# Lab #4 Capacitors and Inductors. Capacitor and Inductor Transient Response

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1 Capacitor and Inductor Transient Response Capacitor Theory Like resistors, capacitors are also basic circuit elements. Capacitors come in a seemingly endless variety of shapes and sizes, and they can all be represented by the following symbol. + v c (t) - Note the curved line in the symbol for the capacitor shown in Figure 1. You will sometimes see a capacitor symbolized by two parallel lines instead of one curved one. This is poor practice because that symbol is normally reserved for a relay. Many capacitors have a polarity associated with them. On a circuit diagram, this is sometimes symbolized with a small + next to the flat line. The curved line of the capacitor symbol is usually associated with the more negative voltage. It is critical that the polarity requirements of a capacitor are observed, or the capacitor is likely to fail in a violent, and possibly, explosive fashion. Capacitors also have a maximum voltage that can be applied across the terminals before the electrical insulation between the plates breaks down. Unlike resistors, which dissipate electrical energy in the form of heat, capacitors store energy in the form of an electric field. The amount of energy stored in the capacitor (in Joules) is given as 1 W CV 2 2 where C is the value of capacitance in Farads, and V is the voltage across the capacitor in Volts. The current and voltage in a capacitor (as seen in Fig. 1) are related by (1) dv i(t) C (2) dt t 1 and v(t) idtv(t 0). C (3) t 0 i c (t) Figure 1: Typical Capacitor Circuit Symbol. 1

2 One conclusion that can be drawn from the above integral is the fact that if a capacitor is charged to some initial voltage, it will remain at that voltage forever if there is nothing that provides a current path for discharge. Thus, for safety reasons, discharge capacitors with a resistor before touching any circuit with capacitors present. t=0 R V s C + v c (t) - i c (t) Figure 2: Capacitor Charging Circuit. Assuming the capacitor didn t have an initial voltage across it at t=0 when the switch is closed, the voltage across the capacitor in Fig. 2 over time is given as: v (t) V (1 e c s t/ τ ) (4) where,, is the time constant of the circuit. The time constant is given by: τ RC. (5) A time constant of a circuit is an important property of a circuit. It provides a useful measure of how fast a circuit responds to change. In the above equation, when the time is equal to one time constant, the exponential is raised to the power negative one.. It is customary to measure this point on the charge or discharge curve to determine experimentally. For two time constants, the power is negative two, and so on. After one time constant, the voltage across the capacitor is 63.2% of its final value and after five time constants has 99.3% of its final value. Similarly, we can solve for the current in Fig. 2 at any instant after the switch closes as: i (t) c Vs e R t/τ. (6) When the initial voltage on the capacitor is non-zero the voltage across the capacitor over time is given by: (t) s + - s e -t (7) Where: V 0 is the initial voltage across the capacitor and V S is the source voltage at time 0+. 2

3 i (t) - e -t (8) Equation 7 can be written in terms of the initial and final voltage across the cap. (t) ( )+ ( )- ( ) e -t (9) Where: ( ) is the initial voltage across the capacitor at time 0+ and ( ) is the final or steady state value of the source voltage. Inductor Theory Like resistors, inductors are also basic circuit elements. The impedance of an ideal inductor is given in equation 0. Like the capacitor the impedance of an ideal inductor is completely imaginary and like the capacitor the voltage across the inductor and the current through the inductor are not in phase. Also an ideal inductor has magnitude impedance at Hz and an magnitude impedance at Hz. The circuit symbol for an inductor is given in Figure 1. + v L (t) - (10) i L (t) Figure 1: Typical Inductor Circuit Symbol Unlike resistors, which dissipate electrical energy in the form of heat, Inductors store energy in the form of a magnetic field. The energy stored in an inductor is given as: where L is value of inductance in Henrys and I is the current in Amps flowing through the inductor. The voltage and current for an inductor are related by: (11) ( ) (12) and ( ) ( ) ( ) ( ) (13) 3

4 where ( ) is the initial current flowing in the inductor. One conclusion that can be drawn from the above integral is the fact that if an inductor has an initial current flowing through it the current will flow forever until it is dissipated through some resistance. Charging an inductor with a voltage source through a resistor is similar to charging a capacitor. The main difference is the exponential time constant dictates the current instead of the voltage. ( ) ( ) (14) Where and is the time constant of the circuit. t=0 R VS I L + V L - GND GND Figure 2: RL circuit time constant measurement. The time constant of the circuit shown in figure 2 is an important property of the circuit. It provides a useful measure of how fast a circuit responds to change. You may recall that a capacitor voltage changes by 63.2% from the initial to final voltage during 1 time constant. The inductor current also changes by 63.2% in one time constant. Similarly the voltage across the inductor is an exponential discharge. ( ) (15) Inductance is a fundamental circuit property like resistance and capacitance. A typical Inductor is made with wire wrapped to form a coil. The inductance is proportional to the square of the number of turns in the coil. Thus more turns of wire in the inductor results in more inductance. One of the primary ways to vary the inductance is to change the number of turns. The inductors in the lab are covered so you can t see the turns of wire. The turns of wire have an undesired property. The wire used to achieve the desired inductance has a resistance associated with it. We don t want resistance we want inductance. This parasitic resistance cannot usually be ignored. A typical model of 4

