Phasors and AC(sec. 31.1) Resistance and reactance(sec. 31.2) RLC series circuit(sec. 31.3) Power in AC circuits(sec. 31.4) Resonance in AC circuits (sec. 31.5) Transformers (sec. 31.6) Alternating Current Ch. 31 C 2010 J. F. Becker

Learning Goals - we will learn: ch 31 How phasors make it easy to describe sinusoidally varying quantities. How to analyze RLC series circuits driven by a sinusoidal emf. What determines the amount of power flowing into or out of an AC circuit. How an RLC circuit responds to emfs of different frequencies.

Phasor diagram -- projection of rotating vector (phasor) onto the horizontal axis represents the instantaneous current.

Graphs (and phasors) of instantaneous voltage and current for a resistor. i(t) = I cos  t (source) v R (t) = i(t) R v R (t) = IR cos  t where V R = IR is the voltage amplitude. V R = IR Notation: -lower case letters are time dependent and -upper case letters are constant. For example, i(t) is the time dependent current and I is current amplitude; V R is the voltage amplitude (= IR ).

Graphs of instantaneous voltages for RLC series circuit. (The phasor diagram is much simpler.)

Graphs (and phasors) of instantaneous voltage and current for an inductor. i(t) = I cos  t (source) v L (t) = L di / dt v L (t) = L d(I cos  t )/dt v L (t) = -I  L sin  t v L (t) = +I  L cos (  t ) where V L = I  L (= IX L ) is the voltage amplitude and  = is the PHASE ANGLE (angle between voltage across and current through the inductor). X L =  L E L I V L L I

Graphs (and phasors) of instantaneous voltage and current for a capacitor. i(t) = I cos  t (source) i(t) = dq / dt = I cos  t Integrating we get q(t) = (I/  ) sin  t and from q = C v C we get v C (t) = (I/  C) sin  t v C (t)=(I/  C) cos (  t ) where V C = I/  C (= IX C ) is the voltage amplitude and  = is the PHASE ANGLE (angle between voltage across and current through the inductor). X C = 1/  C I C E I C V C

Graphs (and phasors) of instantaneous voltage and current showing phase relation between current (red) and voltage (blue). Remember: “ELI the ICE man”

Crossover network in a speaker system. Capacitive reactance: X C =1/ w C Inductive reactance: X L = w L

Phasor diagrams for series RLC circuit (b) X L > X C and (c) X L < X C.

Graphs of instantaneous voltages for RLC series circuit. (The phasor diagram is much simpler.)

Graphs of instantaneous voltage, current, and power for an R, L, C, and an RLC circuit. Average power for an arbitrary AC circuit is 0.5 VI cos f = V rms I rms cos f.

The average power is half the product of I and the component of V in phase with it. Instantaneous current and voltage: Average power depends on current and voltage amplitudes AND the phase angle f :

Graph of current amplitude I vs source frequency w for a series RLC circuit with various values of circuit resistance. The resonance frequency is at w = 1000 rad / sec (where the current is at its maximum)

A radio tuning circuit at resonance. The circles denote rms current and voltages.

Transformer: AC source is V 1 and secondary provides a voltage V 2 to a device with resistance R. TRANSFORMERS can step-up AC voltages or step- down AC voltages.  2 /  1 = N 2 /N 1 V 1 I 1 = V 2 I 1   =    e = - d F B / dt

(a) Primary P and secondary S windings in a transformer. (b) Eddy currents in the iron core shown in the cross- section AA. (c) Using a laminated core reduces the eddy currents.

Large step-down transformers at power stations are immersed in tanks of oil for insulation and cooling.

A full-wave diode rectifier circuit. (LAB)

See Review C 2010 J. F. Becker