**Sinusoidal Response of RLC Circuit:**

Consider a Sinusoidal Response of RLC Circuit consisting of resistance, inductance and capacitance in series as shown in Fig. 12.20.

Switch S is closed at t = 0. At t= 0, a sinusoidal voltage V cos (ωt + θ) is applied to the RLC series circuit, where V is the amplitude of the wave and θ is the phase angle. Application of Kirchhoff s voltage law to the Sinusoidal Response of RLC Circuit results in the following differential equation.

Differentiating the above equation, we get

The particular solution can be obtained by using undetermined coefficients. By assuming

Substituting ip, i′_{p} and i″_{p} in Eq. 12.33, we have

Comparing both sides, we have

Sine coefficients.

Cosine coefficients

Solving Eqs 12.38 and 12.39, we get

Substituting the values of A and B in Eq. 12.34, we get

Putting

To find M and Φ we divide one equation by the other

Squaring both equations and adding, we get

The particular current becomes

The complementary function is similar to that of DC series RLC circuit. To find out the complementary function, we have the characteristic equation

The roots of Eq. 12.42, are

By assuming

K_{2} becomes positive, when (R/2L)^{2} > 1/LC

The roots are real and unequal, which gives an overdamped response. Then Eq. 12.42 becomes

The complementary function for the above equation is

**i _{c} = c_{1} e^{(K}1^{ + K}2^{)t } + c_{2} e^{(K}1^{ – K}2^{)t}**

Therefore, the complete solution is

**i = i _{c} + i_{p}**

**= c**

_{1}e^{(K}1^{ + K}2^{)t }+ c_{2}e^{(K}1^{ – K}2^{)t}K_{2} becomes negative, when (R/2L)^{2} < 1/LC

Then the roots are complex conjugate, which gives an underdamped response. Equation 12.42 becomes

**[D – (K _{1} + jK_{2})] [D – (K_{1} – jK_{2})]i = 0**

The solution for the above equation is

**i _{c} = e^{K}1^{t} [c_{1} cos K_{2}t + c_{2} sin K_{2}t] **

Therefore, the complete solution is

**i = i**

_{c}+ i_{p}**i = e ^{K}1^{t} [c_{1} cos K_{2}t + c_{2} sin K_{2}t] **

K_{2} becomes zero, when (R/2L)^{2} = 1/LC

Then the roots are equal which gives critically damped response. Then, Eq. 12.42 becomes (D – K_{1}) (D – K_{1}) i = 0.

The complementary function for the above equation is

**i _{c} = e^{K}1^{t} (c_{1} + c_{2}t)**

Therefore, the complete solution is

**i = i _{c} + i_{p}**

**i = e ^{K}1^{t} (c_{1} + c_{2}t)**