Physics 24100 Electricity & Optics Lecture 19 Chapter 29 sec. 1,2,5 Fall 2017 Semester Professor Koltick
Series and Parallel R and L Resistors and inductors in series: R series = R 1 + R 2 L series = L 2 + L 2 Resistors and inductors in parallel: R parallel = 1 R 1 + 1 R 2 1 L parallel = 1 L 1 + 1 L 2 1
Alternating Current θ φ m = NBA cos ωt ℇ = dφ m dt = NBAω sin ωt But we can arbitrarily re-define when t = 0 and then write this as ℇ = NBAω cos ωt
Alternating Current in a Resistor ℇ = NBAω cos ωt = ℇ peak cos ωt V R = IR I peak = V R peak R
Power Dissipated by the Resistor Instantaneous power: P = I 2 R = I peak 2 R cos 2 ωt
Power Dissipated by the Resistor Average power P av = 1 T T P t dt 0 = I peak T 2 R T cos 2 ωt dt 0 Peak value Average value
Root-Mean-Squared Current We can define I RMS = 1 2 I peak 2 = I peak 2 Now we can write P av = I RMS 2 R For AC circuits, we can use the usual DC formulas, provided we use RMS currents and voltages.
Example How much power is dissipated through the resistor if the voltage source is a square wave?
Standard AC Voltage in North America + peak ~180 V ~110 V 1 2 0.707 - peak
Not a world standard
AC vs DC Power Distribution Westinghouse and Tesla promoted AC power distribution. Edison promoted DC power distribution.
AC Power Distribution Grid Power loss: P = I 2 R More efficient to distribute power with high voltage, low current. Transformers convert high voltage AC to low voltage AC.
Two-Phase AC Power Usually, two voltage sources are provided that are 180 out of phase: 340 V 170 V Voltage -170 V -340 V Voltage measured between either phase 1 or phase 2 and common is 120 V (RMS). Voltage measured between phase 1 and phase 2 is 240 V (RMS).
Inductors in AC Circuits I peak = ℇ peak ωl I rms = ℇ rms ωl Potential difference across the inductor: V L = L di dt ℇ = ℇ peak cos ωt di = ℇ peak L I t = ℇ peak ωl cos ωt dt sin ωt
Inductive Reactance For a resistor, I rms = ℇ rms R For an inductor, I rms = ℇ rms ωl The quantity X L = ωl is sort of like a resistance that depends on the frequency. It s called the inductive reactance It has units of ohms Inductors store (and release) energy they don t dissipate energy. Average power delivered to an inductor is zero
Inductive Reactance V L = L di dt Potential difference across an inductor is largest when the current is increasing rapidly. Potential difference leads the current by 90
Inductive Reactance Inductive reactance depends on both ω and L Smaller inductance and higher frequency gives the same reactance Smaller inductors are physically smaller, cost less, and are more efficient (lose less energy due to heating) Industrial applications use motor-generators to produce higher frequency (400 Hz) AC power: 60 Hz 400 Hz
Capacitors in AC Circuits I peak = ℇ peak ωc I rms = ℇ rms ωc Potential difference across the capacitor: V C = 1 I t dt C 0 ℇ = ℇ peak cos ωt I t = C dℇ dt = ℇ peak ωc sin ωt t
Capacitive Reactance We want to be able to write I rms = ℇ rms X C But we already had I rms = ℇ rms ωc The capacitive reactance is defined X C = 1 ωc Units: ohms, just like R and X L Capacitors store and release energy they don t dissipate energy Average power delivered to a capacitor is zero
Capacitive Reactance I V C Potential difference across an inductor is largest when the current is decreasing rapidly. Potential difference lags the current by 90
Phasors We can keep track of the magnitude and phase of currents and voltages using phasor diagrams: y Projection onto y-axis represents the value of an AC quantity. r θ = ωt x = r cos ωt y = r sin ωt x
Phasors in a Resistor Circuit I R and V R are in phase: V R = ℇ peak sin ωt I R = ℇ peak R sin ωt
Phasors in a Capacitive Circuit Voltage lags the current by 90 V C = ℇ peak sin ωt I C = ωcℇ peak sin ωt + 90 = ωcℇ peak cos ωt
Phasors in an Inductive Circuit Voltage leads the current by 90 V C = ℇ peak sin ωt I C = ℇ peak ωl = ℇ peak ωl sin ωt 90 cos ωt
Lecture 19-17 Summary: Resistance and Reactances Time-varying emf V emf V max sin t Time-varying emf V R with resistor VR ir IRsin t Resistance R R Time-varying emf V C with capacitor X C 1 VC ic sin( t 90 ) C X Time-varying emf V L with inductor VL X il sin t 90 L L X C L Capacitive Reactance X C Inductive Reactance X L
Lecture 19-18 Summary: Phase and Phasors