2 Objectives: After completing this module, you should be able to: Calculate the equivalent capacitance of a number of capacitors connected in series or in parallel.Determine the charge and voltage across any chosen capacitor in a network when given capacitances and the externally applied potential difference.
3 Electrical Circuit Symbols Electrical circuits often contain two or more capacitors grouped together and attached to an energy source, such as a battery.The following symbols are often used:GroundBattery-++Capacitor-
4 Series CircuitsCapacitors or other devices connected along a single path are said to be connected in series. See circuit below:Series connection of capacitors. “+ to – to + …”+-BatteryC1C2C3Charge inside dots is induced.
5 Charge on Capacitors in Series Since inside charge is only induced, the charge on each capacitor is the same.Charge is same: series connection of capacitors.Q = Q1 = Q2 =Q3BatteryC1C2C3+-Q1Q2Q3
6 Voltage on Capacitors in Series Since the potential difference between points A and B is independent of path, the battery voltage V must equal the sum of the voltages across each capacitor.Total voltage V Series connection Sum of voltagesV = V1 + V2 + V3BatteryC1C2C3+-V1V2V3•AB
7 Equivalent Capacitance: Series Q1= Q2 = Q3+-C1C2C3V1V2V3V = V1 + V2 + V3Equivalent Ce for capacitors in series:
8 Example 1. Find the equivalent capacitance of the three capacitors connected in series with a 24-V battery.+-2 mFC1C2C324 V4 mF6 mFCe for series:Ce = 1.09 mF
9 Example 1 (Cont.): The equivalent circuit can be shown as follows with single Ce. +-2 mFC1C2C324 V4 mF6 mFCe = 1.09 mF1.09 mFCe24 VNote that the equivalent capacitance Ce for capacitors in series is always less than the least in the circuit. (1.09 mF < 2 mF)
10 For series circuits: QT = Q1 = Q2 = Q3 Example 1 (Cont.): What is the total charge and the charge on each capacitor?+-2 mFC1C2C324 V4 mF6 mF1.09 mFCe24 VCe = 1.09 mFQT = 26.2 mCQT = CeV = (1.09 mF)(24 V);For series circuits: QT = Q1 = Q2 = Q3Q1 = Q2 = Q3 = mC
11 Example 1 (Cont.): What is the voltage across each capacitor? +-2 mFC1C2C324 V4 mF6 mFVT = 24 VNote: VT = 13.1 V V V = 24.0 V
12 Short Cut: Two Series Capacitors The equivalent capacitance Ce for two series capacitors is the product divided by the sum.3 mF6 mF+-C1C2Example:Ce = 2 mF
13 Parallel capacitors: “+ to +; - to -” Parallel CircuitsCapacitors which are all connected to the same source of potential are said to be connected in parallel. See below:Voltages: VT = V1 = V2 = V3Parallel capacitors: “+ to +; - to -”C2C3C1+-Charges: QT = Q1 + Q2 + Q3
14 Equivalent Capacitance: Parallel Parallel capacitors in Parallel:C2C3C1+-Q = Q1 + Q2 + Q3Equal Voltages: CV = C1V1 + C2V2 + C3V3Equivalent Ce for capacitors in parallel:Ce = C1 + C2 + C3
15 Example 2. Find the equivalent capacitance of the three capacitors connected in parallel with a 24-V battery.C2C3C12 mF4 mF6 mF24 VQ = Q1 + Q2 + Q3VT = V1 = V2 = V3Ce for parallel:Ce = ( ) mFCe = 12 mFNote that the equivalent capacitance Ce for capacitors in parallel is always greater than the largest in the circuit. (12 mF > 6 mF)
16 Example 2 (Cont.) Find the total charge QT and charge across each capacitor. 2 mF4 mF6 mF24 VQ = Q1 + Q2 + Q3Ce = 12 mFV1 = V2 = V3 = 24 VQT = CeVQ1 = (2 mF)(24 V) = 48 mCQT = (12 mF)(24 V)Q1 = (4 mF)(24 V) = 96 mCQT = 288 mCQ1 = (6 mF)(24 V) = 144 mC
17 Example 3. Find the equivalent capacitance of the circuit drawn below. 4 mF3 mF6 mF24 VC2C3Ce = 4 mF + 2 mFCe = 6 mFC14 mF2 mF24 VC3,6Ce6 mF24 V
18 Example 3 (Cont.) Find the total charge QT. Ce = 6 mFC14 mF3 mF6 mF24 VC2C3Q = CV = (6 mF)(24 V)QT = 144 mCC14 mF2 mF24 VC3,6Ce6 mF
19 This can also be found from Q = C3,6V3,6 = (2 mF)(24 V) Example 3 (Cont.) Find the charge Q4 and voltage V4 across the the 4-mF capacitor.C14 mF3 mF6 mF24 VC2C3V4 = VT = 24 VQ4 = (4 mF)(24 V)Q4 = 96 mCThe remainder of the charge: (144 mC – 96 mC) is on EACH of the other capacitors. (Series)This can also be found from Q = C3,6V3,6 = (2 mF)(24 V)Q3 = Q6 = 48 mC
20 Example 3 (Cont.) Find the voltages across the 3 and 6-mF capacitors. Q3 = Q6 = 48 mCC14 mF3 mF6 mF24 VC2C3Note: V3 + V6 = 16.0 V V = 24 VUse these techniques to find voltage and capacitance across each capacitor in a circuit.
21 Summary: Series Circuits Q = Q1 = Q2 = Q3V = V1 + V2 + V3For two capacitors at a time:
22 Summary: Parallel Circuits Q = Q1 + Q2 + Q3V = V1 = V2 =V3For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.