9.1 First-order RC Circuits. RC circuits are frequently applied to control timing in frequency

generating circuits, including oscillators like the astable multivibrator from a previous laboratory

(astable means two unstable states). Figure 3 shows a schematic diagram that induces a step

response in an RC circuit. In this circuit, a capacitor C is connected through a resistor R to a DC

voltage source V, at time t = 0. The most general statement of the problem allows the capacitor

to be precharged to a voltage (often through a circuit that is not shown) to a voltage that could be

positive, negative, or zero at a time just before the switch is thrown, referred to as v(0"). The

resulting response is called the step response because the action of closing the switch produces a

jump or "step" of voltage. Because of the requirement of continuity for capacitor voltages, the

capacitor voltage the instant before the switch is thrown must equal the voltage the instant after

the switch is thrown, or v(0) = v(0+¹) = V..

t = 0

V₂

R

ww

V,u(t)

Figure 3. Circuit for inducing a step response in an RC circuit.

C

R

C + v

Figure 4. Alternate circuit for inducing a step response in an RC circuit.

Figure 4 shows an equivalent representation of the RC circuit where the switch closing at

t = 0 is replaced with the source voltage multiplied by the unit step function, u(t). Recall that

the unit step function u(t) is 0 for negative values of t and 1 for positive values of t. The unit

step function performs an equivalent operation to a switch, but mathematically it is more

convenient to analyze.

Pre-lab Exercise #3. Write the equation that results after applying KCL at the node between the

capacitor and resistor in Fig. 4./n