Alternating Current (AC) Circuits

Transcription

1 Alternating Current (AC) Circuits We have been talking about DC circuits Constant currents and voltages Resistors Linear equations Now we introduce AC circuits Time-varying currents and voltages Resistors, capacitors, inductors (coils) Linear differential equations 73

2 Recall water analogy for Ohm s law (a) Battery (b) Resistor 74

3 Now we add a steel tank with rubber sheet (a) Battery (b) Resistor (c) Capacitor 75

4 Water enters one side of the tank and leaves the other, distending but not crossing the sheet. At first, water seems to flow through tank. How to decrease capacitance of tank? Make rubber sheet (a) smaller or (b) thicker. 76

5 Charge, like water is practically incompressible, but within a small volume (closely spaced plates) charge can enter one side and leave the other, without flowing across the space between. The apparent flow of current through space between the plates (the displacement current ) led Maxwell to his equations governing EM waves. 77

6 Basic Laws of Capacitance Capacitance C relates charge Q to voltage V C = Q V Since Q = I dt, V = 1 C I dt I = C dv dt Capacitance has units of Farads, F = 1 A sec / V + _ 78

7 Charging a Capacitor with Battery V B Voltage across resistor to find current ( ) = V B V ( C t) I t R Basic law of capacitor ( ) = C dv ( t) C I t dt Differential Equation yields exponential diminishing returns as cap becomes charged V C V C ( ) ( t) + RC dv C t dt t ( ) = V B 1 e t RC = V B 79

8 What determines capacitance C? Area A of the plates Distance d between the plates Permittivity ε of the dielectric between plates. C = ε A d Alignment of dipoles within dielectric between plates increases capacitor s ability to store charge (capacitance). Permittivity of a vacuum ε F m 1. 80

9 Disk (Ceramic) Capacitor Non-polarized Low leakage High breakdown voltage ~ 5pF 0.1μF Electrolytic Capacitor High leakage Polarized Low breakdown voltage ~ 0.1μF 10,000μF Types of Capacitors Supercapacitor (Electrochemical Double Layer) New. Effective spacing between plates in nanometers. Many Farads! May power cars someday. 3 digits ABC = (AB plus C zeros) 682 = 6800 pf 104 = 100,000 pf = μ 81

10 Inductor (coil) Water Analogy inductance is like inertia/momentum of water in pipe with flywheel. heavier flywheel (coil wrapped around iron core) adds to inertia/momentum. 82

11 Joseph Henry Invented insulation Permitted construction of much more powerful electromagnets. Derived mathematics for self-inductance Built early relays, used to give telegraph range Put Princeton Physics on the map 83

12 Basic Laws of Inductance Inductance L relates changes in the current to voltages induced by changes in the magnetic field produced by the current. I = 1 L V dt V = L di dt Inductance has units of Henries, H = 1 V sec / A. 84

13 What determines inductance L? Assume a solenoid (coil) Area A of the coil Number of turns N Length of the coil Permeability μ of the core L = µ N 2 A Permeability of a vacuum μ H m 1. 85

14 Energy Stored in Capacitor I = C dv dt P = VI = VC dv dt E = P dt E = C V dv E = 1 2 CV 2 86

15 Energy Stored in Caps and Coils Capacitors store potential energy in electric field E = 1 2 CV 2 independent of history Inductors store kinetic energy in magnetic field E = 1 2 L I 2 independent of history Resistors don t store energy at all! the energy is dissipated as power = V I 87

16 Generating Sparks What if you suddenly try to stop a current? V = L di dt goes to - when switch is opened. use diode to shunt current, protect switch. Nothing changes instantly in Nature. Spark coil used in early radio (Titanic). Tesla patented the spark plug. 88

17 Symmetry of Electromagnetism (from an electronics component point of view) I = C dv dt V = 1 C I dt V = L di dt I = 1 L V dt Only difference is no magnetic monopole. 89

