Series and Parallel Circuits Ver. 1.2 In this experiment we will investigate the properties of several resistors connected in series and parallel. Our purpose is to verify the simple equations for the equivalent resistance (R eq ) in series and parallel connections, and to verify the current relations when a potential is placed across a network (circuit) connecting resistors in series or parallel with each other. Resistors are said to be in series if each terminal end of a resistor is connected to a terminal end of another resistor such that only one path for the passage of electrical charges (current) exists. The diagram below shows three resistors connected in series. i R 1 i R 2 i R 3 i Since only one path exists, any charge q passing through the network must pass through each resistor. Therefore the current i ( =Δq/Δt) will be the same through all the resistors. In this experiment we will check this property. Now since Ohm s Law tells us the current potential relationship is given by V = ir The voltage across the series network of resistors should be the sum of the individual voltage drops across the resistors or V tot = ir 1 + ir 2 + ir 3 + etc. V tot = i(r 1 + R 2 + R 3 + etc.) = i(σr i ) If we factor out the common term i, the equivalent resistance becomes ΣR i. That is, the effective resistance is the resistance that would allow the same current to flow and would have the same potential across it as V tot. R eq = R 1 + R 2 + R 3 + etc. Several resistors are said to be in parallel if each one represents an independent path for the transfer of charges between two points The diagram below shows three resistors connected in parallel across a potential V. i i 1 i 2 i 3 V R 1 R 2 R 3 From Ohm s Law we know that the current through each resistor depends on the value of the resistance and the potential difference across the resistor. When a charge q leaves the potential source, a fraction q 1 passes through resistor R 1, another fraction q 2 passes through R 2 and the remaining charge q 3 passes through R 3. Conservation of electrical charge implies: q = q 1 + q 2 + q 3. Therefore in a time interval Δt the currents satisfy the relationship: q/δt = (q 1 + q 2 + q 3 )/ Δt = q 1 /Δt + q 2 /Δt + q 3 /Δt i = i 1 + i 2 + i 3 60
Now because all the resistors are in parallel the potential (voltage) difference across each of them is the same so that Ohm s Law gives us: R 1 i 1 = V R 2 i 2 = V R 3 i 3 = V Therefore i = i 1 + i 2 + i 3 = V/R 1 + V/R 2 + V/R 3 = V(1/R 1 + 1/R 2 + 1/R 3 ) = V/R eq or 1/R eq = 1/R 1 + 1/R 2 + 1/R 3 when we have a parallel configuration. PROCEDURE: 1.) Series connections properties. Select three resistors, 1000 Ω, 1500 Ω and 3300 Ω and using the ohmmeter measure their actual resistance using the DVM. Now connect them in series in the circuit shown in the diagram below. Use the DC power output on the blue box to a 20V across the series connection. The circuit diagram labels A, B, C, and D represent parts of the circuit where we wish to measure the current in the circuit. In order to measure the current flowing through the circuit an ammeter must be inserted at each of these points. Your lab instructor will show you how to connect the ammeter at each point in order to measure the current. CIRCUIT DIAGRAM A B C R 1 R 2 R 3 DC + BLOCK DIAGRAM D DC + V A R 1 B R 2 C R 3 D Our goal here is to verify the current is the same at each point in a series circuit. In general the procedure consists of these steps : 1. Set the function to A (for ammeter), and the range of your DVM ammeter at 1 A. 2. Break the circuit at A, B, C or D by disconnecting one at a time, the wire that connects the resistors and adding a wire to the second resistor. The wire closest to 61
the potential source is put into the ammeter input on the DVM, and the second wire in inserted into the COM. 3. Now apply the power to the circuit and lower the range scale to an appropriate value. Once you have completed your measurements of the currents, you can move onto the voltage (potential difference) measurements. In this part of the experiment you will measure the potential difference across each resistor and the potential difference across the entire circuit. In general the procedure consists of these steps : 1. Set your DVM to the DC voltage, and the range to the highest available. 2. Set the probes of the DVM across the voltage source and record the voltage of the. 3. Now place the probes across each resistor and record its potential drop. Lower the range scale to a more sensitive range if necessary. 2.) Parallel connections properties. Using the same three resistors, connect them in parallel in the circuit shown in the diagram below. Again use the DC power output on the blue box to a 20V across the series connection. The circuit diagrams labels A, B, C, D and E show five locations in the circuit where we wish to measure the current. Again in order to measure the current flowing through the circuit an ammeter must be inserted at each of these points. CIRCUIT DIAGRAM A i tot i 1 i 2 i 3 DC + R 1 R 2 R 3 B C D E BLOCK DIAGRAM i tot i tot A i 1 i 2 i 3 DC + R 1 R 2 R 3 i tot B C D E Our goal here is to verify how the currents divide in a parallel circuit. In general the procedure consists of the same steps : 1. Set the function to A (for ammeter), and the range of your DVM ammeter at 1 A. 2. Break the circuit at A, B, C, D or E by disconnecting the wire that connects the resistors and adding a wire to the second resistor. The wire closest to the potential source is put into the ammeter input on the DVM, and the second wire in inserted into the COM. 3. Now apply the power to the circuit and lower the range scale to an appropriate value. 62
Once you have completed your measurements of the currents, you can move onto the voltage (potential difference) measurements. In the case of the parallel connection we expect the voltage to be the same across each resistor, verify that this is true. 1. Set your DVM to the DC voltage, and the range to the highest available. 2. Set the probes of the DVM across the voltage source and record the voltage of the. 3. Now place the probes across each resistor and record its potential drop. Lower the range scale to more sensitive range if necessary. PRESENTATION: Series Connection Verify: V tot = ir 1 + ir 2 + ir 3 + etc. = V 1 + V 2 + V 3 + etc. R eq = R 1 + R 2 + R 3 + etc. do this by calculating R eq and calculating i. Parallel Connection Verify: i = i 1 + i 2 + i 3 i 1 = V/R 1 i 2 = V/R 2 i 3 = V/ R 3 1/R eq = 1/R 1 + 1/R 2 + 1/R 3 do this by calculating R eq and calculating i. ERROR ANALYSIS: The usual considerations regarding agreement of theoretical and experimental numbers. Also comment on errors due to the calibration of the meters. CONCLUSION: Draw some conclusions about how well we verified the rules for series and parallel circuits. 63
DATA SHEET RESISTORS: R 1 R 2 R 3 Ω Ω Ω ± ± ± SERIES CONNECTION: i A V tot i B V R1 i C V R2 i D V R3 PARALLEL CONNECTION: i tot A V i 1 V R1 i 2 V R2 i 3 V R3 i tot E 64