Lecture 20 Lc & Rlc Circuits & Ac Circuits I

LECTURE 20 LC & RLC CIRCUITS & AC CIRCUITS I Lecture 20

¨ Reading chapter 29-1 and 29-3 to 29-4 ¤ LC circuits ¤ RLC circuits (without AC generator) ¤ Transformers ¤ AC circuit with resistor ¤ RMS LC circuits

¨ Consider an LC circuit with initial charge Q0 stored in the upper capacitor plate and with the switch open. At t = 0, the switch is closed. ¨ � + � = 0 ⇒ � + = 0 ⇒ � + = 0 1 Q t Q cos t, where ( ) = 0 ω ω = LC dQ I t Q sin t ( ) = = −ω 0 ω dt ¨ Q lags I by 90°.

¨ Total energy remains constant. Energy in LC circuits

¨ The sum of the electric and magnetic energies stored in a LC circuit at any given time is the same as the initial electric energy stored in the capacitor.

¨ Energy does not dissipate through capacitors or inductors. 2 2 1 Q 1 2 1 Q0 Ue + Um = + LI = 2 C 2 2 C

+Q0 -Q0

-Q0 +Q0 Example: 1

¨ You have a 10 mH inductor. What capacitor should you use with it in series to make an oscillator with a frequency of 920 kHz? (This frequency is near the center of the AM radio band.) Demo: 1

¨ Tesla coil ¤ Produces 0.5 million volts at 350 kHz ¤ Demonstration of resonance

Spark gap

L L’ Lp L Cs 110Vac Cp s ZAP! 110V to 15kV ac ac 1 1 transformer ω0 = = L C L C p p s s RLC circuits

¨ Consider an RLC circuit with an initially charged capacitor and with the switch open. After the switch is closed, if R is small, Q and I oscillate with the natural frequency, ω0, but the oscillation is damped. 1 ω0 = LC ¨ The initial energy stored in the capacitor dissipates through the resistor. The higher the resistance of the resistor, the more damping the circuit has. Demo: 2

¨ LC oscillation

¨ Damped RLC oscillation Alternating current

¨ When you plug a lamp into a wall socket, the voltage and current supplied to the light bulb vary sinusoidally.

¨ The voltage oscillates with a fixed frequency of 60 cycles per second in the US.

¨ The current therefore also periodically reverses direction, so the wall socket provides an alternating current (ac). Quiz: 1

10 AC generator

¨ An alternating current (ac) generator produces an emf given by E = E cosωt peak

where Epeak is the largest value attained by the voltage during a cycle, ω is the angular frequency, and t is time.

¨ In circuit diagrams, an ac generator is represented by the symbol: Resistors in AC circuits

¨ The potential drop across the resistor, VR, and current through the circuit, I, are given by

VR = VR peak cosωt V VR R peak I = = cosωt ≡ Ipeak cosωt R R

¨ VR and I are in phase. rms current and voltage

¨ The voltage and current in an ac circuit both average to zero, making the average useless in describing their behavior.

¨ We use instead the root mean square (rms). ¨ In general, the rms value of x is given by � = �

¨ The rms values of sinusoidally varying current and voltage are given by: I V I = max max rms Vrms = 2 2

¤ 120 volts is the rms value of household ac.

¨ The rms or max current is related to the rms or max voltage by

Vrms Vmax Irms = Imax = R R Power delivered to the resistor

¨ The power, P, and average power,

Pav, delivered to the resistor are given by

2 2 2 P = I R = Ipeak Rcos ωt

1 P = I 2 R = I 2 R av ( )av peak 2

P = I 2 R = E I av rms rms rms Quiz: 2

15 Example: 2

¨ A resistor with a resistance of R = 3.33 kΩ is connected to a generator with a peak voltage

of Vpeak = 141 V. a) Find the average power delivered to this circuit. b) Find the maximum power delivered to this circuit. Transformers

¨ A transformer is a device used to increase or decrease the voltage in a circuit without a loss of power. ¨ The voltages across the primary and secondary coils are related by the numbers of turns of the coils. ¨ Since the power in both circuits must be the same, if the voltage is lower, the current must be higher. � � � = = � � �

V2 Quiz: 3 through 5

18 Demo: 3

¨ Transformers ¤ Turn ratios of 1:1, 1:2, and 2:1. Example: 3

¨ The primary coil of a step-down transformer has N1 = 250 turns and is connected to an outlet of V1 = 120 V. The secondary coil is to supply I2 rms = 20 A at V2 = 9.0 V. a) Find the rms current in the primary coil. b) Find the number of turns in the secondary coil. Demo: 4

¨ Transformers and power lines ¤ To minimize the P = I2R heat loss in electric transmission lines, we use a high voltage and a low current. ¤ For safety, power is delivered to consumers at lower voltages and therefore with higher currents. ¤ Because of this ease of stepping the voltage up or down with transformers, AC rather than DC is in common use. Electrical fuse and circuit breaker

¨ Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance.

¨ A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts.

¨ A circuit breaker is similar, except that it is a bimetallic strip that bends enough to break the connection when it becomes too hot. When it cools, it can be reset. Polarized and grounded plugs

¨ ~0.1 A of current through a body could be fatal.

¨ With a polarized plug, the case of an appliance is always connected to the low-potential side, essentially at ground. The high-potential extends only from the wall outlet to the switch when the appliance is turned off. Polarized plug

¨ With a three-prong grounded plug, the round prong is connected to the ground and the case of an appliance. If a high-potential wire accidentally touches the case, the current flows through the third prong.

Grounded plug GFCI outlets

¨ The ground fault circuit interrupter (GFCI) protects against electric shock. ¤ Wire 1: from the outlet to the appliance ¤ Wire 2: from the appliance to the outlet

¨ If there is no current leakage, the net current though the iron ring is zero.

¨ If there is a leakage, the current in wire 2 is less, and sensing coil detects the change in magnetic flux.