Page 296 - Handbook of Electrical Engineering

282 HANDBOOK OF ELECTRICAL ENGINEERING

Figure 11.7 Instantaneous current response in a series-connected R-L circuit that is fed by a sinusoidal voltage.

single-phase AC circuit can be used to represent a three-phase circuit in which a line-to-line-to-line
short circuit occurs.

Figure 11.7 shows the single-phase circuit, which is supplied by a sinusoidal voltage v.
The differential equation for the current i that responds to the applied voltage v is,

di
ˆ
Ri + L = v = V sin(ωt + θ)
dt
Where ω = the angular frequency in rad/sec
θ = the angular displacement of v at t = 0
t = the time in seconds
√
ˆ
V = peak value of V the rms applied voltage, i.e. 2V.
The complete solution of this equation can be found by several methods e.g. Laplace transforms,
method of undetermined coefficients, see Reference 3. The solution for i is,

V ˆ −Rt
i = −e L sin(θ − φ) + sin(ωt + (θ − φ)) (11.5)
Z
where

√ 2 2 2
Z = (R + ω L )
ωL −1 X

−1
φ = tan = tan
R R
and
X = ωL the inductive reactance.

The exponential term has its maximum positive value when θ − φ equals −π/2radians.
Therefore the maximum value occurs when θ = φ − π/2.