e-t/RC = V0(1-e-t/RC) I R VC V0

20. RC circuit • Charging curve for R = 1k, C = 1mF, RC = 1ms I=CdVcap/dt ~ 0

21. How long does it take to charge up the capacitor? • Depends on RC (the time constant) • t=RC: VCAP(t) = 0.63 V0 • t=5RC: VCAP(t) = 0.99 V0 • t=5RC, the capacitor is almost fully charged up • (to fully charge up a capacitor 100% takes infinitely long time, not practical) • R = 1k, C = 1mF, takes around 5ms to charge up

22. Example • Initially VC = 5V, V0=15V, find VC(t). • Destination=15 • Distance=15-5=10 • Answer: VC(t) = 15 - 10e-t/RC I R VC(0)=5V V0=15V

23. Example: discharging a capacitor • Initially VC = 15V, V0= 0V, what is VC(t)? • Destination=0V • Distance=0-15=-15V • Solution: VC(t) = 0 - (-15)e-t/RC = 15e-t/RC I R VC(0)=15V V0=0V

24. How long will it take to discharge the capacitor? • VC(t) = V0 e-t/RC • t=RC: VC(t) = 0.37 V0 • t=5RC: VC(t) = 0.01 V0 • Again, take 5RC to discharge the capacitor

25. Example • Initially VC = 15V, V0= -5V, what is VC(t)? • Solution: VC(t) = -5 - (-20)e-t/RC = -5 + 20e-t/RC I R VC(0)=15V V0=-5V Distance destination

26. AC circuit analysis • If V1 is an AC voltage, what is V2? R C V2 V1

27. The importance of sinusoids • In real world, interesting signals are time-varying • if we pass the time-varying signal to a linear system, what is the output response? Linear System (resistor, capacitor, and inductor) ?

28. Sunglass example • Sunglass is used to filter sun light • Sun light is a varying signal, how to analyze it? Sun light

29. Principle of superposition!! • Sun light = sum of light with different wavelengths • Each individual light has a frequency and a wavelength + Sun light

30. Characteristic of a filter • The filter (sunglass) can be characterized by its frequency response, that depends on only two parameters • The attenuation factor (V2/V1) • The phase (angle) change q q V1 V2

31. Input-output (transfer) characteristic of a light filter • Magnitude plot • Phase plot 1.0 0 frequency p frequency -p

32. Input : divide sun light into sinusoidal light wave • Output : sum all individual sinusoidal responses • Apply the same approach to electrical signal Filtered Sun light + + Sun light

33. Fourier theorem: signal = sum of sinusoids = sum of sinusoids Input Output = sum of sinusoids’ response Input Output response decompose sum Sinusoidal responses Linear System sinusoids

34. How to represent a sine wave (sinusoid)? • A sine wave • (unit in radian) • Make , so that is a function of time, • Unit of w : Radian/second • We have • For sinusoidal signal, one cycle = 2p radian • If the rate of rotation = f cycle/s, then w=2pf (rad/s)

35. Example : square wave = sum of sine waves !! • Periodic square wave f(wo) at frequency wo • 1kHz square wave = 1kHz + 3kHz +5kHz + . . . (sine waves) Fundamental freq Harmonics

36. Your ear • Why our inner ear has this shape? Shape of a cochlea A stretch out cochlea

37. Sinusoidal response of capacitor • Let V = A cos wt (w = 2pf) • I = C dV/dt = - wC Asin wt • Impedance = , complex relation because V and I are not in phase I leads V by 900 I ~ V V I

38. j - complex number notation • V and I are not in phase • Apart from the ratio of magnitude between V and I • Also need to know the phase angle between V and I • To keep track with the magnitude and phase information • sin and cosine are tedious to use • Simpler to use a the j-notation

39. A mathematician game • The board • 2 dimensional complex plane • x-axis represents the real axis • The player • a vector (e.g. 1) on the plane • The rule • j represents an operation that rotates a vector anticlockwise by 90o on the complex plane x 1

40. j • j*1 = j (meaning rotate vector 1 by 90o) • j*j=j2 = -1 (rotates the vector j by 90o) • Therefore j 1 j2 1 1 j3 1

41. -j = rotate 900 clockwise • -j = j3 = rotate 2700 or 900 clockwise 1 j3 1 = -j

42. j - complex number notation • a+jb • Sum of vector a and jb • Represent a vector with a magnitude and phase • Magnitude = • Phase q = • a vector with magnitude = 1 making angle q with real axis jb q a q 1

43. Euler formula • Euler formula • Proof by series expansion of ex ejq q 1

44. Complex sinusoid ejwt • Represents a rotating vector that rotates at a rate of f cycles/sec ( ) Direction of rotation ejwt wt 1

45. Real part of complex sinusoid • R{ . } denotes the real part of something • Example • V = A cos wt = R { A ejwt } because • V = R { A ejwt } = R { A(cos wt + j sin wt) } = A cos wt • If V = A cos (wt+f), then V = R { A ej(wt+f) } • Without loss of generality, we shall assume f=0 so as to simplify our calculation

46. Impedance of a capacitor • To recover V and I, take the real part • V = R { A ejwt } = A cos wt • I = R { jwC Aejwt } = R { jwC A (cos wt + j sin wt) } = - wC A sin wt (Ohm’s law for capacitor)

47. Impedance of a capacitor (XC) • magnitude of impedance = • Small w=>low frequency => ZC is large => open circuit • Large w=>high frequency => ZC is small => short circuit

48. Circuit analysis • If V2 is a varying signal, what is V2? R C V2 V1

49. Impedance of capacitor • Definition • Impedance of capacitor = • By KVL • So that • Hence

50. Transfer function T(w) • Transfer function T(w)= Output / Input = • Note that the output response depends on the frequency of the input sinusoid