|
Consider an ac circuit with only an inductor connected across the terminals of an ac source (the resistor is there to prevent shorting the source). Since the induced emf in the inductor is is given by L dI / dt,
Where dI / dt is the time derivative of the current.
Kirchhoff's loop rule yields
v - L di / dt = 0
Where v = instantaneous voltage and i = instantaneous current.
Since v = V(max) sin wt
Where V(max) = the maximum output voltage of the source and w = angular frequency = 2 * Pi * frequency of source.
We get
L di / dt = V(max) sin wt (1)
Integrating the above expression (for those of you who know calculus, the constant of integration can be ignored because it is dependant on initial conditions which are not important in this case) yields
i(L) = - ( V(max) / wL ) cos wt (2)
Where i(L) is the instantaneous current in the inductor.
Exploiting the trig identity cos wt = - sin ( wt - Pi / 2 ) we can express equation 2 as
i(L) = ( V(max) / wL ) sin ( wt - Pi / 2) (3)
Comparing this expression with that of equation (1) shows that the current in the inductor lags the voltage across the inductor by a phase difference of Pi / 2 radians or 90 degrees so that the voltage reaches its maximum value one quarter of an oscillation period before the current reaches its maximum value.
This lag occurs because the voltage across the inductor v(L) is proportional to di / dt, and v(L) is greatest when the current is changing most rapidly. In a sinusoidal curve, di / dt is maximum when the curve goes through zero, where the slope is at a maximum therefore v(L) is at its maximum when the current is zero.
On the other hand, the current reaches its maximum values when cos wt = 1 (from equation (2)) so that
I(max) = V(max) / wL (4)
Where wL = X(L) = inductive reactance (expressed in units of ohms)
So that I(max) = V(max) / X(L) (5)
From equation (5) we see that the maximum current decreases as the inductive reactance increases for a given applied voltage. Unlike resistance, however, reactance is dependant on the frequency in addition to the characteristics of the inductor so that the reactance of the inductor increases as the frequency of the current increases. This occurs because at higher frequencies, the instantaneous current changes more rapidly than at lower frequencies, causing an increase in the in the induced emf related to a given peak current.
From expression (1) and expression (5) we see that the instantaneous voltage drop across the inductor is given by
v(L) = L di / dt = V(max) sin wt = I(max) X(L) sin wt
The above expression can be viewed as Ohm's law for an inductive circuit.
|
