Why Would You Want To Store Energy in Capacitors?

        Storing energy is very important.  You count on the energy stored in your gas tank if you drove a car to school or work today.  That's an obvious case of energy storage.  There are lots of other places where energy is stored.  Many of them are not as obvious as the gas tank in a car.  Here are a few.

• You're reading this on a computer, and the computer keeps track of the date and time.  It does that by keeping a small part of the computer running when you think that the computer is turned off.  There's a small battery that stores the energy to keep the clock running when everything else is turned off.

• If you own a stereo or television that you have to plug into the wall plug, then you should realize that the wall plug voltage becomes zero 120 times a second.  When that happens, the system keeps running because there are capacitors inside the system that store energy to carry you through those periods when the line voltage isn't large enough to keep things going!

        Capacitors can't really be used to store a lot of energy, but there are many situations in which a capacitor's ability to store energy becomes important.  In this lesson we will discuss how much energy a capacitor can store.  Your goal for this lesson is this.

• Given a capacitor that is charged,

• Be able to compute amount of energy stored in the capacitor.

        Capacitors are often used to store energy.

• When relatively small amounts of energy are needed.

• Where batteries are not desired because they might deteriorate.

• For larger power/short duration applications - as in power supply filters, or to keep power up long enough for a computer to shut down gracefully when the line power fails.

How Much Energy Can Be Stored In A Capacitor?

        To calculate how much energy is stored in a capacitor, we start by looking at the basic relationship between voltage and current in a capacitor.  First, here is the circuit symbol for a capacitor with the current into the capacitor and the voltage across it defined.

Then, the relationship between the current and the voltage in the capacitor is given by:

        Once we have this relationship, we can calculate the power - the rate of flow of energy into the capacitor - by multiplying the current flowing through the capacitor by the voltage across the capacitor.  And, once we have the power - the rate of flow of energy into the capacitor - we can eventually calculate the energy stored in the capacitor.  We know:

• Given the expression for the power:

• P(t) = i(t)v(t)

• And given the expression for the current:

• i(t) = C dv(t)/dt

• We can use the expression for current in the power expression:

• P(t) = (C dv(t)/dt) v(t)

• And, we can recognize that power is simply rate of energy input.

• P(t) = dE/dt = (C dv(t)/dt) v(t)

• Now, the derivative of energy can be integrated to find the total energy input.  Setting up the integral we have:

• dE = C v(t) dv(t)

• Then, integrating we have:

• 

• After integrating we have:

• E(t) = (C/2)v²(t)

• or simply:

• E = (Cv²/2)

Some Things To Note About The Energy Storage Formula

        Note the following about the energy stored in the capacitor.

• The energy stored in a capacitor is proportional to the capacitance.

• The energy stored in a capacitor is proportional to the square of the voltage across the capacitor.

• The expression for the energy stored in a capacitor resembles other energy storage formulae.

• For kinetic energy, with a mass, M, and a velocity, v, the stored energy is E = (Mv²/2)

• For potential energy, with a spring constant, K, and an elongation, x, the stored energy is E = (Kx²/2)

Note also the following observations.

• Since the square of the voltage appears in the energy formula, the energy stored is always positive.

• You can't have a negative amount of energy in the capacitor.

• That means you can put energy into the capacitor, and you can take it out, but you can't take out more than you put in.

And some final points to note.

• Power in to the capacitor can be negative.

• Voltage can be positive while current is negative.

• Imagine a capacitor that is charged.  You could charge a capacitor by putting a battery across the capacitor, for example.  Then, if you placed a resistor across the capacitor, charge would leave the capacitor - current would flow out of the capacitor - and the energy in the capacitor would leave the capacitor only to become heat energy in the resistor.

• When energy leaves the capacitor, power is negative.

• When you use capacitors in a circuit and you analyze the circuit you need to be careful about sign conventions as defined in the illustration we used above - which is repeated here.