Resistor-Capacitor Circuit (RC circuit)

Introduction

An RC circuit is made of different combinations of capacitors and resistors. This kind of circuit has certain frequency response and thus can be used to reduce the amplitude of signals of certain input frequencies leaving others almost unaffected (Catsoulis, pp 121 - 123). In other words RC circuits built in different ways can allow to pass, say, low(high) frequencies cutting off high(low) frequencies (this kind of RC circuit is called low-pass(high-pass) filter) or they can allow to pass signals with a certain frequency range (so called band-pass filters) (Nilsson, pp 23 - 4).

Discussion

Importance of learning of this type of circuits is determined but their wide area of applications: radio receivers, audio systems (e.g. low pass audio filter is used preselect low frequencies before amplification in a subwoofer) and even AC generators (Catsoulis, pp 121 - 123). Frequency dependent characteristics of RC the combined resistors and capacitors is due to the ability of a capacitor to store charge. While the capacitor in Figure 5-la, is charging, the voltage over the capacitor VC is increasing as in figure 5-lb until it reaches the power supply voltage VS (Horowitz &, Hill, pp 108 - 120). The current passing through the resistor R is initially large (figure 5-2) but it tends toward zero as the capacitor reaches full charge. As the capacitor discharges the voltage across the capacitor decreases.

Fig. 5-1 Charging Capacitor in RC circuit with DC power source.

When the capacitor is first connected to the DC voltage source, there is a rush of charge current that is limited only by the value of resistor R. RC=t is known as a time constant of the RC circuit. Fig. 5-2 Charging Capacitor current (IC) in RC circuit (Catsoulis, pp 121 - 123).



Fig. 5-3 Primitive RC circuit (elements are connected in series). r - internal resistance of the AC power source, V0 - generator's voltage, R and C are elements of the RC circuit.

Charging the capacitor (capacitor is initially uncharged):

If the switch S is moved to position "a" at time t =0, a time dependent current, I(t), begins to flow, and the charge q on the capacitor increases. We can apply Kirchhoff's second rule (conservation of energy) to get:

VR + VC = Vo

or, using Ohm's Law (VR = IR) and the definition of capacitance (C = q /VC),

IR + q /C = Vo .

Further, we can relate q to I (I = dq /dt ):

R(dq /dt ) + (1/C)q = Vo .

This is a differential equation which we can rearrange as dq / (q  VoC) = (1/RC)dt ,

Integrating both sides, where the lower limits are q = 0 at t = 0 and the upper limits are

q = q(t) at t = t gives q(t) = VoC (1  e-t / RC ) .

Since C = q / VC, then VC(t) = q(t)/C and the voltage across the capacitor during charging is

VC(t) = Vo (1  e-t / RC)(1)

Check: ...