Alternating Current Circuits
1. Pure Resistive Circuit:
Let us consider a pure ohmic resistance R is connected across on Ac source, whose instantaneous value of voltage at instant t is given by
V = Vm sin ωt …………..(1)
Since, the resistance has been connected directly across the AC source, at every instant the potential difference across the resistance is equal to the applied voltage of the AC source i.e.
VR = V
or IR = Vm sin wt
or I =( Vm/R) sin wt …………..(2)
Where, I represents the instantaneous value of current at instant t.
Comparing, equation(2) with the standard equation of current Im =( Vm/R) ………………….(3)
i.e. the peak value of current depends only upon the value of resistance R.
And, φ= 0 ………..(4)
i.e. current and voltage are in same phase in pure resistive circuit.
2. Pure Inductive circuit :
Let at instant t, the current flowing in the circuit is I and it is changing at the rate dI/dt. Then back emf generated across the coil is [- L (dI/dt)]
Applying Kirchhoff’s Loop Law on the circuit
So, the instantaneous value of the current in the circuit is given by
Comparing eq (2) with standard eq. for current
(i) Im = Vm/ωt ………(3)
If we compare the above equation with eq. of Im in pure resistive circuit, we get “ the quantity ωL behave exactly in the same way as resistance behaves in pure resistive circuit”. So ωL is the effective value of resistance offered by the coil to the alternating current flowing through it. It is represented as
XL = ωL ………….(4)
XL is known as “Inductive Reactance”. If ν is the frequency of ac source, by eq (4)
XL = 2π ν L ……….(5)
i.e. XL ∝ ν
i.e. Reactance of a coil depends upon frequency (ν) of the source, where as Resistance of a body is constant (at a given temperature).
(ii) φ = -π/2
i.e. current in pure inductive circuit lags behind the voltage by phase angle π/2.
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