Lab 5: Basic Direct-Current Circuits Edited 11/20/14 by Joe Skitka, Stephen Albright, WL, JCH & DGH Objective This set of mini-experiments aims to verify rules governing the addition of resistors and capacitors in series and in parallel as well as the time decay of resistor-capacitor (RC) and resistor-inductor (RL) circuits. Due to there being multiple experiments without a common goal, a full lab report is not required. Instead, students are asked to do the calculations and write up answers to questions posed at the end of the manual. This write-up need not have the usual lab report format, e.g. there need not be experiment section or introduction sections. Students are welcome to complete this write-up in class if there is time. Introduction So far this course has covered basic electrostatics, electrostatic motors, electric fields, the force of an electrostatic field, charge distribution, capacitance, dielectrics, the force between current carrying conductors, and the charge to mass ratio of the electron. This lab exercise will explore the practical electrical components that have been developed from these basic principles and how they can be used in simple direct current (DC) circuits. This lab will also explore the basic DC motor. Standard measuring instruments will be used. The components explored in this lab include resistors ( ( ) and batteries/power supplies ( voltmeters ( ) and ohmmeters ( ) ), capacitors ( ), inductors ). These will be explored with ammeters ( ), Fixed (as opposed to variable) resistors, capacitors and inductors in various configurations will be used to establish the relations between resistance, voltage, and current in simple DC circuits. Resistance, R, is measured in ohms (Ω), capacitance, C, is measured in farads (F), inductance, L, is measured in henries (H), voltage, V, is measured in volts (V) and current, I, is measured in amps (A). Your results will be compared to that predicted by theory. Note: Refer to “Introduction to Electrodynamics", by David J. Grifths, 3rd edition or "Electricity and Magnetism", by Edward Purcell for equations relating current and voltage for resistors, capacitors and inductors connected in series and in parallel. Part I - Resistance Circuits In the following schematics: represents a voltmeter; it measures the potential difference (V) across its terminals represents the voltmeter mounted in the power supply – values are displayed on the front panel of the power supply represents an ammeter; it measures the current (I) passing through a circuit element represent an ohmmeter; it measures inherent resistance (Ω) – Note: ohmmeters can only be used when no power is applied to the circuit element being measured R1, R2 and R3 are resistors represents the power source, in this case a power supply represents an on off switch Partially constructed circuits are provided for convenience (refer to figures 1, 2, 3, and 4). In order for current to flow, it is necessary to ensure that there is continuity between the positive and negative terminals of the battery, i.e., that the circuit is complete, as shown in the recommended configurations in schematics 1, 2, 3, and 4. A) Simple Series Circuit: Verify that resistors add as expected in series, Kirchoff’s law, and conservation of current in magnetostatics. The following procedure is recommended to answer the questions posed on this circuit. You are welcome to attempt to answer the questions without an explicit procedure if you have time though. Because the individual resistors may have resistances slightly different than their stated values, it is recommended that you use an ohmmeter to measure the resistance of R1, R2 and R3 separately. An ohmmeter is built into the Agilent (orange) multimeter. Readings should be taken before the circuit is set up as resistance is an intrinsic property of the component, not of its place in the circuit. Schematic 1 depicts the recommended configuration from which all relevant values can be measured simultaneously. Set the power supply to approximately 3 volts DC as read on the power supply voltmeter (its voltage, Vp, is displayed on its front panel, so no additional voltmeter is needed). Record the current (I) and the voltage across R1 (VR1), R2 (VR2), and R3 (VR3). Figure 1 Schematic 1 Data Analysis: Uncertainty estimates are optional in this lab. So, while it is not possible to use uncertainty in determining the validity of the principles below, use intuition about how large the error is in estimating whether your measurements are likely consistent with the expectations described. i. Assess Kirchoff’s law in the context of this circuit. Kirchoff’s law is that the voltage around a complete circuit sums to zero. ii. Using Ohm’s law, verify the conservation of current