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), rootmean-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 1.4*cos(2π0.04t+π/8)

1.5

1.0*cos(2π0.05t)

1

voltage (V)

0.5

0 0

20

40

60

80

100

-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 t0

f (t ) =

4

π

∞

∑ ( −1)

n =1,3,5,"

n −1 2

cos ( 2π nt t0 ) 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

V (t )

V0 i (t ) = = cos ( 2π f 0t ) = i0 cos ( 2π f 0 t ) R R • the average alternating voltage, Vavg, is _______ • the rms (root mean square) voltage is the square root of the average of (V(t) - Vavg)2 T

1 2 2 cos Vrms = V ( 2π f0t ) dt = 0 ∫ T 0

• the rms current is given by the rms voltage and Ohm's law

irms = 2.3 : 4/12

Vrms 0.707V0 = = R R

AC Power • instantaneous power is given by the product of the voltage and current

P ( t ) = V ( t ) i ( t ) = V0i0 cos 2 ( 2π f0t )

note that the instantaneous power ranges between ____ and V0i0 • average power is given by the product of the rms voltage and current T

T

0

0

Vi Vi 1 Pavg = ∫ P (t )dt = 0 0 ∫ cos 2 ( 2π f 0t ) dt = 0 0 = T T 2 • when the voltage and current differ by a phase angle of φ the average power is given by

Pavg = Vrms irms cos φ 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

C = ε0

A d

A and d must be in meters!

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 air 1.00059 polystyrene 2.56 paper 3.7 SrTiO3 233

dielectric strength (Vm-1) 3×106 24×106 16×106 8×106

• 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 electrolytic (big) electrolytic (small) polyester (orange drop) ceramic 2.3 : 7/12

capacitance range 100 μF - 120,000 μF 1 μF - 2,500 μF 1,000 pF - 1 μF 10 pF - 4,700 pF

Multiple Capacitors with a parallel connection each capacitor sees the same voltage

QT = Q1 + Q2 = C1V + C2V = ( C1 + C2 )V + !

V

C1

+ !

C2

+ !

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)

+ !

C1

+ !

2.3 : 8/12

VT = V1 + V2 =

Q Q ⎛ 1 1 ⎞ + =⎜ + ⎟Q C1 C2 ⎝ C1 C2 ⎠

V + !

C2

equivalent to increasing d

Current and Voltage with a Capacitor What is the alternating current through the capacitor? V(t)

C i

V (t ) = V0 cos ( 2π f 0t )

Q = CV dQ dV i (t ) = =C dt dt i (t ) = −2π f 0CV0 sin ( 2π f 0t )

• 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

V0 cos ( 2π f 0t )

−2π f 0CV0 sin ( 2π f 0t ) 2.3 : 9/12

=

cot ( 2π f 0t ) −2π f 0C

Capacitive Reactance • in order to obtain a constant that relates ac voltage and current, it is necessary to use a complex wave ( j = −1 )

V ( t ) = V0 e j 2π f0t where

e± jθ = cos θ ± j sin θ

• solve for current using complex waves

i (t ) = C XC =

dV ( t ) dt

V (t ) i (t )

=

= j 2π f 0CV0 e j 2π f0t V0 e j 2π f0t

j 2π f 0CV0 e j 2π f0t

=

1 −j = j 2π f 0C 2π f 0C

• 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 amplitudes is given by the circuit impedance • resistance, capacitance and inductance are considered to be vectors in the complex plane • impedance is the sum vector, Z • the relationship between voltage and current amplitudes is given by the magnitude |Z|

10 :H 1 MHz

100 S 1 nF

Im

Z = ⎡⎣ R + ( X L + X C ) ⎤⎦ ⎡⎣ R − ( X L + X C ) ⎤⎦ • the phase angle between voltage and current is given by

⎛ XC + X L ⎞ ⎟ R ⎝ ⎠

φ = tan −1 ⎜

XL = 63j

R = 100

φ = -44E

Re

XC = -160j Z = 139 Ω

2.3 : 11/12

Impedance of Single Components • impedance across a resistor

Z R = RR* = • phase angle across a resistor

⎛0⎞ ⎟= ⎝R⎠

φ = tan −1 ⎜ • impedance across a capacitor

ZC =

−j +j 1 = 2π f 0C 2π f 0C 2π f 0C

• phase angle across a capacitor

2.3 : 12/12

⎛ −1 ⎜ −1 2π f 0C φ = tan ⎜ ⎜ 0 ⎜ ⎝

⎞ ⎟ ⎟= ⎟ ⎟ ⎠