2.3 Sinusoidal Signals and Capacitance • many signals occur over the frequency range of 0.1 Hz to 10 kHz • definition of alternating current (ac), root- mean-square (rms) values, and ac power • capacitance and capacitors • electrical properties of combined capacitors • voltage and current across a capacitor • advantage of using complex waves in calculations • reactance relates ac current and voltage • impedance as a vector in the complex plane 2.3 : 1/12 Time-Varying Voltages as Cosines 1.5 1.4*cos(2π0.04t+π/8) 1.0*cos(2π0.05t) 1 0.5 0 0 20406080100 voltage (V) -0.5 -1 -1.5 time (s) A cosine signal is written as, V(t) = V0 cos(2πf0t + φ), where V0 is the amplitude, f0 is the frequency, and φ is the phase angle. The period is given by, _________. The peak-to-peak voltage is _______. +φ is called a phase lead, while –φ is a lag 2.3 : 2/12 Why Cosines? The mathematician Fourier has shown that any temporal signal measured in the laboratory can be written as a sum of sines and cosines, or alternatively, as phase-shifted cosines. frequency content of a square wave 4 ∞ n−1 cos( 2π nt t ) 2 0 t ft()=−∑ () 1 0 π n=1,3,5," n When a signal is composed entirely of cosines with periods longer than ___________ (0.1 to 0.01 Hz), it can be treated as direct current for the purposes of electronic circuitry. When a signal gets above _________ it begins to behave more like an electromagnetic wave than a simple electrical voltage. We will deal with ac signals from 0.1 Hz to 10 kHz. 2.3 : 3/12 Alternating Current and RMS Values • alternating current can be obtained from an alternating voltage and Ohm's law Vt( ) V0 it()== cos2()ππ ft = i cos2 () ft RR 00 0 • the average alternating voltage, Vavg, is _______ • the rms (root mean square) voltage is the square root of the 2 average of (V(t) - Vavg) T 1 22 VVftdt= cos() 2π = rms T ∫ 00 0 • the rms current is given by the rms voltage and Ohm's law VV0.707 i = rms ==0 rms RR 2.3 : 4/12 AC Power • instantaneous power is given by the product of the voltage and current 2 Pt( ) == Vtit( ) ( ) Vi00cos() 2π ft 0 note that the instantaneous power ranges between ____ and V0i0 • average power is given by the product of the rms voltage and current TT 1 Vi2 Vi PPtdtftdt==()00 cos() 2π == 00 avg TT∫∫0 2 00 • when the voltage and current differ by a phase angle of φ the average power is given by PViavg= rms rms cosφ which means that the average power goes to zero when the voltage and current differ by __________ 2.3 : 5/12 Capacitance • capacitance is the ability of two parallel conductors to hold charge at a given electric potential, C = Q / V • capacitance is given in units of farads, where F = CV-1 • commercially available capacitors range from 100,000 μF to 10 pF (note that the units ___ and ___ are almost never used) • two parallel plates of area, A, and separation, d, have a capacitance given by A C = ε A and d must be in meters! 0 d for A = 1 cm2 and d = 0.1 mm, C = 8.85 pF • when the capacitor plates are separated by an insulator, the capacitance is given by where κ is the dielectric constant of the insulator (note that κ = ε/ε0) 2.3 : 6/12 Common Capacitors • common insulating materials material dielectric constant dielectric strength (Vm-1) air 1.00059 3×106 polystyrene 2.56 24×106 paper 3.7 16×106 6 SrTiO3 233 8×10 • electrolytic capacitors are composed of a sheet of foil inserted into a conducting liquid, with insulation provided by an oxide layer • electrolytic capacitors have + and − leads, and if connected backwards the oxide layer dissolves with explosive results • common capacitor types type capacitance range electrolytic (big) 100 μF - 120,000 μF electrolytic (small) 1 μF - 2,500 μF polyester (orange drop) 1,000 pF - 1 μF ceramic 10 pF - 4,700 pF 2.3 : 7/12 Multiple Capacitors with a parallel connection each capacitor sees the same voltage QQQCVCVCCVT = 121+= + 2 =( 12 +) + C1 ++C2 V ! !! equivalent to increasing A with a series connection the amount of charge separated by each capacitor has to be the same (charge is added to the top capacitor and removed from the bottom capacitor) QQ⎛⎞11 + VVV=+ = + = + Q C1 T 12 ⎜⎟ ! CC12⎝⎠ CC 12 + V ! + equivalent to C ! 2 increasing d 2.3 : 8/12 Current and Voltage with a Capacitor What is the alternating current through the capacitor? C V(t) i QCV= dQ dV it()== C dt dt it()=− 2ππ fCV00 sin2() ft 0 Vt()= V00 cos2()π ft •as f0 → 0, i → 0 and as f0 →∞, i → -∞ • the current is -90° out of phase with the voltage • because of the -90° phase difference no power is dissipated in the capacitor (this is different behavior than a resistor) • because of the phase change, the ratio of voltage divided by current is not a constant Vftftcos( 2π ) cot() 2π 00= 0 −−2sin22πππf00CV() ft 0 fC 0 2.3 : 9/12 Capacitive Reactance • in order to obtain a constant that relates ac voltage and current, it is necessary to use a complex wave ( j = − 1 ) jft2π 0 ± jθ Vt()==± Ve0 where e cosθ j sinθ • solve for current using complex waves dV( t) jft2π it()== C j2π fCVe 0 dt 00 jft2π 0 Vt() Ve0 1 − j XC == = = itjft2π 0 j22ππ fC fC () jfCVe2π 00 00 • the constant is called ________________ and has units of ohms •as f0 → 0, XC → -∞j, and as f0 →∞, XC → 0 • although we will not be using circuits with inductors, it is worth noting that _______________ is given by XL = j2πf0L, where L is the inductance of a coil 2.3 : 10/12 Impedance and the Complex Plane • the relationship between voltage and current 10 :H amplitudes is given by the circuit impedance • resistance, capacitance and inductance are 1 MHz 100 S considered to be vectors in the complex plane • impedance is the sum vector, Z 1 nF • the relationship between voltage and current amplitudes is given by the magnitude |Z| Im Z =++⎣⎦⎣⎦⎡⎤⎡⎤RR()()X L XXCC −+X L • the phase angle between XL = 63j voltage and current is given by R = 100 φ = -44E Re −1 ⎛ XC + X L ⎞ φ = tan ⎜ ⎟ ⎝ R ⎠ XC = -160j Z = 139 Ω 2.3 : 11/12 Impedance of Single Components • impedance across a resistor * ZR = RR = • phase angle across a resistor −1 ⎛⎞0 φ = tan ⎜⎟= ⎝⎠R • impedance across a capacitor −+jj 1 ZC == 22ππf00CfCfC 2 π 0 • phase angle across a capacitor ⎛⎞−1 ⎜⎟2π fC φ = tan−1 ⎜⎟0 = ⎜⎟0 ⎜⎟ ⎝⎠ 2.3 : 12/12.