18 Inductance adds like Resistance Series L S = L 1 + L 2 Parallel L P = 1 1 L 1 +1 L 2 90

19 Capacitance adds like Conductance Series C S = 1 1 C 1 +1 C 2 Parallel C P = C 1 + C 2 91

20 Distribution of charge and voltage on multiple capacitors To find the charge in capacitors in parallel Find total effective capacitance C Total Charge will be Q Total = C Total V Same voltage will be on all caps (Kirchoff s Voltage Law) Q Total = VC Total = Q 1 + Q 2 V = V 1 = V 2 Q 1 = VC 1 Q 2 = VC 2 Q Total distributed proportional to capacitance 92

21 Distribution of charge and voltage on multiple capacitors To find the voltage on capacitors in series Find total effective capacitance C Total Charge will be Q Total = C Total V Same charge will be on all caps (Kirchoff s Current Law) Q Total = VC Total Q Total = Q 1 = Q 2 V 1 = Q 1 C 1 = Q Total C 1 = C Total C 1 V V 2 = Q 2 C 2 = Q Total C 2 = C Total C 2 Voltage distributed inversely proportional to capacitance V 93

22 Lorenz Contraction Length of object observed in relative motion to the object is shorter than the object s length 0 in its own rest frame as velocity v approaches speed of light c. What is Magnetism? = 0 1 v 2 c 2 Thus electrons in Wire 1 see Wire 2 as negatively charged and repel it: Magnetism! 94

23 AC circuit analysis uses Sinusoids 95

24 Superposition of Sinusoids Adding two sinusoids of the same frequency, no matter what their amplitudes and phases, yields a sinusoid of the same frequency. Why? Trigonometry does not have an answer. Linear systems change only phase and amplitude New frequencies do not appear. 96

25 Sinusoids are projections of a unit vector spinning around the origin. 97

26 Derivative shifts 90 to the left Taking a second derivative inverts a sinusoid. 98

27 Hooke s Law F = ma F = kd d = +, a constant Sinusoids result when a function is proportional to its own negative second derivative. Pervasive in nature: swings, flutes, guitar strings, electron orbits, light waves, sound waves 99

28 Orbit of the Moon Hook s Law in 2D velocity is perpendicular to displacement acceleration is negative of displacement 100

29 Complex numbers Cartesian and Polar forms on complex plane. Not vectors, though they add like vectors. Can multiply two together (not so with vectors). 101

30 Complex Numbers How to find r 102

31 Phasor - Polar form of Complex Number 103

32 Cartesian and Polar forms (cont ) 104

33 Complex Conjugates 105

34 Multiplying two complex numbers rotates by each other s phase and scales by each other s magnitude. 106

35 Dividing two complex numbers rotates the phase backwards and scales as the quotient of the magnitudes. 107

36 How to simplify a complex number in the denominator this was + (wrong) rotate backwards real part imaginary part 108

37 Multiplying by j rotates any complex number 90 z = a + jb z = jz = b + ja z b im z This can be recognized as multiplication by a rotation matrix, where z = jz = a + j b re a a b = a b Dot product projects point (a,b) onto rotated axes (circled). 109

38 Examples 110

39 111

40 112

41 The squiggly bracket: not an algebraic expression. y itself is real: the coordinate on the imaginary axis 113

42 Rotating by + or

43 115

44 2π, not π, is the magic number 116

45 Now make the phasor spin at ω = 2π f Note: frequency can be negative; phasor can spin backwards. 117

46 Multiplying by j shifts the phase by 90 j = e jπ 2 θ = 90 solution to Hooke s Law Just like a sinusoid: shifts 90 with each derivative. 118

47 All algebraic operations work with complex numbers What does it mean to raise something to an imaginary power? Consider case of e./0 with ω = 1 Euler s Identity 119

48 Consider e jωt Its derivative de jωt dt graphically. = jωe jωt im jωe jωt e jωt re is rotated by 90 and scaled by ω at all times. Thus it spins in a circle with velocity ω, and since e jωt = 1 when t = 0, e jωt = cosωt + jsinωt Euler s Identity 120