through each of the resistors. iii. Verify the addition rule for resistors in series. This can be done by using I and Vp to determine the effective resistance of the circuit and comparing this with the measured individual resistances. B) Simple Parallel Circuit: Verify that resistors add as expected in parallel, Kirchoff’s law, and conservation of current in magnetostatics. Refer to suggestions from part A and schematic 2 for guidance Figure 2 Schematic 2 Data Analysis: i. Assess Kirchoff’s law in the context of this circuit. Kirchoff’s law is that the voltage around a complete circuit sums to zero. ii. Using Ohm’s law, verify the conservation of current through the resistors. iii. Verify the addition rule for resistors in parallel. This can be done by using I and Vp to determine the effective resistance of the circuit and comparing this with the measured individual resistances. Part II - Resistance and Capacitance (RC) Circuits This lab exercise will explore the discharge rate of capacitors. Fully charged capacitors will be discharged through a known resistor. The rate of discharge will be examined. The discharge of capacitor C through resistor R is analyzed in the textbook. The charge q remaining in the capacitor, at time t after discharge begins, decreases exponentially with a time constant equal to RC. Since the time-dependent current is I = dq/dt , it is easy to show that the current also decays exponentially. Instead of the constant current measured in Schematics 1&2 the current in an RC circuit is a function of time t: I(t) = I0 e -t /RC ; loge (I/I0 ) = -t/RC (1) Where t = 0 represents the start of discharge. During discharge, the capacitor is the power source. Caution: The capacitors used in this lab have fairly high capacitance and it may be possible to get a shock from one even after power to it has been removed. Always turn off the power source and discharge all capacitors before touching the circuit connections. An insulated shorting conductor is supplied for this purpose - be sure to hold the shorting conductor by its insulated handle. A) Simple Series RC Circuit: The main goal of this exercise is to measure the decay constant (1/RC) of the circuit. This will be done by fully charging the capacitor (like filling a balloon) and suddenly turning off the power supply (letting the balloon empty). The amount of time it takes for the capacitor to lose a certain portion, e = 2.718, of its charge is the time constant. A stopwatch is provided to observe this. Schematic 3 is recommended for this exercise, but are welcome to attempt to answer the questions however you would like. Note C1 & C2 are 330 microfarads each. Don’t forget to measure the resistance of the resistor individually. Construct a suitable circuit like the one shown in schematic 3. Be sure to include the switch in the circuit as its position is not labeled on the circuit plate. Note that you should use Agilent multimeter, not the one you were using before, as it is more precise and easier to read quickly. Be sure to connect this using its Com and μA ports. The microamp scale is appropriate for the measurements being made here. With the switch box, S, turned off, set the power supply to approximately 18 volts DC as read on the power supply voltmeter (Vp). Turn switch S on and wait for the current to become constant. For this specific circuit, the current which will pass through the ammeter immediately after the power source is turned off, I0, should be exactly equal to the current going through it immediately before the power is turned off. This is because the resistance, R, is the same in both configurations. If there had been a resistor on the left half of the circuit in schematic 3 (near the power supply), the current after the switch is opened would actually had been larger. You can use the value of I0 to determine values of I0/2, I0/4, I0/8 and I0/16 in advance. Simultaneously turn switch S off and start a stopwatch. Record the time required for the current to fall to I0/2, I0/4, I0/8 and I0/16. Note: As the current decreases, it will pass through 50 μA. At approximately that value, the meter will change scales. The readings before and after the scale change are accurate, it is just helpful to be aware the change will occur. Figure 3 Schematic 3 Data Analysis: Remember to specify estimated uncertainties in your experimental measurements when comparing measured and calculated values. When you are asked to “verify” or “assess” a certain physical principle, this does not mean you necessarily expect perfect agreement. If measurements disagree with expectations, explain why or find your error. i. Investigate the validity of exponential decay of the charge, Eq. (1), using a semi-log (loglinear) plot. ii. Determine the value of