# The RC Circuit

concept

Up until now we've been dealing with circuits that don't change with time. We've assumed that once you connect the circuit up things have certain voltages and currents and that's just how it is.
In the circuits we've seen up until now that's basically true, however in real circuits (and especially circuits with capacitors) there is some "start up" time called "transients" where the values build up to their values.
This is mostly the case because resistors take some amount of time to develop their charge that is based on the current. The higher the current the faster the charge builds up until it hits the value we've been calculating previously.
In this topic we'll start to look at circuits with resistors and capacitors, specifically we'll look at what happens immediately after putting the circuit together (remember that's called transients).
That means we'll look at what the voltage across a resistor is after 1 second, 2 seconds etc. as well as other properties of the circuit.

When the switch on the circuit below is closed the current I starts out at 10mA (just like it would if the capacitor were a short circuit) but then drops exponentially to 0mA (the current through a capacitor after transients is always 0 in DC circuits).
Conversely the voltage across the resistor begins at 10V (like it would if the capacitor was a short circuit) and exponentially drops to 0V (from Ohm's law we know that since the current is 0 the voltage drop across the resistor must be 0).

On the other hand when the switch on the circuit above is closed the voltage across the capacitor begins at 0V and logarithmically increases to \(10\mu\)V (the DC value we calculated in previous topics).

fact

A capacitor-resistor circuit is characterised by what we call a "time constant" whose symbol is \(\tau\). The time constant of a series RC circuit is \(\tau = RC\).

fact

The voltage across a capacitor in an RC circuit is given by:
\(\Large V_c = V(1-e^{\frac{-t}{\tau}})\)
Where \(V\) is the voltage across the resistor-capacitor branch and \(\tau\) is the time constant from above.

fact

The voltage and current of the resistor in an RC circuit will fall 63.2% of its initial value within the first time constant, 86.5% in two time constants, 95% in three time constants, 98.2% in four time constants and 99.3% in five time constants.
The voltage across a capacitor in an RC circuit will rise to 63.2% of its final value within the first time constant, 86.5% in two time constants, 95% in three time constants, 98.2% in four time constants and 99.3% in five time constants.

If the RC circuit involves a complex arrangement of capacitors and resistors try to reduce them to a single capacitor and a single resistor to determine the time constant.
You may notice that according to the above percentages the voltages and currents NEVER reach their steady state values that we calculated in previous topics. If we want to refer to these steady state values we will say that a circuit's switch was closed a "long time" ago.

practice problems