RL Circuit Equipment: Capstone with 850 interface, 2 voltage sensors, RLC circuit board, 2 male to male banana leads approximately 80 cm in length 1 Introduction The three basic linear circuit elements are the resistor, the capacitor, and the inductor. This lab is concerned with the characteristics of inductors and circuits consisting of a resistor and an inductor in series (RL circuits). The primary focus will be on the response of an RL circuit to a step voltage and a voltage square wave. 2 Theory 2.1 Inductors An inductor is a 2-terminal circuit element that stores energy in its magnetic field. Inductors are usually constructed by winding a coil with wire. Initially, as the current increases in the coil, a magnetic field begins building across the coil. When the current stops flowing, the inductor will oppose the current change (Lenz's law) and try to keep the flow of current. This will be done producing a current in the opposite direction by the magnetic field it had stored. To increase the magnetic field inductors that are used for low frequencies often have the inside of the coil filled with a magnetic material. (At high frequencies such coils can be too loosy.) Inductors are the least perfect of the basic circuit elements due to the resistance of the wire they are made from. Often this resistance is not negligible, which will become apparent when the voltages and currents in an actual circuit are measured. If a current I is flowing through an inductor, the voltage VL across the inductor is proportional dI to the time rate of change of I, or dt . We may write dI V = L ; (1) L dt where L is the inductance in henries (H). The inductance depends on the number of turns of the coil, the configuration of the coil, and the material that fills the coil. A henry is a large unit of inductance. More common units are the mH and the µH. A steady current through a perfect inductor (no resistance) will not produce a voltage across the inductor. The sign of the voltage across an inductor depends on the sign of dI=dt and not on the sign of the current. The figure on page 2 shows the relationship between the current and voltage for a resistor and an inductor. 1 General Physics II Lab: RL Circuit Relationship between current and voltage in a resistor and inductor The arrow indicates the positive direction of the current through the two devices (resistor and inductor). That is, if the current is in the direction of the arrow, then I > 0. [Note that just because the arrows point down doesn't mean that the current is flowing in the downward direction. The current can also flow in the upwards direction, but in that case I < 0.] As mentioned above, the coil of wire making up an inductor has some resistance RL. This resistance acts like a resistor RL that is in series with an ideal inductor. (An ideal inductor is one with no resistance.) In this situation, the voltage across the inductor is dI V = L + R I: (2) L dt L 3 RL Circuits A series RL circuit with a voltage source V(t) connected across it is shown in Fig. 1. Figure 1: The loopy arrow indicates the positive direction of the current. The + and − signs indicate the positive values of the potential differences across the components. The voltage across the resistor and inductor are designated by VR and VL, and the current around the loop by I. The signs are chosen in the conventional way. I is positive if it is in the direction of the arrow. Kirchoff's law, which says that the voltage changes around the loop are zero, may be written VL + VR = V: (3) 2 General Physics II Lab: RL Circuit V0 R Voltage V across the resistor V R 0 V0 L Voltage V across the inductor V L 0 0 time Figure 2: Voltages VR across the resistor R and VL across the inductor L as a function of time after the source voltage is switched from V = 0 to V = V0. dI Assuming RL = 0 and letting VL = L dt and VR = IR Eq.(3) becomes dI L + RI = V: (4) dt When the applied source is shorted. The solution to the equation (V(t)=0=a short circuit) is − t I(t) = I0e L=R , where I0 is the current through the circuit at time t=0. This solution leads − t − t immediately to VR = I0Re L=R and VL = −I0Re L=R . The solution decays exponentially with a time constant of L/R. − t When a constant voltage is appled to an RL circuit e L=R describes the time dependence of the circuit. It is only necessary to use physical intuition and put the appropriate 2 constants into the solution. One guiding principle is that the current and VR do not change the instant the voltage across an RL circuit is changed. VR and I are continuous. The entire voltage change must appear across the inductor. After a constant voltage V is applied, the current in the circuit will exponentially approach V/R if the inductor has no resistance, or V/(R+RL) if the inductor has a resistance RL. Consider an RL circuit where V=0 and there is no current. We assume that RL = 0. If at time t=0 a constant voltage V is put across the circuit, VL and VR are given by, for t≥0, − t − t VL = V e L=R and VR = V (1 − e L=R ): (5) This behavior is illustrated in Fig. 2. The current is not plotted, but remember that the current is proportional to VR. Initially all of V appears across L because the current just before and after the application of V is zero. As the current exponentially builds up the voltage across the resistor increases and the voltage across the inductor decreases. If we wait a time T/2 where T/2 is many ∼ time constants (reminder time constants are L/R, 2L/R, 3L/R, etc) we will have VR = V and ∼ VL = 0. If now V is set equal to 0 (this is equivalent to shorting the circuit) VL and VR will be given by − t − t VL = −V e L=R and VR = V e L=R : (6) The response of an RL circuit to a constant voltage which is at first V and then 0 can be observed by applying a square wave to the circuit where one leg of the square wave has zero voltage and the other has voltage V. QUESTION. How would Fig. 2 be modified if the inductor had some 3 General Physics II Lab: RL Circuit resistance? QUESTION. If T L=R, what is the response of an RL circuit to a symmetric square wave that oscillates between +V and -V (assume RL = 0)? If a high frequency square wave, such that T L=R, is applied to an RL circuit, the changes in current and VR are minimal. There is not enough time for the current through the inductor to change much before the voltage is reversed. If the square wave is not symmetric with respect to ground the average VR will be the average voltage of the square wave, assuming RL = 0. 4 Measuring RL In this section the DC value of RL will be measured. Run the Capstone software. Go to the hardware setup window. Click on the output of the 850 interface and select Output Voltage - Current Senor. In the tools column, select Signal Generator icon and click on the 850 output 1. In waveform, pick DC, program it for 2 volts DC, and click on Auto. Next, setup a digits display and in select measurements pick Output Current. Program the digits display for at least 2 decimal places. Connect the output of the 850 interface directly across the coil (see Fig. 3) and click Record and then Stop. Inductor Resistors Inductor Core Figure 3: Picture of the circuit board From your data calculate RL, the resistance of the coil. Remove the wires from the coil. During your experiments compare the value of RL to the values of the resistances used in the RL circuits. Why do we need to measure RL? 4 General Physics II Lab: RL Circuit 5 Experiments In this section, the response of an RL circuit will be examined experimentally using the signal generator, 2 voltage sensors, and the oscilloscope display. In Capstone, go to the Hardware Setup window. Click on the output of the 850 interface and select Output Voltage - Current Senor. Program the analog inputs of the 850 interface for the two voltage sensors. Next, drag the scope icon to the center of the white screen. Resize the windows by using the orange tack. Add two additional y axis's to the scope by clicking the Add new y-axis to scope display icon twice. On the scope display, click Select Measurement on each axes, and set one as Output Voltage, and the other two as the analog voltages. In the experiment, various resistors will be connected in series with an 8.2 mH inductor. Hook the circuit and voltage sensors up as shown in Fig. 7, paying attention to the polarities of the leads and terminals. The polarities are very important in this experiment. A positive voltage from a voltage sensor will correspond to a positive voltage as given by the convention of Fig. 2. The input square wave to the RL circuit will be in the Output Voltage channel, VL and VR will be on the other two channels.