Transcript phase angle
AC Circuits Physics 102 Professor Lee Carkner Lecture 24 PAL #23 Alternating Current 240 W lightbulb, Vrms = 120 V, 60 Hz the rms current Vrms = IrmsR, Irms = Vrms/R = 120/240 = the maximum current Imax = (2)½Irms = (2)½(0.5) = the maximum power Pmax = I2maxR = (0.707)2(240) = the average power Pav = I2rmsR =(0.5)2(240) = the power at time equals 1/120 second I = Imax sinwt = Imax sin(2pft) = Imax sin [(2)(p)(60)(120)-1] = P = RC Circuits A capacitor will act like a resistor with reactance: What if we have a capacitor and a resistor in a circuit together? The voltages can be thought of as vectors each with its own phase angle V2 = V2R + V2C An AC – RC Circuit Phase Diagram Impedance We can write the voltages in terms of the currents: If the resistor and the capacitor are in series they each have the same current, which we can factor out We can rewrite as: Where: Z = (R2 + X2C)½ Today’s PAL (Part 1) Consider a 10 W resistor connected to a 1 Hz, DVmax = 10 V, AC power source: What is the rms voltage? What is the reactance (or resistance)? What is the rms current? What is the maximum current? What is the phase shift between current and voltage? What is the current when the voltage is zero? Phase Angle They are separated by a phase angle f If we plot the voltages we see, cos f = IR/IZ = R/Z Vectors and Phase Angle Phase and Power We know that power can be written P = IV We can re-write power in terms of f by using: R = Z cos f Pav = IrmsVrms cos f The average power depends not just on the magnitude of I and V but also their phase If they are shifted 90 deg (p/2) they “average” out to zero power Phase and Resistance Since cos f = R/Z, we can think of cos f as the ratio of resistance to the total impedance If cos f is small, R is small relative to Z However, we also know that if cos f is large, power is large Only the resistor dissipates power in a RC circuit V, I , f and Power Today’s PAL (Part 2) Consider a 10 F capacitor connected to a 1 Hz, DVmax = 10 V, AC power source: What is the rms voltage? What is the reactance (or resistance)? What is the rms current? What is the maximum current? What is the phase shift between current and voltage? What is the current when the voltage is zero? Inductors and AC The changing current produces an induced back emf in the inductor (DVL) The induced voltage is maximum when the current is zero (since that is where it is changing the most) The voltage leads the current by 90 degrees (V is max 1/4 cycle before I) AC Circuit With Inductor Inductive Reactance We can define the way in which an inductor impedes the current with the inductive reactance: XL = wL We can relate the current and the potential difference across the inductor with: Compare to the capacitive reactance: XC = 1/(wC) Reactance and Frequency Phase for R, L and C The phase angle for a circuit with just one R, L or C is as follows: For just resistor: f = For just capacitor: f = Voltage is max 1/4 cycle after current For just inductor f = Voltage is max 1/4 before current Today’s PAL (Part 3) Consider a 10 H inductor connected to a 1 Hz, DVmax = 10 V, AC power source: What is the rms voltage? What is the reactance (or resistance)? What is the rms current? What is the maximum current? What is the phase shift between current and voltage? What is the current when the voltage is zero? RCL and AC For a series circuit, all elements have a common current 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 RLC Circuit RLC Impedance Z = (R2 + (XL - XC)2)½ The voltages for the inductor and capacitor are 180 degrees opposed and so subtract DV = IZ Next Time Read 21.14 Homework Ch 21, P 64, 65, 69, 70