RC (ohms × farads = seconds). iii. Verify the addition rule for capacitors in parallel using this result. B) Simple Parallel RC Circuit: Once again, the main goal of this exercise is to measure the decay constant (1/RC) of the circuit, this time with the capacitors in parallel configuration (see figure 4). Refer to suggestions from part A and schematic 4 for guidance. Figure 4 Schematic 4 Data Analysis: Remember to specify estimated uncertainties in your experimental measurements when comparing measured and calculated values. When you are asked to “verify” or “assess” a certain physical principle, this does not mean you necessarily expect perfect agreement. If measurements disagree with expectations, explain why or find your error. i. Investigate the validity of exponential decay of the charge, Eq. (1), using a semi-log (loglinear) plot. ii. Determine the value of RC (ohms × farads = seconds). iii. Verify the addition rule for capacitors in series using this result. Part III - Resistance and Inductance (RL) Circuits This lab exercise will explore the charge and discharge rate of inductors. Inductors will be charged and discharged through a resistor with a varying applied voltage. Charge and discharge rates are examined in the textbook. Due to the magnetic field generated by the inductor, the current through an inductor resists change. Therefore, when the voltage across an inductor suddenly changes, the current will exponentially approach its new value with time constant equal to L/R. Explicitly, the current as a function of time for charging and discharging from an applied voltage V with resistor R in series is Icharge(t) = V/R(1- e -t R/L) (2) Idischarge(t) = I0 e -t R/L (3) Where t = 0 represents the start of the charge/discharge. During discharge, the inductor acts as a current source. Due to the limitations of the properties of real inductors, an RL circuit cannot be built that allows measurement of a change in current with a stopwatch. To allow measurement, an oscilloscope (or scope) will be used. Oscilloscopes are electronic devices that graphically display electrical signals. The scope measures voltage very rapidly and plots the values against time. Imagine plotting the measurements in the RC experiment against time – this is essentially what a scope does, just much more rapidly. The voltage will be supplied by a function generator, an electronic device that generates signals. In this case, it will simply be a square wave – the signal switches between 0 and 10V. The frequency of this switching is 600 Hz, so the voltage turns on/off every 1.67 ms. While use of function generators and scopes will be required in subsequent lab courses, both will be setup already; they only need to be turned on and adjusted according to specific instructions to be given. Schematic 5 Simple Series RL Circuit: If you are confused by the use of the oscilloscope, consult the TA. 1) Use an ohmmeter to measure the resistance of resistor R in Schematic 5 2) Construct the circuit in Schematic 5. You may use one of the 100 Ohm resistors from the first plate of part 1. Also, ensure that the black output of the function generator is on the same side of the resistor as the black input to the scope (scope ground). Use the 15mH commercial inductor manufactured by Miller (not the longer, homemade coil). 3) Turn on both the function generator and the scope. The waveform on the scope should resemble Figure 5. Figure 5 4) There should be yellow lines running vertically across the screen. These are cursors. Their positions are controlled by the Multipurpose A and B knobs on the front panel of the scope. The box in the upper right hand side of the screen displays their positions in time, the voltage of the waveform when they intersect, and the differences between the cursors. Put cursor A at the maximum of one discharge (right before the discharge begins). Record this voltage as V0. Note: I0 = V0 / R. Note that the discharge has an identical form as the charge stage. You may use either; however, be sure to subtract the asymptotic value of the exponential trend to get a meaningful fit. 5) On the same discharge curve, put cursor B where the voltage is 1/2, 1/4, 1/8, and 1/16 the maximum voltage and record the time from cursor A – the time from the beginning of the discharge. 6) Using data collected from the charge and discharge cycles, calculate the time constant L/R. 7) Using the time constant and the measured value of resistance, calculate the inductance of the inductor. How does the measured value compare to the value written on the inductor (15 milliHenry)? 