PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Direct Current and Transient Analysis 131

ANALYTICAL Solution

For t < 0, vC(0) = 2 A * 4 Ω = 8 V.

For t > 0, the nodal differential equation is given by

C

dv t

dt

vt

R

CC()
() 0

where τ = RC. T h e s olut io n t o t h e pr e c e d i n g d i f f e r e nt i a l e q u at io n i s of t h e for m vC(t) =

Ae−t/τ, and by satisfying the initial conditions at t = 0, the following is obtained:

vC(0) = 8 = A, then vC(t) = 8e−t/τ V and iR(t) = vC(t)/R = 2 e−t/τ (amp).

R.2.105 The RC and RL transient responses are modeled by fi rst-order differential equa-

tions. The RLC transient response leads to a second-order differential equation

where the roots of its characteristic equation may be complex numbers. Those

roots are referred to as the complex natural frequencies of the circuit. The location

of the roots on the complex plane determines the form of the transient response

of the electric network. Roots located on the left half of the plane represent decay-

ing exponentials, whereas roots on the right half of the plane represent growing

exponentials.

Complex conjugate roots can be associated with oscillations (sinusoids); and if

they are located in the left half of the plane, they are decaying; and in the right

half of the plane, they represent growing sinusoids. When the roots are purely

imaginary, they are located on the imaginary (jw) axis of the complex plane and

represent sustained oscillations.

R.2.106 Oscillations occur in an RLC circuit when R is small, or the ideal case is when

R = 0, then oscillations are sustained. Observe that the resistance R controls the

energy dissipated (losses) in the form of heat (friction) and is commonly referred to

as the damping coeffi cient.

R.2.107 Three distinct cases are encountered in the solution of second-order systems. They

are

Overdamped

Critical damped

Underdamped

These cases are analyzed next for both the series and parallel RLC confi gurations.

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FIGURE 2.30

RC network of R.2.104.

R = 4 Ω

Switch opens at t = 0

I = 2 A C = 1 F

+

−

+

−