The mechanical work of rotating a coil in a magnetic field produces electrical work i.e. moving charges.

An alternating emf varies with time and is produced by an alternating voltage power supply.

emf = e = V_msin2pft = V_msinwt

An alternating emf will produce an alternating current (a.c.)

The frequency of an alternating emf (and current) is the number of complete cycles an emf goes through each second.

Frequency, f, is measured in hertz (Hz) and the angular frequency, w, is measured in radians per second (rd s^-1).

Adding two p.d.’s with the same frequency

Take two alternating potential differences with the same frequency

e₁ = V_m1 sin(2pft + a) = V_m1 sin(wt + a)

e₂ = V_m2 sin(2pft + b) = V_m2 sin(wt + b)

The sum of p.d.'s is also a p.d. with the same frequency but differs in amplitude and initial phase.

e₃ = e₁ + e₂ = V_m1 sin(wt + a) + V_m2 sin(wt + b) = V_m3 sin(wt + g)

a, b and g are "phase angles".

"Phasors" (not vectors) because the angles are not real in space but "phase angles".

An alternating current in the ciruit with R only

i = I_osin(wt)

and instantenous p.d. across R becomes

v_R = e = V_mR sin(wt) = I_oRsin(wt)

Phase amplitude diagram for p.d. and current for the a.c. ciruit with resistor only.

p.d. across resistor is in phase (or "in step") with its current.

P_{instantaneous} = Ri² = R I_o² (sinwt)²

Derivation

is called the r.m.s. current because it is the square root of the mean of the square of the current.

The r.m.s. value is effectively a d.c. current which gives the same heating effect as the a.c. current.

Using r.m.s. current we may predict r.m.s. voltage.

The r.m.s. 240 V, 50 Hz mains electricity supply is connected in series with a 1 kW heater. Find the rms and instantaneous values of

1. the p.d. across the heater, and
2. the current in the circuit, and
3. the power dissipated in the heater.

Solution

Reactive Elements in AC circuits

Coils and capacitors react against AC without dissipating power.

Parrallel plate capacitor stores charge

Alternating current flows in capacitive circuit, where direct current would not flow.

The charge Q stored on the plates = CV, where V is the applied voltage

Q = CV_o sin wt

I = dQ/dt = d/dt (CV_o sin wt) = wCV_o cos wt

I_{o =} wCV_o

The reactance X_c = V_o/I_o = 1/wC

Power dissipation in a capacitor

Instantaneous power p(t) = i(t)v(t)

P_av = total energy dissipated in the time interval t = 0 to t = T (period of oscillation)

Coils in AC circuits

Changing current in a coil produces changing magnetic field and e.m.f. is induced, which opposes the current change: self induction.

In DC circuits coils retard current growth and prolong current decay.

Magnetic field B in the coil is proportional to current I and F_B is proportional to I,

define self-inductance L:

F = LI

self induced e.m.f:

the unit of self-inductance is henry (H)

Energy E stored in an inductor (coil)

E = ½ LI²

Coil connected to AC power supply

where w = 2pf

The peak voltage V₀ = I₀wL

The reactance, X_L , is defined as wL

The p.d. across the coil is ahead of the current in the circuit by one quarter of a cycle: 90^o

Similarly to average power in the capacitor the average power in the coil is zero.

Resonant circuits

LCR series circuit

Define circuit impedance, Z,

when X_L = X_C, Z = R and the circuit is in resonance

Resonant frequency f₀ (or w₀ = 2pf₀)

Example 1: resonant circuits
Solution

Example 2: real inductor
Solution