Ch 33. Alternating Current Circuits 33.1 AC Sources AC generator: Sinusoidal voltage (emf): = Vmax sin t Vmax Maximum voltage or valtage amplitude Angular frequency.
=2 f = 2 /T
(In US, f = 60 Hz and T = 1/60 seconds)
33.2 Resistors in an AC Circuit Kirchhoff's loop rule: -vR = 0 Thus, the instantaneous voltage drop across the resistor: vR = Vmax sin t
The current: iR =vRR=VmaxRsin t
iR = Imax sin t
Imax =VmaxR
• vR and iR are always in phase.
Phasor Diagram: A phasor is a vector whose length is proportional to the maximum value of the variable it represents (Vmax or Imax). The phasor rotates counterclockwise at the angular speed . The projection of the phasor onto the vertical axis represents the instantaneous value of V(t) or I(t). • Example: Phasors at t = T/8 and 9T/8 • The power dissipated is time dependent:
P(t) =V (t)I(t) = RI(t)2 = RImax2 sin2 t
P is always 0, but is not a constant.
• The average power dissipated:
Pav = RI2 = R I 2( )
2= RIrms
2
Pav = RIrms2
Irms is called "the root mean square of current".
But, Pav = RImax2 sin2 t =
RImax2
2= R
Imax2
2
Thus, I rms=Imax2= 0.707Imax
For voltage: Vrms=Vmax2= 0.707Vmax
Example: 120 V ac voltage: Vrms = 120 V,
Vmax = 2Vrms = (1.414)(120V ) =169.7V
33.3 Inductors in an AC Circu
Kirchhoff's loop rule: Ldi
dt= 0
Ldi
dt= Vmax sin t
The solution: iL =VmaxLcos t =
VmaxLsin t
2
iL = Imax sin t2
The current and voltage are out of phase by /2 or 90°. For a sinusoidal applied voltage, The current in an inductor always lags behind the voltage across the inductor by 90°.
Imax =VmaxL=VmaxXL
XL = L "Inductive reactance". Unit has a frequency dependence.
33.4 Capacitors in an AC Circuit Kirchhoff's loop rule:
vC =Q
C= 0
Q = CVmax sin t
The current:
iC =dQ
dt= CVmax cos t = CVmax sin t + 2( )
iC = Imax sin t + 2( ) Ic and Vc are out of phase. For a sinusoidally applied emf, the current always leads the voltage across a capacitor by 90°.
Imax = CVmax =VmaxXC
XC =1
C, "Capacitive reactance", Unit: .
Summary
R circuit L circuit C circuit
v = Vmax sin t v = Vmax sin t v = Vmax sin t iR = Imax sin t iL = Imax sin t 2( ) iC = Imax sin t + 2( )
Imax =VmaxR
Imax =VmaxXL
Imax =VmaxXC
XL = L XC =1
C
Pav = RIrms
2 Pav = 0 Pav = 0
Vrms =Vmax2= 0.707Vmax
Irms =Imax2= 0.707Imax
Example: In an R circuit, v = 200 sin t and R = 100 . • Find Vrms:
Vmax = 200 V,
Vrms =Vmax2=200V
2= 141V
• Find Imax:
Imax =VmaxR
=200V
100= 2.0A
• Find Irms:
Irms =VrmsR
=141V
100= 1.41A
Example: In an L circuit, L = 25 mH, Vrms = 150 V, and f = 60Hz. (a) Find the inductive reactance XL: = 2 f = 2 (60) = 377 /s XL = L = (377 /s)(25 10-3 H) = 9.43 (b) Find rms current:
Irms =VrmsXL
=150V
9.43= 15.9A
Example: In a C circuit, C = 8 μF, f = 60 Hz, and Vrms = 150 V. • Find the capacitive reactance XC: = 2 f = 2 (60) = 377 /s XC = 1/ C = 1/(377 /s)(8 10-6 F) = 332 • Find Irms and Imax:
Irms =VrmsXC
=150V332
= 0.451A;
Imax = Irms 2 = 0.451 2 = 0.639A
• Find I(t): IC (t) = Imax sin t + 2( ) = 0.639sin 377t + 2( )