SERIES RL CIRCUIT VM The series RI circuit behaves very RA similarly to RC circuits in that the current and voltage are time-dependent That is, the voltage and current passing through different sections of the circuit R change with time. We will be exploring AM RLS how this circuit behaves within this lab and we will be considering the internal resistance of both the ammeter and inductor. We can characterize the voltage by using Kirchhoff's voltage law and Figure 1: Series RL circuit with internal resistance of Ohm's law. The voltage of the circuit the ammeter and inductor included. will be V - IR - IRA + VI = 0, (2) where RA is the internal resistance of the ammeter, and V, is the voltage across the inductor. V, has two components, one from the emf generated by the inductor (Equation 1) and another from its internal resistance (through Ohm's law). We can characterize the inductor's voltage as V = -LO - IRI. (3) dt Combining equations 2 and 3, we get a first order differential equation as di + RIJ - V - = 0, (4) dt L where Ry = R + RA + Ry or the total resistance of the series circuit. You can solve equation (4) by substituting an integrating factor in the form of a V = I(t) = 7 -+ Ce it We can use the initial condition that / (0) = 1, = 0 to solve for the constant C and get I(t ) = = (1 -z z) (5) When looking at equation S, we can determine a time constant for our circuit in a similar manner to Lab 3. The time constant is defined as the amount of time it takes the current to decrease by (1 - e-1) = 0.632 or by 63.2%. The quotient of has units of seconds when R is in ? and L RT is in Henrys (you can check this for yourself) and is called the time constant of the circuit.So the time constant for the RL circuit, including internal resistance, is given as If we didn't consider the internal resistance of our ammeter or inductors within the circuit, the time constant would take a similar form as I = Using Ohm's law along with our result for the current, we can solve for the voltage across the resistor Va(t) as VA (t ) = RV R.It - RT (1 - 8 1) (6) The important things to understand from equation 6 are the t - 0 and t - co limits which give R lim Va(t) = 0 & lim Va (t) = V- We will be studying the RL series circuit experimentally and verifying equations 5 and 6 within Part 1 of this lab. If we didn't consider any internal resistances within this circuit, the current would have a similar shape, but the voltage would take on a negative exponential which approaches zero 35 [ - 00.a. Waveform=DC. b. DC Voltage=1V. c. Set the frequency of measurement to the "Common Rate" of 1kHz (this will record a value for voltage and current every millisecond). Note: we are measuring the voltage over the resistor ONLY (as opposed to the voltage of the entire circuit). 4. Once your circuit is made, with the switch OPENED, turn on the power supply and start recording. After a few seconds (=2s), close the circuit and stop recording a few seconds after. a. Note: Capstone has trouble tracking large amounts of data, so chances are it will automatically stop recording after about 6s. 5. Your voltage and current may seem to show an instantaneous jump the moment you close the circuit. Zoom in on the x-axis over the range when the switch is first closed to a few milliseconds afterwards (200ms). To do this, you can either: a. Right click on the time axis, go to "Axis Scaling" and manually enter the min max values. b. Click and drag on the x-axis from left to right. This should show you a curve similar to what the theory predicts. As a sanity check, the moment the circuit is closed (to), our inductor should have a voltage of VI (t = to ) = Vr 6. Calculate the time constant using the Current versus Time graph. According to theory, the time constant is measured as the length of time it takes the CURRENT to reach 63.21% of its maximum value. a. Record the value of your time constant in the report for this resistance and voltage in table 1. b. Measure the value for voltage as time gets very large with each resistor and record that value in Table 2 of your report. You only need to do this when your initial voltage is set to 4V. STOPCHECKPOINT (Ask the TA to check your graphs and your value for t at IV) 7. Open the circuit and repeat steps 3-6, but with a voltage source of 2V and 4V. Record the average value for your time constant along with the standard deviation from your different voltages at that resistance in Table 2 of the report. Calculate the inductance using the average value of your time constant and the TOTAL resistance of the circuit. You can calculate the total resistance (R) of the circuit by considering the internal resistance of the ammeter (R ), the internal resistance of the inductor (R,), and the resistor's value. The total resistance will be R, = R + RA + R.9. FOR THE 4V SOURCE ONLY, title your graph based on the resistors value. You can change the title by double clicking on the bottom left where it says "[Graph Title Here]". Copy and paste your graph from Capstone into your word report. 10. Repeat steps 3-9 with resistors of value 330 and 500 and then calculate the average inductance of the calculated inductances for each resistor. Since our inductor and ammeter have an associated internal resistance with them, the voltage over time is predicted to asymptote to a value of V(t - 00) = VI RT where Vy is the total voltage provided by the power supply, R is the resistors value, and R, is the total resistance of the resistor, internal resistance of the ammeter, and internal resistance of the inductor combined (see Theory section). Question 1. Do we need to consider the internal resistance of our voltmeter when we calculate inductance? Explain. Question 2. In Lab 3, we used a similar method to calculate the time constant for a parallel RC circuit when the capacitor was discharging. Is this average value more accurate compared to the average value calculated in Lab 3? Explain why. Hint: will Capstone allow us to measure at a higher frequency? Question 3. Is the behaviour of voltage versus time as time gets very large consistent with what the theory predicts? What about the moment the circuit is closed? Compare and contrast between the different resistor values