# RC Circuits

Following on from our discussions of capacitance we come to RC Circuits aka Resistor-Capacitor circuits which are circuits which consist of resistors, capacitors and a power source.

This will be our first practical example of a transient circuit where we will see the output voltage ramp up in time until reaching a steady-state value or decreasing in time asymptoting to zero depending on the circuit configuration.

Note that a circuit consisting of a resistor and a single type of energy storage component, e.g. a capacitor, is also known as a first order circuit.

## The RC Charging Circuit:

The circuits shown below are the classic example of an RC charging circuit where the voltage across the capacitor ramps up once the switch is closed:

As the switch is closed current begins to flow from the voltage source towards the plates of the capacitor to charge it with the voltage across the capacitor increasing with time with a typical curve as shown below which is exponential in nature approaching a steady state voltage equal to the voltage source:

From the point of view of the resistor once the switch is closed the entire source voltage is dropped across it which then exponentially decays with time as the voltage across the capacitors plates increases taking a higher proportion of the source voltage than the resistor with time until the entire voltage drops across the capacitor:

We can also graph the charging current with time which shows that initially the current is high due to the capacitors electric field being zero (essentially functioning as a short-circuit) before slowly ramping up acting to impede the current flow:

It can be shown with calculus that the voltage across the capacitor with time for the circuit above is given by:

Where Vc denotes the charged voltage across the capacitor, t is time, and RC is the circuit resistance multiplied by capacitance.

Note: The capacitor voltage is denoted in lower case to highlight that the variable varies with time while the source voltage is upper cases to denote that it remains constant with time.

### Time Constant:

The RC factor in the equation above is known as the time constant for the circuit. It is a figure of merit which denotes the time in seconds taken for the capacitor to charge to approximately 63% or discharge to 37% of the source voltage.

It is sometimes referred to as the time delay of the circuit as it impacts the time taken for the output to respond to the input. A higher time constant means it takes longer for the output to react and reach the steady-state value.

Thinking about this intuitively, if the capacitance is higher then more charge will be able to spread across the plates which will take longer to charge and more resistance will reduce the charging current which will also increase the charging time.

We refer to the time constant with the letter τ, hence the equation can be reduced to:

We consider the transient period of the charging to be the time from 0 to 4τ with the steady state period occurring after (99.3% charged).

## Examples:

1. Determine the time constant for the circuit below:

2. Determine the time constant for the circuit below:

To determine the time constant for the circuit above we need to calculate the total resistance and capacitance in the circuit. The total resistance is simply R1+R2 but what is the total capacitance? Recall that capacitance adds in parallel, hence the total capacitance is equal to C1 + C2. Hence the time constant for this circuit is given by:

3. Determine the time constant for the circuit below:

For this circuit the resistance is more complex to determine, so let's start with the capacitance :) We have C1 and C2 in series which means we calculate them in the same way as parallel resistors and as we only have two we can use our shortcut formula: To determine the resistance we need to calculate the resistance "seen" by the terminals of the capacitor, i.e. we calculate the Thevenin Resistance assuming the sources are zeroed. We can consider the circuit below: Which we can then simplify further by combining the series resistances of R1 and R2 and R3 and R4: From here the resistance seen is simply (R1+R2)||(R3+R4) which for this case is simply 2.5k as they're both the same value we can halve it :) Now we can multiply this resistance by the capacitance found earlier to calculate the time constant as shown below:

4. a) Determine the voltage across the capacitor in the circuit below for time t after the circuit has been switched on:

Recall that the equation for the voltage across a capacitor is given by: The first step is to determine the time constant for the circuit by considering the circuit once the switch has been flicked on. In this case it is simply R multiplied by C: The next step would be to determine what the steady-state voltage across the capacitor would be initially and once completely charged. Initially it would be equal to 0V with the switch open and once closed it would eventually be equal to the supply voltage Vs, i.e. 5V. We can then substitute these into the formula to calculate the voltage across the capacitor for time t:

b) Sketch the voltage vs time curve:

To sketch the curve we can calculate the times for 1, 2, 3, 4 and 5 time constants and then equate the corresponding voltages and connect the points with an exponential curve approximation: We can present these in a table:

## The RC Discharging Circuit:

The circuits shown below are the classic example of an RC discharging circuit where the capacitor acts as a voltage source with voltage decreasing exponentially with time as the switch is closed eventually completely discharging:

If we assume the capacitor is initially charged to 5V, then the discharge curve will look as shown below:

It can be shown with calculus that the voltage across the capacitor with time for the circuit above is given by:

Where Vc denotes the charged voltage across the capacitor, t is time, and τ is the time constant.

For the discharging circuit shown, the time constant is 0.2s and the charged voltage is 5V hence the voltage across the capacitor is given by:

## Questions:

1. a) Determine the time constant for the circuit below:

0.5s

b) Determine the transient voltage across the capacitor once the switch has been closed.

vc(t) = 5(1-e^-2t)

c) Sketch the voltage vs time curve for the voltage across the capacitor.

d) Determine the voltage across the capacitor once the switch has been closed for a long time.

5V

e) Determine the voltage across the capacitor after 1 time constant.

3.161V

2. a) Determine the time constant for the circuit below:

0.64s

b) Determine the transient voltage across the capacitor once the switch has been closed.

vc(t) = 8(1-e^(-1.5625t))

c) Determine the voltage across the capacitor once the switch has been closed for time t.

8V

d) Determine the voltage across the capacitor after 5 time constants.

7.946V

3. a) Determine the time constant for the circuit below assuming the equivalent capacitance was charged to 5V:

0.076s

b) Determine the transient voltage across the capacitor once the switch has been closed.

vc(t) = 5e^(-13.158t)

c) Sketch the voltage vs time curve for the voltage across the capacitor.

e) Determine the voltage across the capacitor after 2 time constants.

0.677V

## Lab:

The best way to get a feel for how RC circuits work intuitively is to simulate RC circuits with different values to see how the time constant makes an impact as well as how the location of the resistor and capacitor makes a difference to the curve.

Let's begin by assembling our classic RC circuit in LTSpice:

Let's then create a voltage source which acts as a pulse. A pulse is a waveform that starts at 0V, rises to a set voltage and then reduces back to 0V at a time later. We can define a pulse as shown below via right clicking on the voltage source:

We can then run a simulation for say 100ms and left click on both the voltage source and top end of the capacitor to plot both voltages together:

This circuit simulates the scenario of closing a circuit containing a resistor, capacitor and voltage source. As we can see the voltage begins quickly ramping which reaches around 63% of 3.3V for a time equal to a single time constant and then begins ramping up more slowly with time before eventually settling to the 3.3V.

Now we can simulate the discharge side by extending the period (say out to 200ms) to include the off time of the pulse (which begins after 100ms):

As you can see the voltage quickly ramps down to 37% after a single time constant before then slowly ramping down with time before eventually settling to 0V.

I'll leave it as an exercise for you to determine the time constant to confirm that the voltage reaches 63% on the rise and falls to 37% on the fall :)
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