7.10 Summary

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7.10 Summary

• 1. The analysis in this chapter is applicable to an y circuit that can be reduced to an equi valent circuit comprising a resistor and a single energy-storage element (inductor or capacitor). Such a circuit is first-order because its behavior is described by a first-order differential equation. When analyzing RC and RL circuits, one must always keep in mind that the capacitor is an open circuit to steady-state dc conditions while the inductor is a short circuit to steady-state dc conditions.
• 2. The natural response is obtained when no independent source is present. It has the general form

$$x(t) = x(0)e^{-t/\tau}$$

where x represents current through (or v oltage across) a resistor, a capacitor, or an inductor, and x(0) is the initial value of x. Be cause most practical resistors, capacitors, and inductors always have losses, the natural response is a transient response, i.e. it dies out with time.

3. The time constant τ is the time required for a response to decay to 1∕e of its initial value. For RC circuits, τ = RC and for RL circuits, τ = L∕R.

4. The singularity functions include the unit step, the unit ramp func tion, and the unit impulse functions. The unit step function u(t) is

$$u(t) = \begin{cases} 0, & t < 0 \\ 1, & t > 0 \end{cases}$$

The unit impulse function is

$$\delta(t) = \begin{cases}\n0, & t < 0 \\ \text{Undefined}, & t = 0 \\ 0, & t > 0\n\end{cases}$$

The unit ramp function is

$$r(t) = \begin{cases} 0, & t \le 0 \\ t, & t \ge 0 \end{cases}$$

• 5. The steady-state response is the beha vior of the circuit after an in dependent source has been applied for a long time. The transient response is the component of the complete response that dies out with time.
• 6. The total or complete response consists of the steady-state response and the transient response.
• 7. The step response is the response of the circuit to a sudden application of a dc current or v oltage. Finding the step response of a firstorder circuit requires the initial v alue x( 0 ⁺), the final value x(∞), and the time constant τ. With these three items, we obtain the step response as

$$x(t) = x(\infty) + [x(0^+) - x(\infty)]e^{-t/\tau}$$

A more general form of this equation is

$$x(t) = x(\infty) + [x(t_0^+) - x(\infty)]e^{-(t-t_0)/\tau}$$

Or we may write it as

Instantaneous value = Final + [Initial − Final] e⁻(t−t0)∕^τ

• 8. PSpice is very useful for obtaining the transient response of a circuit.
  • 9. Four practical applications of RC and RL circuits are: a delay circuit, a photoflash unit, a relay circuit, and an automobile ignition circuit.