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2.3 Sinusoidal Signals and Capacitance

2.3 Sinusoidal Signals and Capacitance

2.3 Sinusoidal and

• many signals occur over the frequency range of 0.1 Hz to 10 kHz • definition of (ac), root- mean-square (rms) values, and ac • capacitance and • electrical properties of combined capacitors • and current across a • 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 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 is written as, V(t) = V0 cos(2πf0t + φ), where V0 is the , 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 , 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.

frequencyπ content of a square wave 4 ∞ n−1 cos( 2π nt t ) 2 0 t ft()=−∑ () 1 0 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 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 's law Vt( ) V it()==0 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 VVftdt= 22cos() 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 Vi Vi PPtdtftdt==()00 cos2 () 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 conductors to hold charge at a given , C = Q / V • capacitance is given in units of , 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 , the capacitance is given by

where κ is the constant of the insulator (note that κ =

ε/ε0)

2.3 : 6/12 Common Capacitors

• common insulating materials

material dielectric constant (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

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 ) • because of the phase change, the ratio of voltage divided by current is not a constantπππ

Vftft00cos( 2π ) cot() 2π 0 ()= −−2sin22f00CV 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 0 it()== C j2π fCVe00 dt π jft2 0 Vt() Ve0 1 − j XC == = = it() jft2 0 π j22 fC fC jfCVe2 00π 00

• the constant is called ______andππ has units of •as f0 → 0, XC → -∞j, and as f0 →∞, XC → 0 • although we will not be using circuits with , it is worth

noting that ______is given by XL = j2πf0L, where L is the of a coil

2.3 : 10/12 Impedance and the Complex Plane

• the relationship between voltage and current 10 :H 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 π 2 0

• phase angle across a capacitor

φ ⎛⎞−1 ⎜⎟2π fC = tan−1 ⎜⎟0 = ⎜⎟0 ⎜⎟ ⎝⎠ 2.3 : 12/12

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