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A.C. circuits are circuits through which an alternating current flows. Such circuits are used extensively in power transmission, radio, telecommunication and medicine.
Alternating currents are produced by time dependent alternating voltages given by the relation E = E0 sin ωt. Much of what we learned about d.c. circuits also apply to a.c. circuits. The effects on such voltages on resistors, capacitors and inductors will be discussed.
Nomenclature in A.C Circuits
An Alternating Current (A.C.) is one that varies sinusoidally or periodically, in such a way as to reverse its direction periodically. The commonest form of such an a.c. can be represented by
I = I0 sin 2πft
I0 sin ωt
Where I is the instantaneous current at a time t, I0 is the maximum (or peak) value of current or its amplitude; f is the frequency and ω (= 2πft) is the angular velocity, (ωt) is the phase angle of the current. Alternating is also represented by
V = V0 sin 2πft
= V0 = ωt
Here, v, v0 are the instantaneous and peak (or maximum) values of the voltages or its amplitude
If an a.c. voltage is represented by the relation V = 4 sin 900vπt, the peak voltage V0 = 4
V and 2πft = 900πt or f = 900/2 = 450 Hz. Then ω = 2πf = 900π.
Peak, and r.m.s Values of A.C.
An alternating current (or voltage) varies sinusoidal as shown below which is a sine waveform. The amplitude or peak value of the current I0, is the maximum numerical value of the current.
The root mean square (r.m.s.) value of the current is the effective value of the current.
Root-mean-square current is that steady current which will develop the same quantity of heat in the same time in the same resistance.
The r.m.s. value for the current is given by
The moving iron and the hot-wire meters measure the average value of the square of the current called the mean square current. They are however calibrated in such a way as to indicate the r.m.s. current directly. Thus most a.c. meters read the effective or r.m.s. values. The average value of an a.c. voltage or current is zero.
Resistance in A.C Circuit
At any instance the current through the resistor (R) is I and the voltage across it is V
From Ohm’s law we have that V = IR
Thus the current is given by I = V/R
If we put V = V0 sin ωt, then the current is also given by
I =V/R = V0sin ωt/R
= I0 sin ωt
The voltmeter and ammeter connected in the circuit will read the r.m.s. value of voltage and current.
Hence we can also write that Ir.m.s. = Vr.m.s. /R
The voltage and the current are said to be in phase or in step with each other. This means that both of them attain their maximum, zero and minimum values at the same instant in time.
Find the root mean square value of the sinusoidal voltage with peak value at 260V.
Using V0 = √2 x Vr.m.s.
Given that V0 = 260V, Vr.m.s. =?
V = √2Vr.m.s. ; 260 = √2Vr.m.s.
260 = 1.414Vr.m.s.
Vr.m.s = 260/1.414 = 183.867 = 184V