To obtain Coefficient A1 and A2 SECOND ORDER General solutions To obtain Coefficient A1 and A2

General Characteristic Equation: for series and for parallel

the underdamped response , If 3. 2. the overdamped response 1. Three types of response: the underdamped response , If 3. 2. the overdamped response 1. w a < = > the critically damped response Once we know the type of the response, we can write its general solution.

General Solution for Overdamped Response: General Solution for Critically Damped Response: General solution for Underdamped Response:

To solve for A1 and A2

1 2 Overdamped Response: Expression for v(t) and dv(t)/dt To solve for A1 and A2 1 2

1 2 Critically Damped Response: Expression for v(t) and dv(t)/dt To solve for A1 and A2 1 2

Underdamped Response: Expression for v(t) and dv(t)/dt To solve for A1 and A2 1 2

A Comparison Problem in mathematics Find x(t) for t>0.

A Comparison Problem in circuits : Problem 1 i v (0) = 10V v 1 H R = 5 + 24 V v (0) = 10V v i (0) = 2A - Find v(t) for t>0.

A Comparison What are the differences? The differential equation is not given. We have to find from the circuit, in particular from the source free circuit. For a series circuit, it always has the same form. Memorize it. 2. v(0+) is given but dv(0+)/dt is not. We have to find dv(0+)/dt from the circuit by using dv(0+)/dt = i(0+)/C.

A Comparison Problem in circuits: Problem 2 i v R = 1 W 1 H i + 24 V 0.25 F v 1 W - The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.

A Comparison What are the differences? The differential equation is not given. We have to find from the source free circuit. Since this circuit is a series RLC circuit, we can write it directly. 2. v(0+) and dv(0+)/dt are not given. We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.

A Comparison Problem in circuits: Problem 3 i v R = 1 W 1 H i + 24 V 0.25 F v 1 W - The switch has been opened for a long time and it is closed at t = 0. Find i(t) for t > 0.

A Comparison What are the differences? The differential equation is not given. We have to find from the source free circuit. Since this circuit is a general second order; like it or not, we have to derive the differential equation from the source free circuit. Then, obtain the characteristic equation. 2. v(0+) and dv(0+)/dt are not given. We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.

Steps to solve this problem R = 1 W 1 H i + 24 V 0.25 F v 1 W - The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.

1. Draw the circuit for t = 0- “The switch has been closed for a long time and it is open at t = 0” This statement means that the circuit is in steady state at t = 0-. Therefore, C is open and L is shorted. 1W i + v 24 V 1W - Find i(0) and v(0)

2. Draw the circuit for t = 0+ This is a starting point the circuit to experience transient. Therefore, C is not open and L is not shorted. We know that i(0-) = i(0+) and v(0-) = v(0+) 1W i(0+) = + 24 V 0.25 F v(0+) = - Find dv(0+)/dt or/and di(0+)/dt

3. Draw the circuit for t = ∞ At t = ∞ the circuit reaches steady state again. Therefore, C is open and L is shorted. 1W i + 24 V v - Find v(∞) or/and i(∞)

4. Draw the source free circuit for t >0 Voltage source is shorted and current source is opened. 1W 1H i + 0.25F v - Find the differential equation for the source free circuit. Then its characteristic equation. Since the circuit is RLC series, we can directly write its characteristic equation. Determine the type of the response.

5. Write the general solution for the circuit for t > 0. + 24 V 0.25 F v -

6. Find A1 and A2 1W i + 24 V 0.25 F v -

7. Find other circuit quantities for t > 0. vL 1W + - i + 24 V 0.25 F v -

Example 1 A series RLC circuit has R = 10 kW, L = 0.1 mH, and C = 10 mF. What type of damping is exhibited by the circuit. Example 2 A parallel RLC circuit has R = 10 kW, L = 0.1 mH, and C = 10 mF. What type of damping is exhibited by the circuit.

(a) Overdamped (b) Critically damped (c) Underdamped If R = 20 W, L = 0.6 H, what value of C will make an RLC series circuit: Example 3 Example 4 The responses of a series RLC circuit are Determine the values of R, L, and C. V mA

Example 5 Find i(t) in the circuit of Fig. 8.10. Assume that the circuit has reached steady state at t = 0-. Fig 8.10

Example 8.9 Find v(t) for t > 0 in Fig. 8.29.

Example 8.9 Find v(t) for t > 0 in Fig. 8.29.

Problem 8.56 Find i(t) for t > 0 in Fig. 8.102.