Circuits with inductors and alternating currents Chapter 20 #45, 46, 47, 49
RL circuits Ch. 20 (last section) Symbol for inductor looks like a spring. An inductor is a circuit element that has a large inductance. (usually a coil of wire). Because of the inductance in a RL circuit, the current changes gradually. When you close a switch, the current takes time to go from zero to its maximal value.
The time it takes for the current to rise depends on the time constant. In a RC circuit, time constant ( ) depended on the resistance and the capacitance. = RC In a RL circuit, = L/R See graph of current vs. time on pg. 684 After each time constant, the current rises to 63.2% of the remainder to the max value. I R (1 e t / ) I R (1 e Rt / L )
RL circuit The larger the time constant the longer it takes for the current to reach its maximum value. = L/R increase inductance, increase increase resistance, decrease Note the difference between RL and RC circuits.
Behavior of the inductor Take a RL circuit with the switch open. (fig. 20.27) When the switch is closed, the current is changing the most rapidly. The back emf from the inductor is largest at this time. As the current approaches the steady state value, the back emf is reduced. After a long time, the inductor behaves like a ordinary wire. Do example 20.8.
Ch. 21 AC circuits concept # 2, 8, 13, 15 problems# 1, 5, 7, 9, 11, 13, 14, 18, 19, 29, 37, 39, 43, 45, 47
AC circuit consists of an AC generator or other AC source hooked up to electrical devices: capacitors, resistors, inductors The voltage alternates between a maximum and minimum value. The output is a sinusoidal wave v = V max sin 2 ft v = instantaneous voltage f = frequency at which the voltage changes t = time
The current spends just as much time going current in one direction as it does with the current going the opposite direction. The average current is zero. RMS current (I rms ) The square root of the mean of the square I rms = Imax 0. 707 Imax 2
For the AC circuit with the resistor. The current and the voltage peak at the same time. See fig. on pg 697 for a AC circuit with a resistor. They are said to be in phase. applying Ohm s law V V rms = max 0. 707 2 V max V rms = I rms R and V max = I max R Average Power: P ave I 2 rms R
Capacitors in AC circuit Circuit with AC source and capacitor: At initial time there is no charge on the capacitor. The current is maximum then because there is no charge to fight the new charges from moving onto the plates. The current decreases as the charge build up. (the voltage across capacitor increases) When the current reverses direction, the voltage drops because the plates are losing their charge. This process repeats over an over again. See the figures on page 700.
Capacitors in AC circuit The current peaks before the voltage. The current and voltage are not in phase. The voltage lags behind the current by 90 0. Impeding effect of a capacitor is called the capacitance reactance X c. X c = 1/(2 fc) X c depends on the fequency V c,rms = I rms X c
Inductors in AC circuit Inductor is a coil and due to Faraday s law the inductor impedes a changing current. The effective resistance of an inductor is called the inductive reactance, X L. X L = 2 fl depends on frequency. Because of the inductance, the current lags behind the voltage. They are out of phase. See fig. 21.7 on pg 701. The voltage peaks before the current.
RLC circuit Now a resistor, capacitor, and inductor are all in an AC circuit. Because the voltages across all the components are not in phase, we cannot just add them together. They need to be added like vectors. See figures 21.9 and 21.11. The total impedance, Z, of the circuit is 2 2 defined as: Z R ( X L X C ) V max = I max Z see table 21.2
Phase shift between potential difference and current is. tan X L X C R see figure 21.11
Power in AC circuit Energy stored in capacitor is PE c = ½ C( V max ) 2 During half of a cycle the capacitor is charged. During the other half the charge returns to the voltage source. Average power supplied by the source is zero. No power losses occur in capacitor in AC circuit.
Energy stored in inductor is PE L = ½ L(I max ) 2 The current source does work against the back emf of the inductor. When the current decreases the stored energy is returned to the source as the inductor attempts to maintain the current in the circuit. There is no power loss through the inductor. All the power in RLC circuit is converted to internal energy in the resistor. 2 In RLC circuit P av I rms R
power P av I 2 rms R P av I rms V R V R V rms cos P av I rms V rms cos cos = power factor The power depends on the phase shift between the current and voltage. If a large generator has a large inductance, capacitors can be used to shift the phase.
Resonance in series RLC circuit. I rms When the impedance has its minimum value the current is maximized. This happens when: X L = X C Then Z = R V Z rms R This happens when the frequency of the circuit is just right. Called the resonance frequency. 2 V rms ( X X ) L C 2
Resonance frequency. To find the resonance frequency, f 0. Set the X L = X C 2 f 0 L = 1/(2 f 0 C) f 0 2 1 LC See fig 21.13. At f = f 0, the current spikes. If the resistance was zero, the current would become infinite.
Transformer Used to change the magnitude of an AC voltage. Works on basis of Faraday s Law. Simple transformer is made up of 2 coils wrapped around a soft iron core. When the current through the primary coil changes, the flux through the secondary coil changes producing an induced emf. The voltages in each coil is proportional to the number of turns in the coil.
Transformer V 1 = -N 1 B / t and V 2 = -N 2 B / t Assuming no flux is lost in the iron core, the term B/ t is common to both coils. V 2 = (N 2 /N 1 ) V 1 When N 2 >N 1 the voltage is increased, step up transformer. When N 2 < N 1 the voltage is decreased, step down transformer.
The power input to the primary coil equals the power output at the secondary coil for an ideal transformer. (No power loss) I 1 V 1 = I 2 V 2 So for a step up transformer the voltage is increased, but the current is decreased. In a real transformer, there is some power loss due to eddy currents in the iron core. Real transformers have efficiencies ranging from 90% - 99%.
Power lost due to the currents in transmission lines is lost as resistive heating. P = I 2 R Want the current to be as small as possible to reduce power loss. More economical to transmit electric power using high voltage and low current.