1.4. The Source-Free Parallel RLC Circuits

Applying KCL at the top node gives;

takig derivative with respect to t and dividing by C results in;

𝑑𝑑𝑡

𝑑2

𝑑𝑡2s𝑠2

1.4. The Source-Free Parallel RLC Circuits

Overdamped Case (negative and real

Critically Damped Case ( and

1.4. The Source-Free Parallel RLC Circuits

Under Damped Case (

𝑠1𝑎𝑛𝑑𝑠2are complex

;

1.4. The Source-Free Parallel RLC Circuits

Example 1.5.

In the parallel circuit of fig. , find for t>0, assuming

Consider these cases; R=1.923 Ω, R=5 Ω, R=6.25 Ω.

İf R=1.923 Ω

Example 1.5. Since ( in this case, overdamped.

The roots of the characteristic equation are;

Apply initial conditions to get and

At t =0

Example 1.5.

Must be differeantiated

At t =0

With and the solution gets…

When R=5 Ω

Example 1.5.

remains 10; Since α= , the responce isCritically damped…

Apply initial conditions to get and

Example 1.5.

Must be differeantiated

With and the solution gets…

Example 1.5. İn the last case R=6.25 Ω…

Solution is…

Response for three degrees of damping

1.5. Step Response of Series RLC Circuits

This equation has two compenants;

Natural response;

Forced response;

1.5. Step Response of Series RLC Circuits İf we set contains only natural response (

(t) can be expressed as three conditions;

The forced response is the steady-state or final value of The final value of the capacitor voltage is the same as the

source voltage .

1.5. Step Response of Series RLC Circuits Thus the complate solution…

Example 1.6. For the circuit in Fig., find for t>0. Consider these cases:R=5Ω, R=4Ω,R=1Ω.

Example 1.6.

Example 1.6. is the forced response or steady-state response. It is final value of the capacitor voltage. =24 V.

For t>0 , current i,

Take

de

riva

tive

of

v(t)

Example 1.6. Finally,

but we have to solve i(t)…

1.6. Step Response of Parallel RLC Circuits

Example 1.7.

Solution:

For t<0, the switch is open The circuit partitioned into two independent subcircuits.

Capacitor voltage equal the voltage of 20Ω resistor.

Example 1.7. For t>0, the switch is closed We have parallel RLC circuit. The voltage source is off or short-circuited.

Example 1.7.

The final value of I…Using initial conditions we get

Example 1.7.

From i(t) we obtain v(t)…