49 Voltages and Currents are Real 121

50 Cosine is sum of 2 phasors 122

51 Sine is difference between 2 phasors 123

52 Trigonometry Revealed 124

53 125

54 Complex Impedance - Capacitor represents orthogonal basis set 126

55 Complex Impedance - Inductor 127

56 Complex Impedance - Resistor 128

57 Impedance on the Complex Plane 129

58 Taxonomy of Impedance 130

59 Series Capacitors and Inductors Two capacitors in series: Z = 1 jωc jωc 2 = C 1 + C 2 jωc 1 C 2 = jω C C 2 Two inductors in series: Z = jωl 1 + jωl 2 = jω ( L 1 + L ) 2 131

60 Parallel Capacitors and Inductors Two capacitors in parallel: Z = 1 jωc 1 + jωc 2 = 1 ( ) jω C 1 + C 2 Two inductors in parallel: Z = jωl 1 jωl 2 = jω L 1 L 2 132

61 Same rules as DC circuits now AC sources 133

62 Impedance of a Passive Branch RC circuit 134

63 LC circuit - Resonance At resonance, impedances add to zero and cancel. Analogous to spring and weight system Energy in passed between magnetic and electric fields, as in electromagnetic wave. 135

64 Adding R to LC damps the ringing Like dragging your feet on the swing. Energy being passed from magnetic to electric field eventually dissipated by resistor as heat. 136

65 Tank Circuit permiability and permitivity around loop ^ 137

66 current is now through loop rather than around it How can impedance be infinite through the parallel LC circuit when each of the components can pass current? At the resonant frequency the currents trying to pass from the antenna to ground are shifted 90 in opposite directions and thus are 180 out of phase and cancel. No net current! This null point is an example of destructive interference, how lenses work with light (described by phasors 3D space). Flute. 138

67 Phasor Notation In BioE 1310, complex exponentials may be described unambiguously with shorthand notation re jθ "r θ " Unfortunately when applied to real voltages and currents, the same notation is widely used by engineers to mean sin or cos, peak, or RMS. Thus, A θ may mean (among other things) v( t) = A 2 sin( ωt +θ ) or v( t) = Acos( ωt +θ ) 139

68 Phasor Notation Ambiguity (cont ) This ambiguity is allowed to continue because linear systems change only magnitude and phase. Thus a given network of coils, capacitors, and resistors will cause the same relative change in v( t) = A 2 sin( ωt +θ ) as it does in v( t) = Acos( ωt +θ ) so it doesn t matter which definition of A θ they use for real signals, so long as they are consistent

69 Sample Problems with Phasor Notation Using our unambiguous definition of phasor notation, r θ = re jθ Express the following as a complex number in Cartesian form (x + jy): ( 4 45 )( 6 45 ) = = j In other words, multiply the magnitudes and add the phases. For division, divide the magnitudes and subtract the phases = 2 60 = 1 j 3 141

70 Another look at Superposition Asin( ωt) + Bcos( ωt) combine to form a sinusoid with frequency ω, and how any sum of sinusoids with frequency ω amplitude frequency phase i A cos( ωt +θ ) i i any sinusoid of frequency ω is a sinusoid with frequency ω 142

71 Example: cos(t) + sin(t), ω = 1 cos( t) sin( t) cos( t) + sin( t) = 2 cos t π 4 is sinusoid of same frequency. 143

72 Recall complex conjugate pairs of phasors cos( t) = e jt + e jt 2 sin( t) = e jt e jt 2 j e jt 2 _ e jt 2 j e jt 2 e jt 2 j (recall 1 ) j = j positive (solid) and negative (dashed) frequency. 144

73 Adding cos and sin conjugate pairs (black) sin( t) cos( t) creates single conjugate pair (gray). cos( t) + sin ( t ) = 2 cos t π 4 hypotenuse 145

