Example-Connections of Capacitors
Let’s do an example related to the connections of capacitors. Assume that we have a circuit with a power supply which generates v volts of potential difference connected to capacitor c 1. Let’s say c 2, c 3, c 4, and c5 this way. Let’s say we have another capacitor over here with capacitance of c 6. We’d like to figure out the equivalent capacitance of this circuit.
Let’s give some numerical values to these capacitors. Say c 1 is equal to 1 microfarad, c 2 is equal to 2 microfarads, and c 3 is equal to 2 microfarad. c 4 is equal to 3 microfarad and c 5 is equal to 5 microfarad. Finally, c 6 is equal to 1 microfarad. When we look at our circuit, we can immediately realize that c 3, c 4, and c 5, these three, are connected in parallel. So the equivalent of this parallel combination is connected in series to c 1, c 2, and c6. As we solve these types of problems or analyze these types of circuits, we can go ahead and calculate the equivalent capacitance of the connections of the capacitors within the circuit and at every stage, we redraw the circuit.
First let’s go ahead and calculate the equivalent capacitance of this parallel combination. In other words, just take this unit out and replace it with the equivalent capacitance. As we do that, let’s redraw the circuit. Our circuit is then going to be equal to c 1 over here, c 2 over here, and here is the equivalent of this parallel combination, and let’s call that one as c equivalent 1, and moving on, c 6 over here. They are all connected now in series. Since these three c 3, c 4, and c 5 are connected in parallel, we can instantly calculate their equivalent capacitance for a parallel connection, c equivalent is directly the sum of the capacitances of each capacitor, so c equivalent is c 3 plus c 4 plus c 5. C equivalent 1 will therefore be equal to c 3 is 2 microfarads, plus c 4 is 3 microfarads, plus c 5 is 5 microfarads, therefore c equivalent is going to be equal to 10 microfarad.
Once we determine c equivalent 1, now we can easily go ahead and calculate the equivalent capacitance of these four capacitors which are connected in series. For the series connection, we know that the inverse of the equivalent capacitance is equal to sum of the inverses of each capacitance in the combination. Therefore we will replace all these four with a single one, and that will be the total equivalent capacitance of this circuit. Our final simplified circuit will take this form.
1 over c equivalent is therefore going to be equal to 1 over c 1 plus 1 over c 2 plus 1 over c equivalent 1 plus 1 over c 6. If we substitute the numerical values, 1 over c equivalent is equal to c 1 was equal to 1 microfarad and c 2 is 2 microfarads, so we have 1 over 1 plus 1 over 2 plus 1 over c equivalent 1 and that is 10 microfarads, 1 over 10, plus c 6 and c 6 was equal to 1 microfarad, therefore 1 over 1.
If we have common denominator over here, that will be 10, so we will multiply this by 10, this one by 5, both numerator and denominator, this will be multiplied by 1 and this will be multiplied again by 10 in order to have a common denominator. Therefore 1 over c equivalent is going to be equal to 10 plus 5 plus 1 plus 10 divided by 10. So 1 over c equivalent will be equal to 10, 20, 25, 26 over 10 and from here, solving for c equivalent, which we will take the inverse of this, we’re gonna have 10 over 26 or we can simplify this as 5 over 13 microfarads as the equivalent capacitance of this circuit.