Lab. 1: Simple Linear Circuit Analysis

Transcription

1 Lab. 1: Simple Linear Circuit Analysis Philippe Piot (February 9th, 27) 1. Ohm's Law The circuit shown in Figure 1 was built with resistance R=1 and then 1 kω. For these two values of the resistance, the voltage was varied from to 1 V and the corresponding measured current was recorded. Tables 1 and 2 present the obtained experimental data. Figure 1: Circuit built to verify Ohm's law. The voltage source provides a tunable DC voltage, and the resistance was set to 1 and 1 kω. Set Voltage (V) Measured Current (ma) Table 1: Measured current dependence versus voltage for R=1 kω.

2 Set Voltage (V) Measured Current (ma) Table 2: Measured current dependence versus voltage for R=1 kω. The data presented in Tables 1 and 2 are plotted in Figures 1 and 2 respectively. For both set of data, di the relation between current and voltage is linear. The slopes of these lines,, directly give the dv di resistance value via dv = 1 R provide the value of the resistance.. A linear regression of the data give the slope and its inverse directly Figure 2: Measured current versus set voltage for R=1kΩ in circuit shown in Figure 1.The squares are the experimental measurement, and the red line the result of a linear regression.

3 di For the data presented in Figure 2, we computed the slope to be dv = ma/v and the corresponding resistance is 1.6 kω, i.e. within.6% in agreement with the expected value of 1 kω. di For data shown in Figure 3, we computed the slope to be dv = ma/v. The inferred resistance value is 9.96 kω, i.e. within.4% in agreement with the expected value of 1 kω. Figure 3: Measured current versus set voltage for R=1kΩ in circuit shown in Figure 1. The squares are the experimental measurements, and the red line the result of a linear regression. In summary we have demonstrated Ohm's law, by (1) verifying that the voltage drop across a resistor is proportional to the current flowing through the resistor, and (2) showing, for two cases of resistances, that the coefficient of proportionality is the resistance associated to the resistor. 2. Nonlinear Effects The resistor shown in Figure 1 was replaced by an incandescent bulb, and a set of data of current versus set voltage was recorded. The raw data are gathered in Table 3 and a corresponding plot is presented in Figure 4. The relationship between the current and voltage is no more linear. As the voltage increases, the current rate of increase is decreasing: the local slope of the curve decrease indicating that the resistance (which is related to the inverse of the slope) increases with the voltage.

4 Set Voltage (V) Measured Current (ma) Table 3: Measured current dependence versus voltage for the circuit of Figure 1. The resistance as been replaced with a incandescent light bulb. Figure 4: Measured current versus set voltage for the case where the resistance shown in Figure 1 is replaced by an incandescent bulb. 3. Series Circuit The circuit shown in Figure 5 was built. The DC voltage provided by the voltage source was set to 5 V. The voltage across each resistor was measured; see Table 4 for a summary of the measurements. The current was measured to be I=.64 ma by inserting an ampère-meter in series between the source and R 1. Because the circuit consists of one closed path with resistors only, the current at any location in the circuit is.64 ma.

5 Figure 5: Circuit built to verify the Kirchoff's voltage law. The resistance values are R1=1, R2=2.2 and R3=4.7 kω. Resistor Voltage across resistor (V) R 1 R 2 R Table 4: Voltage measured across each resistor for the circuit presented in Figure 5. The current flowing through the circuit was measured to be.64 ma. From Table 4, we easily verify the Kirchoff's voltage law: the sum of the voltage across each resistors is = 5.3 V in agreement with our set voltage on the source (5 V). We can also experimentally determine the equivalent resistance of the series circuit: from V circuit = R circuit I circuit, we find R circuit =5 / =7.81 kω, a value in decent agreement with the expected value 7.9 kω (since R circuit =R 1 R 2 R 3 for a series circuit). 4. Parallel Circuit The circuit shown in Figure 6 was built and an ampère-meter. The DC voltage provided by the voltage source was set to 5 V. The voltage across the branches was measured to be 5. V and the current in each branch was measured by inserting an ampère-meter in series with the resistors in each branch; the measured currents are presented in Table 5. The current was also measured in the main branch and found to be 8.37 ma. Figure 6: Circuit built to verify the Kirchoff's current law. The resistance values are R1=1, R2=2.2 and R3=4.7 kω.

6 Branch containing resistor R 1 R 2 R 3 Current in branch (ma) Table 5: Current measured in each branch for the circuit presented in Figure 5. The voltage across each resistor is approximately 5 V.. From Table 5, we easily verify the Kirchoff's current law: the sum of the current flowing in each branch is is =8.36 ma, a value in close agreement with the measured current upstream of the branches (8.37 ma). We can also experimentally determine the equivalent resistance of the series circuit: from V circuit = R circuit I circuit, we find R circuit =5 / =597 Ω, a value in decent agreement with the expected value Ω (since R circuit =1 / 1/ R 1 1/ R 2 1/ R 3 for a parallel circuit).