AC Circuits Physics 102 Professor Lee Carkner Lecture 24

PAL #23 Alternating Current  240  lightbulb, V rms = 120 V, 60 Hz  the rms current  V rms = I rms R, I rms = V rms /R = 120/240 =  the maximum current  I max = (2) ½ I rms = (2) ½ (0.5) =  the maximum power  P max = I 2 max R = (0.707) 2 (240) =  the average power  P av = I 2 rms R =(0.5) 2 (240) =  the power at time equals 1/120 second  I = I max sin  t = I max sin(2  ft) = I max sin [(2)(  )(60)(120) -1 ] = 0   Completed 1/2 cycle, I back to zero

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, 1 C)1, 0, -1 D)0, 1, 0 E)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, 2 C)0, 1 D)1, 0 E)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)0 C)its maximum value D)√2 times its maximum value  times its maximum value

AC Circuit Elements   Resistors (Resistance, R)  Capacitors (Reactance, X C )   We can combine them together to get the impedance (Z)   V = IR, or V = IX, or V = IZ

Resistors and AC  For AC circuits we also define 3 different values of V and I   The instantaneous (I = I max sin  t)   Ohm’s law holds (V = IR) for each  If the circuit has no inductors or capacitors, I and V are in phase

AC Circuit with Resistor

AC and Capacitors  The ”resistance” of a capacitor is the reactance, X 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  X L =  L   Creating a rapidly changing magnetic field and thus a strong back emf  We can relate the current and the potential difference across the inductor with:  V L = IX L

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  We can add the resistances and both reactances of the circuit in the same way to get a grand total “resistance”  Z = (R 2 + (X L - X C ) 2 ) ½   The total voltage is:   Can think of Z as a generalized resistance for any AC circuit

Phase Angle and Power Factor  The current and voltage at not in phase in an RC circuit  They are separated by a phase angle  defined as:  We know that power can be written P = IV   Can write power as: P av = I rms V rms cos   Note that only the resistor dissipates power 

Next Time  Read  Homework Ch 21, P 64, 65, 69, 70