RESISTANCE AND NETWORKS

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Bryce Campbell
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1 PURPOSE The purpose of this laboratory is to learn to construct simple circuits; and, to become familiar with the use of power supplies and the digital multimeter. to experimentally find the equivalent resistance of parallel and series combinations of resistors; to experimentally confirm current-voltage, branch-loop equations in simple circuits; THEORY Electrical current is a measure of the net charge flowing across a fixed cross sectional area of conductor per unit time. In the MKS system, the current is measured in Coulomb/second, or Amperes. The amount of current through a conductor is a function of the applied electric field and of the geometry and physical properties of the material. If, as in many cases of interest, the current is directly proportional to the applied potential difference, the conductor is said to be linear or ohmic. The linear relation between the potential difference and the current is called Ohm's law. This relation takes the form I = σ V = 1 V R where σ, the constant of proportionality, is a characteristic parameter of the conductor called the conductivity. R, the resistance, is the inverse of the conductance, σ -1. The unit of resistance is the Ohm. Of course, if two ends of a current-carrying conductor are to be kept at a fixed potential difference, work must be spent somewhere else in a circuit to bring the charges that have gone to a lower potential level back up to a higher level. The situation is similar to the following mechanical analogue. A collection of blocks is dragged up to a higher level by a mechanical contraption (for example, a conveyor belt) and allowed to fall after reaching the top. Once the blocks get back to the lower level, they are caught up again by the device, so that the cycle can repeat itself. The work done by the conveyor belt to move a block up to the higher level is the analogue of the "electromotive force" (emf) of a battery. (Note however that, strictly speaking, emf is not work, but work per unit charge.) If the conveyor belt is an ideal machine (no friction, etc.) the work due in bringing each block up will equal the potential energy difference between the high and low levels. If as is usually the case, some energy is lost to friction, the mechanical energy balance will read (Mechanical energy to bring each block up) = (Potential energy difference) + (Energy lost to internal friction) XII-1

2 The electrical analogue (per unit charge) of this energy balance is ε = V + Ir where r is the internal resistance of the generator itself (not R). In this experiment, it is unlikely that the accuracy will be sufficient to notice the effects associated with the internal resistance r. In practice then, r is frequently assumed to be zero. One important observation to be made is that if a substantial amount of current is drawn out of the power supply, the measurement of potential difference between the terminals of the power supply should be made under load conditions, i.e., with charges flowing through your circuit. This will not be the case in these experiments. For a series combination of resistors, the same current flows through each resistor. The equivalent resistance for two resistors in series is R eq = R 1 + R 2 For a parallel combination of resistors, there is equal voltage across each resistor. The equivalent resistance for two resistors in parallel is R = + eq R R or 1 2 R R2 R 1 R eq = R1 + 2 with I = I + I 0 i.e., I = 0 at a branch point. 1 2 = n n XII-2

3 EXPERIMENTAL PROCEDURE A. Measurement of resistance with an ohmmeter Arrange the three resistors by their color codes (nominal values) so that R 1 < R 2 < R 3. Evaluate the resistance (R) for each resistor using the color code, record the colors and their corresponding values. Measure the resistance of each of the three resistors with the ohmmeter. Compare the measured and the nominal values of each resistor and calculate the percent difference. The percent difference should be equal to or less than the coded percent tolerance. The values to be used in the calculations that follow should be those obtained by direct measurement with the multimeter. B. Current-voltage relation for a resistor To prevent damage to instrumentation, be sure the multimeter measuring voltage is selected to measure voltage and the multimeter measuring current is selected to measure current. Construct the following simple circuit, setting the power supply for V PS 5 volts. Here two multimeters are shown. XII-3

4 NOTES: If you are using a single multimeter, you must substitute it in each section of the circuit to measure voltage and current. For either case, it is important to take the presence of the multimeter into account. Measure and record the power supply voltage, V PS, the total current (I T ) and voltage drop (V R ) across R 1 using the multimeter first as an ammeter (I) and then as a voltmeter (V). Compare and discuss the values of resistance measured with the ohmmeter and calculated by Ohm's law. Mention possible sources of error. C. Series Circuit Construct the following series circuit: With the voltage source disconnected, measure the equivalent resistance using the ohmmeter function of the multimeter. With the measured values of the resistors recorded in Section A, calculate the expected value of resistance with the series equation. Using the multimeter, measure and record V 1, V 2, V T, I T. Calculate the indicated values of resistance using Ohm's law. Compare and discuss the values of R 1,calc and R 2,calc with their actual values measured with the ohmmeter in Section A. Also compare R T,calc with that calculated by series equation (R S ) and measured directly with ohmmeter (R eq ). XII-4

5 D. Parallel Circuit Construct the following parallel circuit: With the voltage source disconnected, measure the equivalent resistance using the ohmmeter function of the multimeter. With the measured values of the resistors recorded in Section A, calculate the expected value of resistance with the parallel equation. Using the multimeter, measure and record the V T, I 1, I 2, I T. Calculate the indicated values of resistance using Ohm's law. Compare and discuss the values of R 1,calc and R 2,calc with their actual values measured with the ohmmeter in Section A. Also compare R T,calc with that calculated by parallel equation (R P ) and measured directly with ohmmeter (R eq ). E. Series - Parallel Combination Leaving R 1 and R 2 in parallel, add R 3 in series as shown, thus constructing a series - parallel circuit. With the voltage source disconnected, measure the equivalent resistance with the ohmmeter. With the measured values of the resistors recorded in Section A, calculate the expected value of resistance with the series-parallel equation. XII-5

6 Measure and record: V 1 = V 2, V 3, V T, I 1, I 2, I 3 = I T. Calculate the indicated values of resistance using Ohm's law. Compare and discuss the values of R 1,calc, R 2,calc and R 3,calc with their actual values measured with the ohmmeter in Section A. Also compare R T,calc with that calculated by series-parallel equation (R SP ) and measured directly with ohmmeter (R eq ). XII-6

7 NAME Sec/Group Date TABLE 1 RESISTOR DATA R 1 R 2 R 3 Color Code % Tolerance Nominal val Meas value % difference In tolerance? SINGLE RESISTOR Measured values Compared with I T measured values: V T R 1,calc % difference SERIES CIRCUIT R eq measured with ohmmeter R eq calculated by series eqn. Measured values Compared with I T measured values: V T R T,calc % difference V 1 R 1,calc % difference V 2 R 2,calc % difference PARALLEL CIRCUIT R eq measured with ohmmeter R eq calculated by parallel eqn. Measured values Compared with V T measured values: I T R T,calc % difference I 1 R 1,calc % difference I 2 R 2,calc % difference XII-7

8 SERIES - PARALLEL CIRCUIT R eq measured with ohmmeter R eq calculated by series-parallel eqn. Measured values Compared with I T (=I 3 ) measured values: V T R T,calc % difference V 1 (=V 2 ) I 1 R 1,calc % difference I 2 R 2,calc % difference V 3 R 3,calc % difference XII-8

EE301 RESISTANCE AND OHM S LAW

EE301 RESISTANCE AND OHM S LAW Learning Objectives a. Describe the concept of resistance b. Use Ohm s law to calculate current, voltage, and resistance values in a circuit c. Discuss the difference between an open circuit and a short