74 Fourier Series Applies only to periodic signals x t Inverse Fourier Series + n= ( ) = a n e jnω 0t harmonic number fundamental frequency Fourier coefficient: stationary phasor (complex number) determines magnitude and phase of particular harmonic. Any periodic signal x (t) consists of a series of sinusoidal harmonics of a fundamental frequency. ω 0 Each harmonic with is a pair of complex conjugate phasors with positive and negative frequency. The DC harmonic has a constant value,. n 0 n = 0 146

75 The n th harmonic can also be written as a weighted sum of sin and cos at frequency nω 0, (n 0) A cos( nω t) + B sin( nω t) n 0 n 0 real coefficients creating a single sinusoid of a given phase and amplitude. (same as created by the conjugate phasors at n and n). The zero harmonic n = 0 (DC) is a cosine of zero frequency cos(0t) 147

76 Building a square wave by adding the odd harmonics: 1, 3, 5, 7 An infinite number of harmonics are needed for a theoretical square wave. The harmonics account for the harsher tone of the square wave (buzzer), compared to just the fundamental 1rst harmonic sinusoid (flute). 148

77 Fourier Series: How to find coefficient a n Inverse Fourier Series x t a n = 1 T 0 stationary phasor for harmonic number n T 0 + n= ( ) = a n e jnω 0t Fourier Series x( t)e jnω 0t dt fundamental frequency backwards-spinning phasor. period T 0 = 2π ω 0 Periodic signal x (t) consists of phasors forming the sinusoidal harmonics of. ω 0 Backward-spinning phasor e jnω 0t spins the entire set of phasors in x(t), making the particular phasor e jnω 0t stand still. All other phasors complete revolutions integrating to

78 Fourier Transform Applies to any finite signal (not just periodic) Inverse Fourier Transform x( t) = 1 2π + Fourier Transform ( ) = x t X ω + X ( ω )e jωt dω ( )e jωt dt As before, backwards-spinning phasor makes corresponding component of x(t) stand still. Fourier coefficient a n has now become a continues function of frequency, X(ω), with phasors possible at every frequency. X(ω) is a stationary phasor for any particular ω that determines the magnitude and phase of the corresponding phasor in x(t). e jωt 150

79 The complex exponential e./0 forms an orthogonal basis set for any signal. Each phasor passes through a linear system without affecting the system s response to any other. To understand a linear system, all we need to know is what it does to e./0 This is the linear system s frequency response. A linear system can only change the phase and amplitude of a given phasor, not its frequency, by multiplying it by a stationary phasor H(ω), the frequency response of the system. 151

80 Frequency component X ω e./0 When inverse Fourier Transform builds x (t) x( t) = 1 2π + X ( ω )e jωt dω stationary phasor X ω scales the magnitude and rotates the phase of unit spinning phasor e./0. 152

81 Systems modeled as Filters We describe input and output signals as spectra X ω and Y ω, the amplitude and phase of e./0 at ω. System s transfer function H ω changes the magnitude and phase of X ω to yield Y ω by multiplication. Y ω = H ω X ω H ω = 6 / 7 / 153

82 Systems modeled as Filters H ω = 6 / 7 / Consider system with voltage divider of complex impedances. H ( ω ) = Z 1 Z 1 + Z 2 Same rule applies as with resistor voltage divider. Impedance divider changes the amplitude and phase of X ω e./0. 154

83 Example: RC High-Pass Filter H ω = 6 / 7 / H ( ω ) = R R + 1 jωc = jω RC 1+ jω RC At high frequencies, acts like a piece of wire. At low frequencies, attenuates and differentiates. H ( ω ) 1, ω >> 1 RC H ( ω ) jω RC, ω << 1 RC Key frequency is reciprocal of time constant RC. 155

84 Example: LR Low-Pass Filter H ω = 6 / 7 / H ( ω ) = R R + jωl = 1 1+ jω L R At low frequencies, acts like a piece of wire. At high frequencies, attenuates and integrates. H ( ω ) 1, ω << R L H ( ω ) R jωl, ω >> R L Key frequency is reciprocal of time constant L/R. 156

