AC Circuits

Physics 102Professor Lee

CarknerLecture 23

PAL #23 Alternating Current 240 lightbulb, Vrms = 120 V, 60 Hz

the rms current Vrms = IrmsR, Irms = Vrms/R = 120/240 = 0.5A

the maximum current Imax = (2)½Irms = (2)½(0.5) = 0.707 A

the maximum power Pmax = I2maxR = (0.707)2(240) = 120 W

the average power Pav = I2rmsR =(0.5)2(240) = 60 W

the power at time equals 1/120 second I = Imax sint = Imax sin(2ft) = Imax sin [(2)()(60)(120)-1] = Imax sin () = 0 P = 0 Completed 1/2 cycle, I back to zero

AC Circuit Elements In an AC circuit we get resistance-like effects from three

different elements: Capacitors (Reactance, XC)

We can combine them together to get the impedance (Z) We can then use Ohm’s Law to find the current

For AC circuits we also define 3 different values of V and

I The instantaneous (I = Imax sint) The rms (Irms = 0.707 Imax)

AC Circuit with Resistor

AC and Capacitors The ”resistance” of a capacitor is the reactance, XC

XC = 1/(C) High frequency and large capacitance means less

reactance

The voltage and the current across the capacitor are not in phase

Shift the current sine wave ¼ cycle “backwards” from the in-phase situation

AC Capacitor Phase Lag

Inductive Reactance We can define the way in which an inductor

impedes the current with the inductive reactance:

XL = L

Creating a rapidly changing magnetic field and thus a strong back emf

VL = IXL

Inductors and Phase What is the phase shift between V and I?

look at the slope of the current sine wave

The induced voltage is zero when the current is a maximum (since that is where the current is not changing) The voltage leads the current by 90 degrees (V is max

1/4 cycle before I)

AC Circuit With Inductor

Reactance and Frequency Resistor

Capacitor

Inductor Low current

at high frequency

RCL and AC

Let’s combine all three elements together

If you combine a resistor, capacitor and an inductor into one series circuit, they all will have the same current but all will have difference voltages at any one time Voltages are all out of phase with each other

RLC Circuit

RLC Impedance

Called the impedance (Z)Z = (R2 + (XL - XC)2)½

The voltages for the inductor and capacitor are 180 degrees opposed and so subtract

The total voltage is:

Can think of Z as a generalized resistance for any AC circuit

Phase Angle and Power Factor

They are separated by a phase angle defined as:

cos = IR/IZ = R/Z We know that power can be written P = IV

Can write power as:Pav = IrmsVrms cos

Note that only the resistor dissipates power

Next Time

Read 22.1-22.4, 22.7 Homework Ch 21, P 64, 65, Ch

22, P: 3, 7

Consider a sinusoidally varying current with a maximum value of 1 A. What is the value of the current at ¼, ½ and ¾ of the cycle?

A) ¼, ½, ¾ B) 0, -1, 1C) 1, 0, -1D) 0, 1, 0E) 1, 1, 1

Consider a sinusoidally varying current with a maximum value of 1 A and an angular frequency of . What is the value of the current at time equals ½ second and one second?

A) ½, 1 B) 1, 2C) 0, 1D) 1, 0E) 0, 0

Consider two sine waves with a phase shift of radians. When one wave is at its maximum value, the other is at,

A) its minimum value B) 0C) its maximum valueD) √2 times its maximum valueE) times its maximum value