8) Measure the resistance of the inductor with an ohmmeter. Recalculate the inductance of by assuming the inductor resistance is in series with the resistor. How does this value compare with the one published on the inductor? Data Analysis: Do the following with your data from the charging and discharging cycles: i. ii. Investigate the validity of exponential decay of current through an RL circuit, Eq. (2), using a semilog plot. Determine the value of the time constant, L/R (henries ∕ ohms = seconds). iii. Using the measured value of R, calculate the inductance of the inductor in the circuit. Compare with the value printed on the inductor. iv. Refine this calculation by accounting for the resistance of the inductor. Hint: you don’t need to use an Ohm meter; the data you have should be sufficient. Self-Inductance of a Coil This lab exercise will explore the effects of rapidly changing current in an inductor. Faraday discovered that a time varying magnetic flux induces an electromotive force (emf) in a conductor that opposes the change in flux. We can demonstrate this using an inductor, which in our case will be a small handmade coil. When a current passes through the coil, a magnetic field is produced that generates a magnetic flux through the coil. If the current through the coil is changing, so is the flux . From Faraday’s Law, there will be a voltage drop across the coil given by: 𝑉=− 1 𝑑𝛷 (4) 𝑐 𝑑𝑡 The flux through the coil depends only on the current I in the coil and some geometrical factors. Thus, we can write the induced emf V as dI/dt times a constant (defined as inductance L) specific to the coil: 𝑉 = −𝐿 𝑑𝐼 (5) 𝑑𝑡 For a long solenoid of length , radius r, and N turns of wire, with a uniform field B = 4NI/c inside (the flux through each loop is then r2B), the self-inductance is: 𝐿= 4𝜋2 𝑁2 𝑟 2 𝑙𝑐 2 sec 2 cm−1 (6) The above expression for inductance L is in CGS units. The SI unit of inductance is the “henry”, where 1 henry = 1.11 10-12 sec2 cm-1. In SI units, formula 3 becomes 𝐿= 𝑁2 𝜋𝑟 2 𝜖0 𝑐 2 𝑙 henry (7) Simple Series RL Circuit Part II: 1. Estimate the number of turns, radius and length of the homemade coil. Using a ruler, provided on the central counter of the lab room, will save you time. Calculate the inductance of your coil using Eq. (7). 2. Replace the inductor in Schematic 5 with the homemade inductor. 3. Set the function generator to a triangle wave by pressing the down arrow, in the “Func” section of the panel, once. This should produce a wave that looks like Figure 6. Set the function generator to 10 kHz by pressing the following buttons in the “Data Entry” section: 1, 0, Vpp/kHz. The Vpp/kHz button is located on the extreme right, with an up arrow on it. 4. On the scope screen, there should be a time on the bottom center part of the screen. Rotate the Horizontal Scale knob clockwise until this time reads “40.0 µs.” 5. In this configuration, the oscilloscope is measuring the voltage across the resistor. Since V = IR, this voltage is proportional to the current in the RL circuit. Use the cursors to find dI/dt in amps/sec. Record your value. Figure 6: Voltage Across R 6. Now switch the oscilloscope input so that it is measuring across the homemade inductor instead of the resistor, as shown in Schematic 6. Schematic 6 7. On the bottom, left-hand side of the scope screen, there should be a voltage next to a yellow “1.” Rotate the Vertical Scale knob under “1” clockwise until the voltage on the screen reads “10.0 mV.” The waveform should look similar to Figure 7. Figure 7 The sudden change in voltage across the indicator, VL, comes from the abrupt change in dI/dt: Δ𝑉𝐿 = 2𝐿 𝑑𝐼 𝑑𝑡 (8) 8. Press the Cursors button once to get horizontal lines across the screen. Use the cursors to measure the drop in voltage, VL. Data Analysis: i. Using equation 7, calculate the theoretical inductance of the homemade coil from the measured dimensions and number of turns. ii. Calculate dI/dt from measurements made of the slope of the triangle wave in step 5. iii. Using the measured VL and dI/dt, calculate the inductance of the homemade coil using Equation 8. Compare this value to the one calculated from the coil’s dimensions. Questions: a) What shape would you expect the waveform to have in the limit of no resistance? b) How does the coil’s resistance affect the waveform? Aside: The sudden voltage drop measured above is common when working with inductors and can become quite dangerous with large enough currents. In a real circuit with a large inductor, one must avoid shutting off the current supply suddenly; if the current shuts off suddenly, a huge voltage will develop across the inductor. This voltage can be large enough to destroy other electronics in the circuit or even discharge through the air, seriously injuring or killing someone nearby. A good example that requires attention to this is the large electromagnet used in the upper level labs for pulsed-NMR experiments.