85 Example: RC Low-Pass Filter H ω = 6 / 7 / At low frequencies, acts like a piece of wire. (assuming no current at output) H ( ω ) = 1 jωc R + 1 jωc = H ( ω ) 1, ω << 1 RC 1 1+ jω RC At high frequencies, attenuates and integrates. H ( ω ) 1 jω RC, ω >> 1 RC Key frequency is reciprocal of time constant RC. 157

86 Decibels ratio of gain (attenuation) 1 Bell = 10 db = order of magnitude in power P 1 db 10 log out 10 P in so if P in = 1 W and P out = 100 W è 20 db Since power voltage 2 V 1 db 20 log out 10 V in so if V in = 1 V and V out = 10 V è 20 db Alexander Graham Bell db are pure ratios, no units, as opposed to dbv (voltage compared to 1 V), db SPL (sound pressure level compared to threshold of hearing), etc. 158

87 Magnitude and Phase of Low-Pass Filter Recall low-pass filter: H ω = 9 9:./;< At corner (or cut-off ) frequency, ω C = 1/RC, H ( ω ) = 1 1+ j 1 j 1 j = 1 j 2 Magnitude (Gain/Attenuation) H ( ω ) = 1 j 2 H ( ω ) 3dB = 1 2 H ω Phase ( ) = arctan 1 2 H ( ω ) =

88 Bode Plot of Low Pass Filter (previous slide) Simply a log/log plot of H ( ω ) and H ( ω ) (this one vs. f, not ω) 160

89 Magnitude and Phase of High-Pass Filter Recall high-pass filter: H ω =./;< 9:./;< At cut-off frequency, ω C = 1/RC, H ( ω ) = j 1+ j 1 j 1 j = 1+ j 2 Magnitude H ( ω ) = 1+ j 2 = 1 2 Phase H ( ω ) = arctan H ( ω ) 3dB H ( ω ) =

90 Bode Plot of High Pass Filter (previous slide) Simply a log/log plot of H ( ω ) and H ( ω ) (this one vs. f, not ω) 162

91 Values for AC Voltage Peak V P or Max V M Peak-to Peak V PP Root Mean Square V RMS Any sinusoidal signal V(t) has all three values. Since sin 2 + cos 2 = 1, and since sin 2 and cos 2 must each have the same mean value, each must have a mean value of ½. Or put another way: cos 2 Therefore, for a sinusoid ( ωt) = V RMS = V P 2 ( ) 2 mean = ½ 1+ cos 2ωt 163

92 RMS used to compute AC Power in Resistor When V and I are in-phase (resistor), average power is defined as in DC. Energy is not stored in the resistor, but simply dissipated as heat. Power in a resistor may be computed from V P and or I P for sinusoids, or from V RMS and or I RMS for any signal. For any signal in a resistor: ( P = V RMS I RMS = V ) 2 RMS R P = 1 2 V P I P = 1 2 R because V RMS = V P 2 For sinusoids: and ( ) 2 V P = ( I ) 2 RMS R = 1 ( 2 I P ) 2 R I RMS = I P 2 164

93 AC Power in Capacitor or Inductor Since V RMS and I RMS are 90 outof-phase in capacitor or inductor, the power is 0 cos( ωt)sin( ωt) = sin( 2ωt) 2 Thus no heat is dissipated, all stored energy returned to circuit average = 0 Thus light propagates by passing energy from magnetic to electric fields without loss sin and cos have 0 correlation 165

94 iron core Transformer Extremely efficient at preserving power: V 1 I 1 V 2 I 2 Allows voltage (AC) to be changed: V 2 = N M V 1 Voltages and currents assumed to be sinusoids, generally reported as RMS Permits efficient high-voltage power transmission, with small current: thus little I 2 R energy wasted in long wires. 166

95 World s Fair Chicago 1893 Tesla and Westinghouse (AC) beat Edison (DC). George Westinghouse Nikola Tesla Thomas Edison 167

96 168