Alternating-Current Circuits and Machines February 27 th , 2013

1. Chapter 22 Alternating-Current Circuits and Machines February 27th, 2013

2. LC Circuit • The circuit behaves as a simple harmonic oscillator • The charge is q = qmaxcos (2πƒt) • The current is I = Imax sin (2πƒt)

3. Energy in an LC Circuit electric potential energy • Capacitors and inductors store energy • A capacitor stores electric potential energy, which depends on the charge • An inductor stores magnetic potential energy, which depends on the current • As the charge and current oscillate, the energies stored also oscillate • But their sum, that is, the total potential energy is constant (see dashed line) magnetic potential energy

4. Frequency Oscillations – LC Circuit remember that: XC = reactance of the capacitor XL = reactance of the inductor • In an LC circuit, the magnitudes of the instantaneous voltages across the capacitor and inductor are always equal • Therefore, |VC| = |I XC| = |VL| = |I XL| • Simplifying, XC = XL • This assumed the current in the LC circuit is oscillating and hence applies only at the oscillation frequency • This frequency is the resonance frequency • e.g. metal detector: near a piece of metal L in the coils changes, thus the fres changes

5. LCR Circuit • An LCRcircuit contains a generator, a resistor, an inductor and a capacitor, all in series • From Kirchhoff’s Loop Rule, VAC = VL + VC + VR • But the voltages are not all in phase, so the phase angles must also be taken into account. That’s why phasor diagrams are useful.

6. LCRCircuit – Phasor Diagram • All the elements are in series, so the current is the same through each one • All the current phasors are in the same direction • Resistor: current and voltage are in phase • Capacitor and inductor: current and voltage are 90o out of phase, in opposite directions resistor current in phase inductors current lags the voltage current lags adding all 3 in 1 phasor diagram capacitor current leads the voltage Vtotal = phasor sum = VR + VL + VC = VRat resonance At resonance VC=-VL

7. Resonance • The VC and VLvalues are equal and opposite at the resonance frequency • Only the resistor is left to “resist” the flow of the current • This cancellation between the voltages occurs only at the resonance frequency • LC circuit is a harmonic oscillator, LCR circuit is a “damped” oscillator. The AC source makes it a “driven” harmonic oscillator, in which at the resonance frequency the current is maximum

8. LCR circuit L = 1 H R = 1 kΩ C = 0.01 µF Fresonance = 1500 Hz @resonance off resonance L C R

9. Resonance in the LC and LCR Circuit • At resonance, the LC or LCR impedance is minimum, thus the current is maximum • LC circuit is a harmonic oscillator, LCR circuit is a “damped” oscillator. • The AC source makes it a “driven” harmonic oscillator • The resistor “damps” the current: with a larger R the resonance curve is less intense, but it still peaks at fres smaller R larger R

10. Applications of Resonance • Radios and TVs have LCR circuits. • Tuning the radio changes C, and thus fres in the LCR circuit so the fres matches the frequency of the station you want to listen to • Then, in the radio LCR circuit the source of AC voltage are directly the radio waves transmitted by the radio station, and received by the antenna. The current in the LCR circuit is maximum at fres, but still pretty small, thus you need an amplifier, and speakers. • In an LCR circuit the AC voltage source can be either a generator or an antenna. Think about them interchangeably. • LCRcircuits can be used to construct devices that are frequency selective

11. Real Inductors in AC Circuits • A typical inductor has R≠0 due to the wire • The inductor can be modeled as L in series with R • The current can be calculated using phasors

12. Real Inductor, cont. • The elements are in series, so the current is the same through both elements • Voltages are VR = I R and VL = I XL • The voltages must be added as phasors • The phase differences must be included • The total voltage has an amplitude of

13. Impedance • The impedance, Z, is a measure of how strongly a circuit “impedes” current in a circuit • The impedance is defined as Vtotal = I Z where • This is the impedance for an RL circuit only • The impedance for a circuit containing other elements can also be calculated using phasors • The angle between the current and the impedance can also be calculated

14. Impedance, LCR Circuit • The current phasor is on the horizontal axis • The total voltage is • The impedance is

15. Impedance summary • Vtotal= I Z • for the RL circuit • for the LC circuit • for the LCR circuit

16. Problem 22.53 An LCR circuit contains a resistor with R=4500 Ω, a capacitor with C=5500µF, and an inductor with L=2.2 mH, all connected in series. The AC source has a frequency of 75 kHz. (a) What is the impedance of the circuit?

17. Problem 22.53, cont. An LCR circuit contains a resistor with R=4500 Ω, a capacitor with C=5500µ/F, and an inductor with L=2.2 mH, all connected in series. The AC source has a frequency of 75 kHz. (a) What is the impedance of the circuit? (b) What phase angle does the phasor describing the voltage source make with the current phasor? You are not required to know this!

18. RC or RL circuit High-pass circuit, e.g. the one sending signal to tweeter in a sound system Low-pass circuit, e.g. the one sending signal to woofer in a sound system L C R R L = 100 mH R = 1 kΩ C = 0.1 µF 18

19. Transformers • Transformers are devices that can increase or decrease the amplitude of an applied AC voltage • A simple transformer consists of two solenoid coils with the loops arranged so that all or most of the magnetic field lines and flux generated by one coil pass through the other coil

20. Transformers, cont. • The wires are covered with an insulating layer, so that current cannot flow directly from one coil to the other, even if they touch • An AC current in one coil will induce an AC voltage across the other coil • An AC voltage source is typically attached to the primary coil

21. Transformers, Equations • Faraday’s Law applies to both coils • If the input coil has Nin coils and the output coil has Nout turns, the fluxes in the coils are related by • The voltages are related by remember that the flux in each solenoid is

22. Transformers, final • Depending on the ratioNout/Nin the voltage Vin can be transformed into a higher or lower voltage Vout. • to transform Vin to higher voltage Vout, Nout must be greater than Nin • Transformers do not affect the frequency of the AC, thus fin = fout • Transformers cannot change DC voltages • Since they are based on Faraday’s Law • To transform 120V AC into 9V or 12V DC (e.g. to power computers) a transformer is needed, and also a rectifier (beyond the scope of this course).

23. transformers Vout=1/2Vin Jacob’s ladder Tesla coil Vin Vout Vout Vin

24. Applications of Transformers • Transformers are used to distribute electric power over long distances Plost along the line = Irms2Rline • Rline can’t be reduced. • Thus to minimize losses and environmental impact we want to minimize I • Since Pdelivered= I Vdelivered • We can do this by using Valong the line =500,000 V