What is an RC Circuit?

An $RC$ circuit contains a single resistor, $R$ and a single capacitor $C$. From your course textbook you should already know that a capacitor is a two-terminal device whose voltage, $v(t)$, and current, $i(t)$, satisfy the following relationship,

$$\begin{eqnarray*} i(t) = C \frac{dv(t)}{dt} \end{eqnarray*}$$

This equation says that the current flowing through a capacitor is proportional to the rate at which voltage changes across the device's terminals. The proportionality constant, $C$ is called the device's capacitance and it is measured in units called farads.

Capacitors come in a variety of forms. One of the most common types of capacitors is a ceramic capacitor. A ceramic capacitor is shaped like a disk with two leads coming out of it. A picture of the schematic symbol of the capacitor is shown in figure 1. This symbol consists of two bars (representing the capacitor's two plates) with two leads coming out of them. A picture of a representative ceramic capacitor is also shown in figure 1. Another type of capacitor is the electrolytic capacitor. The symbol for an electrolytic capacitor has one of its plates curved and the top plate is marked with a plus sign (see figure 1). Electrolytic capacitors are constructed using a paper soaked in an electrolyte. This fabrication method gives enormous capacitances in a very small volume. But it also results in the capacitor being polarized. In other words, the capacitor only works with one polarity of voltage. If you reverse the polarity, hydrogen can disassociate from the internal anode of the capacitor and this hydrogen can explode. Electrolytic capacitors always have their polarity clearly marked, often with a bunch of negative signs pointed at the negative terminal. A picture of an electrolytic capacitor is shown in figure 1.

Figure 1: Symbols and drawings of capacitors

An $RC$ circuit is a particularly simple network containing a capacitor. The $RC$ circuit consists of an independent voltage source in series with a resistor, $R$, and a capacitor $C$. The schematic diagram for this circuit is shown in figure 2. Analyzing this circuit means determining the voltage over the capacitor, $v_c(t)$, (as a function of time). The exact solution, of course, depends on two things. These two things are the initial voltage over the capacitor, $V_0$, and the input voltage, $v(t)$, generated by the independent source. In the remainder of this section we state two specific solutions known as the natural response and step response. The derivation of these particular response equations is done in the lecture component of the course.

Figure 2: RC circuit

Natural Response: The first specific solution we'll consider is the voltage over the capacitor under the assumption that the capacitor's initial voltage is $V_0$ and the applied input voltage is zero (i.e., $v(t)=0$ for all $t \geq 0$). This particular solution is called the natural response of the $RC$ circuit and it can be shown to have the form

$$v_{\rm c}^{\rm (nat)}(t) = V_0 e^{-t/RC}$$ (1)

for $t \geq 0$.

It is valuable to plot the general shape of the natural response in equation 1. Note that the voltage has a time dependency that is an exponential function of time. This exponential function, $e^{-t/RC}$ has a negative exponent so that as $t$ increases, the function's value decreases in a monotone (non-increasing) manner to zero. In other words, if we consider $v_{\rm c}^{\rm (nat)}(t)$ for $t \geq 0$, we expect it to start (at time 0) at the voltage $V_0$ and then to taper off to zero as $t$ increases. This particular relationship is shown in figure 3.

Figure 3: Natural Response

Note that the expression, $RC$, has units of time. We generally refer to $RC$ as the time constant of the circuit. In fact, at time $t=RC$, we know that the voltage is $v_{\rm c}^{\rm (nat)}(RC) = V_0 e^{-1} \approx 0.368 V_0$. This means that after one "time constant", the initial voltage on the capacitor has decayed to about one third of its initial value. After three time constants, we expect $v_{\rm c}^{\rm (nat)}(3RC) = V_0 e^{-3} \approx 0.05 V_0$. This is, of course, a very small number and it means that after 4-5 time constants, the voltage over the capacitor is essentially zero. The time it takes to finish discharging the capacitor is determined by our choice for the resistors $R$ and $C$. In other words, the discharge time for the capacitor is determined by the RC constant of our circuit.

Standard capacitor values are on the order of $\mu$F (a large capacitor) to pico-farads. If we were to use a 1 k-ohm resistor in series with a 1 $\mu$F capacitor, the RC constant would be $RC = 1000 \times 10^{-6} = 1$ m-sec. In this case, our source-free circuit would discharge the capacitor in about 4-5 milli-seconds. If we were to use an even smaller capacitor, let's say about 100 pico-farad, then this discharge time would be even shorter. In particular, for a 100 pico-farad capacitor in series with a 1 k-ohm resistor, we would expect a time constant of $RC = 10^3 \times 100 \times 10^{-12} = 10^{-7}$ sec. This is one tenth of a micro-second. So in this case we would discharge a capacitor in about half a micro-second, a very very short time interval.

Step Response: The second specific solution we'll consider is the voltage over the capacitor under the assumption that the capacitor's initial voltage is $V_0$ and the applied input voltage is a step function of magnitude $V$. In other words,

$$\begin{eqnarray*} v(t) = \left\{ \begin{array}{cc} V & t \geq 0 \\ 0 & t < 0 \end{array} \right. = V u(t) \end{eqnarray*}$$

where $u(t)$ is a unit step function. The capacitor's response to this particular "step" input is called the step response of the RC circuit. The step response can be shown to have the following form,

$$v_{\rm c}^{\rm (step)}(t) = \left(V_0 e^{-t/RC} + V(1-e^{-t/RC}) \right)u(t)$$ (2)

Let's assume that $V_0=0$ so that the capacitor is initially uncharged. In this case the step response takes the following simplified form,

$$v_c(t) = V(1-e^{-t/RC})u(t)$$ (3)

for all $t$. Figure 4 plots this function for $t \geq 0$. This figure shows that the initial voltage over the capacitor is zero and then grows in a non-decreasing (monotone) manner until it approaches a steady state voltage of $V$ volts. The rate at which $v_{\rm c}^{\rm (step)}(t)$ approaches the steady state voltage is determined by the time constant $RC$. On the basis of our discussion for the natural response, we expect the capacitor to be fully charged to within 5 percent of its full charge ($V$) within three time constants. After 4-5 time constants, the capacitor should be completely charged to $V$ volts for all practical purposes.

Figure 4: Step Response assuming uncharged initial capacitor

If we do not neglect the initial charge on the capacitor, then the circuit's response is given by equation 2. Notice that this equation is simply the sum of equation 3 and the natural response in equation 1. So we can simply sum the two responses shown in figure 3 and 4 to obtain a plot of the system's total response.

Figure 5 illustrates how these individual parts of the response are combined to form the total response. One of the lighter lines represents the forced response to a step input. The other decreasing light line represents the natural response to an initial voltage on the capacitor. The total response is simply obtained by taking their sum which is shown by the dark trace in figure 5. What we see in this figure is that as time goes to infinity, the initial charge on the capacitor dies away and the total response converges to the steady state voltage $V$.

Figure 5: Total Response of RC Circuit