In passing I’ve referred to the fact that capacitors and inductors are time-dependent components but I never really gave any explanation of this assertion. I’d like to go over a simple case of time-dependent circuitry to clarify exactly what this means and how it differs from time-independent circuitry (resistors, diodes ect.).
Step Response?
Time dependent circuits in a nut shell are circuits which respond to changes in voltage or current over time. These changes can come in many forms but the easiest to wrap your head around (and do the math for) is a step. A step simply refers to any time the voltage (and by extension the current) in your circuit immediately changes from one value to another.
The most basic example of this is when you turn on a power switch on a device. The voltage in the device before the power was connected is typically 0V and after the switch is turned it immediately jumps to the devices operating voltage.
When we look at how the circuit responds to this change (i.e. how the current and voltage in different parts of the circuit change after this step occurs) we are looking at the step response. In a simple resistive circuit the change occurs immediately throughout the circuit however once you introduce capacitors and inductors the story gets a little more complicated.
RC Circuit?
An RC Circuit is a circuit which contains only resistors and capacitors. These Resistors and capacitors can be arranged in series, parallel, or some combination of the two.
Capacitors
Before we get into the math it’s important to understand some of the properties of capacitors to understand how they react in circuits. A capacitor is made up of two conductive plates separated by a non-conductive layer referred to as a dielectric. These plates collect charge as current flows through the capacitor. When there is no charge on the plates current flows freely through the capacitor but as charge builds less and less current can pass through.
Once a capacitor becomes fully charged no current can flow through it and it behaves as an open circuit. Going back to our power switch example, when the power is off there is no charge on the capacitor. Once the switch is turned the current begins flowing around the circuit. This charges the capacitor until it reaches steady state. When the capacitor is fully charged no current flows and the voltage across the capacitor is maximized.
Math Time
Wait! Don’t run, I promise it’s not as bad as it looks. There is a fairly satisfying calculus derivation for this equation but in the interest of keeping things high level I’ve skipped right to the final formula. That means all we have to do is find values for the unknowns above and we can plug them right in. So how do we find these unknowns?
The Time Constant – The first thing we want to find is a value called the time constant. This represents the speed at which the circuit reaches steady state. The time constant is R*C. The negative of the time is divided by the time constant in the exponent. We simply plug in the resistance and capacitance values from the circuit and we’re all set.
Vo – Vo is the starting voltage of the circuit. This would be the voltage before the step takes place. In our basic switching example this would be equal to 0V since there is no voltage across the capacitor before the switch is flipped.
Vs – Vs represents the steady state voltage. This would be the voltage in the circuit after a long time has passed (Once the capacitor is fully charged). To find this we can replace the capacitor with an open circuit and determine the voltage between the two points.
Now that we’ve defined these variables lets have a look at an example to see them in action.
Example
Above is a basic example of a simple RC circuit. What we want to find out is how the circuit responds when the switch is closed at t=0. If we complete the equation given above for this circuit we can find the voltage at any time after the switch is flipped by entering the desired value of t.
The first thing we can fill in is the time constant. As I said above the time constant is equal to RC. This would be (10*100uF) for our circuit. Converting uF to F gives us 10*0.0001 and taking the inverse of this (since time is divided by it) gives us 1000. You can see above I have filled in this value above the exponent.
Next up is Vo. For a simple series circuit like this this step is very easy. Before the switch is closed the battery is disconnected from the circuit so no voltage is present in the circuit. This makes Vo equal to zero and it can be removed from the equation.
Finally we can calculate Vs which represents the steady state voltage of the system. Remember that when the capacitor is fully charged it allows no current to flow through the circuit. Since the voltage drop across a resistor is defined by Ohm’s Law (V=IR) and there is no current flowing through the circuit, there is no voltage drop across the resistor. This means the steady state voltage across the cap is the full battery voltage (9V).
Now we’ve got a fully defined function for the voltage at any time after t=0. The trick here is that the exponent (e^(-1000t)) becomes smaller the larger t becomes. This means the more time that passes the larger the voltage becomes until it reaches steady state. Graphing this function we get the following:
