Alternating Current Circuits

Alternating Current Circuits General Physics II, Spring 2011 Brian Meadows University of Cincinnati Brian Meadows, U. Cincinnati.

Outline • Mutual Inductance – examples • Inductance and capacitance (LC) circuits • LCR circuits • A/C Power Brian Meadows, U. Cincinnati

Recap: Mutual Inductance – Pair of Coils • Flux linkage due to coil 1 in coil 2, N21 2/ i1 so • Likewise • Therefore Brian Meadows, U. Cincinnati

Example – Solenoid and Short Coil • Flux linkage due to solenoid (s) in coil (c) is Nc s c = M is • This flux linkage is • Therefore Turns/m = n Field B = 0 n is Area A=r12 Nc total turns Brian Meadows, U. Cincinnati

Example • Solenoid has n=2000 turns/m and r=5 cm, coil has Nc=50 turns. What is e in coil if dis/dt = 0.1 A/s ? • So • Therefore, EMF induced in coil is Turns/m = n Field B = 0 n is Area A=r12 Nc total turns Brian Meadows, U. Cincinnati

Example - Transformer • Has “primary (NP turns) and “secondary” (NS turns) coils. • Iron form ensures that virtually ALL flux through primary also passes through secondary. • fP/NPandfS/NS • dfP /dtNPanddfS /dt/NS • So eS/eP= NS / NP Brian Meadows, U. Cincinnati

L-C Circuit (without EMF) • Apply Kirchoff rule 2 to series circuit: VL= L d2q/dt2 • As before: • This is similar to SHM equation: • So solution is - + i + L C i = dq/dt VC= q/C - No EMF so   0 Charge q(t) oscillates back and forth between C and L Brian Meadows, U. Cincinnati

+ qmax - qmax Charge Oscillations in Series LC circuit q(t) t 0 Angular frequency of oscillations is . Maximum current is Brian Meadows, U. Cincinnati

Energy Oscillations in LC Circuit • Energy stored in electric field in capacitance is UE = ½ q 2/C • Energy stored in magnetic filed in inductance is UB = ½ L i 2 • Since the instantaneous current is NOTE – q and i are not in phase BUT, if we replace  for i (t) by  - /2, then i (t) would be in phase with q (t) We say that i (t) is /2 ahead of q (t). Brian Meadows, U. Cincinnati

Total Energy in LC Circuit • Energy at time t is UE +UB • Using and • So that the energy at any time is which is constant (as expected). NOTE – UE is like PE and UB is like KE of the charges. Brian Meadows, U. Cincinnati

LC-Circuit Energy Cycle Recall: i (t) is /2 ahead of q (t). • Energy in C goes from maximum when energy in L is zero. • Energy in L goes from maximum when energy in C is zero. Brian Meadows, U. Cincinnati

LC-Circuit Energy Cycle Brian Meadows, U. Cincinnati

Example: LC Circuit (1) Consider a circuit containing a capacitorC = 1.50 F and an inductor L = 3.50 mH.The capacitor is fully charged using abattery with Vemf = 12.0 V and thenconnected to the circuit. Questions: What is the oscillation frequency of the circuit? What is the energy stored in the circuit? What is the charge on the capacitor after t = 2.50 s? Brian Meadows, U. Cincinnati 15

Example: LC Circuit (2) Answer: The oscillation frequency of the circuitis given by The energy stored in the circuit is given by Brian Meadows, U. Cincinnati September 7, 2014 University Physics, Chapter30 16

Example: LC Circuit (3) The maximum charge on the capacitoris given by Thus the energy is Brian Meadows, U. Cincinnati September 7, 2014 University Physics, Chapter30 17

Example: LC Circuit (4) The charge on the capacitor as afunction of time is given by So, at t = 2.50 s we have Brian Meadows, U. Cincinnati September 7, 2014 University Physics, Chapter30 18

L-C-R Circuit • Apply Kirchoff rule 2 to series circuit: VL= L d2q/dt2 • As before this can be written as: • This is analogous to SHM with damping term: • Charge oscillates with peak value qmax that dies down as t increases, settling to zero. - + i + L i = dq/dt VR= iR - C - + VC= q/C Brian Meadows, U. Cincinnati

L-C-R Circuit • For damped SHM, the solution is: where • Therefore: • Where as long as VL= L d2q/dt2 - + i + L i = dq/dt VR= iR - C - + VC= q/C Brian Meadows, U. Cincinnati

L-C-R Series Circuit Brian Meadows, U. Cincinnati

Alternating Current Circuits