The Source-Free Series RLC Circuit

8.3 The Source-Free Series RLC Circuit

An understanding of the natural response of the series RLC circuit is

a necessary background for future studies in filter design and commu- nications networks. Consider the series RLC circuit shown in Fig. 8.8. The circuit is

RL

being excited by the energy initially stored in the capacitor and induc-

tor. The energy is represented by the initial capacitor voltage V 0 and

initial inductor current . Thus, at I 0 t⫽ 0 ,

C 冮 ⫺⬁

v (0) ⫽ i dt ⫽ V 0 (8.2a)

Figure 8.8

A source-free series RLC circuit. Applying KVL around the loop in Fig. 8.8,

i (0) ⫽ I 0 (8.2b)

di t 1

C 冮 ⫺⬁

Chapter 8

Second-Order Circuits

To eliminate the integral, we differentiate with respect to t and rearrange terms. We get

This is a second-order differential equation and is the reason for call- ing the RLC circuits in this chapter second-order circuits. Our goal is to solve Eq. (8.4). To solve such a second-order differential equation requires that we have two initial conditions, such as the initial value of i and its first derivative or initial values of some i and The v . ini- tial value of i is given in Eq. (8.2b). We get the initial value of the derivative of i from Eqs. (8.2a) and (8.3); that is,

With the two initial conditions in Eqs. (8.2b) and (8.5), we can now solve Eq. (8.4). Our experience in the preceding chapter on first-order circuits suggests that the solution is of exponential form. So we let

i ⫽ Ae st

where A and s are constants to be determined. Substituting Eq. (8.6) into Eq. (8.4) and carrying out the necessary differentiations, we obtain

Since i ⫽ Ae st is the assumed solution we are trying to find, only the expression in parentheses can be zero:

s 2 ⫹ s⫹ ⫽ 0 (8.8)

LC

This quadratic equation is known as the characteristic equation of the See Appendix C.1 for the formula to

differential Eq. (8.4), since the roots of the equation dictate the char- find the roots of a quadratic equation.

acter of i. The two roots of Eq. (8.8) are

A more compact way of expressing the roots is

⫽ ⫺a ⫹ 2a 2 s 2 1 ⫺␻ 2 0 , s 2 ⫽ ⫺a ⫺ 2a ⫺␻ 2 0 (8.10)

8.3 The Source-Free Series RLC Circuit

The roots and are s 1 s 2 called natural frequencies , measured in

nepers per second (Np/s), because they are associated with the natural

response of the circuit; ␻ 0 is known as the resonant frequency or

strictly as the undamped natural frequency, expressed in radians per second (rad/s); and a is the neper frequency or the damping factor,

The neper (Np) is a dimensionless unit

named after John Napier (1550–1617), written as

expressed in nepers per second. In terms of and a ␻ 0 , Eq. (8.8) can be

a Scottish mathematician.

s 2 ⫹ 2a s⫹␻ 2

0 ⫽ 0 (8.8a)

The variables s and ␻ 0 are important quantities we will be discussing The ratio a 兾 ␻ 0 is known as the damp- throughout the rest of the text.

ing ratio . z

The two values of s in Eq. (8.10) indicate that there are two pos- sible solutions for i, each of which is of the form of the assumed solu- tion in Eq. (8.6); that is,

1 ⫽A 1 e 1 , i 2 ⫽A 2 e 2 (8.12)

Since Eq. (8.4) is a linear equation, any linear combination of the

two distinct solutions i 1 and i 2 is also a solution of Eq. (8.4). A com-

plete or total solution of Eq. (8.4) would therefore require a linear combination of i 1 and . Thus, the natural response of the series RLC i 2 circuit is

s i t (t) ⫽ A

1 e 1 ⫹A 2 e 2 (8.13)

where the constants A 1 and A 2 are determined from the initial values

i (0) and di (0) 兾dt in Eqs. (8.2b) and (8.5). From Eq. (8.10), we can infer that there are three types of solutions:

The response is overdamped when the roots of the circuit’s characteristic

1. If a 7␻ 0 , we have the overdamped case. equation are unequal and real, critically

2. If a⫽␻ 0 , we have the critically damped case. damped when the roots are equal and

3. If a 6␻ 0 , we have the underdamped case. real, and underdamped when the We will consider each of these cases separately.

roots are complex.