**(i) R-L transients :**

In the R-L circuit shown in Fig. 76

i = V / R [1 –e^{ –(R / L)t}]

The plot of i (exponential rise equation) versus time is shown in Fig. 77.

The time constant (λ)for the above function is the time at which the exponent of e is unity. Thus in this case time constant(λ) is Ll R. At one time constant, the value of i will be

i = (1- e-1) = 1-0.368 = 0.632

At this time current will be 63.2% of its final value.

The voltage across inductance,

vL = L di / dt = Ve^{ –(R /L)t}

and voltage across resistor,

vR = V[1- e – (RIL)t ] … (25)

The exponential rise of resistor voltage and exponential decay of inductor voltage are shown in Fig. 78.

Also, V_{R} + V_{L} = V[1- e^{-(R/L)t} + Ve^{-R/L)t} = V

Power in the circuit elements is given by

PR = V_{2} / R [ 1 – 2e ^{–(R / Lt)t} + e ^{-2 (R / L)t}

PL = V_{2} / R [ e ^{–(R / L)t} – e ^{-2 (R / L)t}

Total power,

P = PR + PL = V_{2} / R [1-e –(R / L)t

**(ii) R-C transients :**

In the R-C circuit shown in Fig. 79,

i = V / R ^{e – t / RC}

Transients voltages across R and C are given by

u _{R} = Ve ^{–t / RC}

u c = V(1 – e^{-t/RC)}

Also, the power in circuit elements is given by

P_{R} = V_{2} / R e ^{-2t / RC}

P_{L} = V_{2} / V_{2} / R (e ^{–t / RC –e 2t / RC})

**(iii) R-L-C transients** :

For R -L-C circuit shown in Fig. 81 the following integro differential equation can be written as follows :

While solving for i, the following three cases are considered

**Case I. ** (R / 2L) ^{2} > 1 / LC

In this case the current is given by

i = C1e ^{at} C1(e^{βt} + C2e ^{-βt})

**Case II. **(R / 2L)^{2} = 1 / LC

Here , i = e at (C1 + C2 t)

**Case III.** (R / 2L)^{2} < 1 / LC

Here , i = e ^{at} (C_{1} cos βt + C_{2} sin βt)

In all the above cases the current contains the factor e^{at} and since ex = – R/2L the final value is zero, assuming that the complimentary function decays in a relatively short time. Fig, 82 shows the value of i for initial values zero and initial slope positive.