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 voltagev.

The differential equation for the currentithat responds to the applied voltagevis,

Ri+L

di

dt

=v=Vˆsin(ωt+θ)

Whereω=the angular frequency in rad/sec
θ=the angular displacement ofvatt= 0
t=the time in seconds
Vˆ=peak value ofV the rms applied voltage, i.e.√ 2 V.

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 foriis,

i=

Vˆ

Z

(

−e

−Rt

L sin(θ−φ)+sin(ωt+(θ−φ))

)

( 11. 5 )

where

Z=

√

(R^2 +ω^2 L^2 )

φ=tan−^1

(

ωL

R

)

=tan−^1

(

X

R

)

and
X=ωLthe inductive reactance.

The exponential term has its maximum positive value whenθ−φequals−π/2radians.
Therefore the maximum value occurs whenθ=φ−π/2.