5 an inductor must then include this parasitic resistance. Such a model is shown in Figure 3. This parasitic resistance will dissipate power and we must consider this when using inductors. Inductor Model + R L i(t) = I m sin( t) L v(t) - Figure 3: Inductor with Parasitic Resistance. The model of the inductor including the parasitic resistance in Fig. 3 holds for most cases. 5

6 Instructional Objectives Analyze the transient response of a simple RC circuit. Analyze the transient response of a simple RL circuit. Procedure Parts needed for this lab: Use the R and C values from the Prelab, 1K resistor, a 100mH Inductor and 3 K resistor. That s it. For all experiments in this lab you will be using a bread-board and the Analog Discovery measurement system. Part 1: Measuring the transient response of an RC network. Before we actually measure the RC time constant there are a few things that need to be determined about the circuit and the measurement instruments. The theory section talks about the initial and final conditions of the voltage on the capacitor. We will investigate these conditions, since they influence the measured results. The initial conditions are not difficult to set or measure. To make it easy to measure τ we force the initial voltage across the capacitor to a known voltage. Then we can use Eq. 7 or 9 to measure τ with the scope. We are going to drive the RC with a very slow square wave. We do this so that the capacitor has time to get extremely close to the voltage that is driving the circuit. This defines the initial and final conditions for us because we wait long enough before the square wave repeats the waveform so it is almost like at time. Another issue we need to deal with is the influence the input impedance of the Analog Discovery has on our measurement since we will use it to measure the τ of the RC circuit. The A1+ and A1- or A2+ and A2- get connected across the resistor and across the capacitor so the input impedance will always be in parallel with the resistor or capacitor. 1. Measure the input impedance of the Analog Discovery A1+ to A1- and the impedance of R1 in figure 4? See following discussion as to how to do this. Analog Discovery input impedance Ω. Resistor value R1 Ω. The Analog Discovery has an active input impedance is 1MΩ. What does active mean? Usually most devices have an input impedance that is not just a resistor that you can measure. This is true for the Analog Discovery. Measure the ch1(1+, 1-) or ch2 (2+, 2-) with your DVM on te resistance setting. You get a value..ω Is it right? Most likely NO! So how do we find out what the input impedance is? For this lab we will 6

7 only consider the input resistance (DC) and not the input impedance which is frequency dependent. Consider the input of the Analog Discovery a black box. All you can do is measure the voltage across the input and the current into it. Then use Ohms law V/I=R to calculate the input impedance. Simple enough if you can measure the small signals to get an accurate measurement. You are in luck your DVM can just barely measure the current and easily measure the voltage. Give it a try using the following schematic and picture as a guide. Set the Analog discovery to output 5VDC from W1. Measure the W1 to GND voltage with the D M. We can t actually measure the voltage from 1+ to 1- because you need 2 meters to set this up. The ammeter will not influence this resistance measurement. The resistance of the meter is much much smaller than the input resistance of the Analog discovery. Now connect the DVM as shown in the circuit to measure the current going into the Analog Discovery. Don t forget to set the meter to ua and move the red lead to the right spot. W1 VDC, Input current ua. alculate the input resistance using these measured values. Z /I Ω How does this compare to the value you got when you simple measured the resistance across 1+ and 1- using the DVM? The input impedance discharges the capacitor while R1 charges it. Does this input impedance discharge the cap at a rate high enough to influence the measureable charging through R1? To determine this compare the R s. If the input impedance is >> than the charging R, R1, there won t be a problem unless you are trying to measure with incredible accuracy. 100:1 ratio is a 1% error. 1000:1 ratio is a 0.1% error. It all depends on the accuracy you need for your tests. 7

8 Do you need to worry about the input impedance when determining? 2. Measure the charging of a capacitor to determine : Build the circuit shown below. Figure 4: RC circuit. Setup the W1 source to put out a 0 to 4V (2V 2V OFFSET ) square wave at a frequency that has about 5 time at 4V and 5 time at 0. Set triggering to C2 Rising edge at about 2V Set the time base to xs/div which allows you to see a charging and discharging waveform on the same trace and which seems to show the initial and final voltages (Slow one). Time base = xs/div. Measure the initial and final voltages, V INIT, V FINAL across the capacitor. Make sure you are convinced the waveform is done rising or falling. Make the time base 10 to 20 times faster just to see what the waveform looks like when you don t wait long enough to find FINAL. 3. Use the cursors to measure the time constant Put cursor at the most negative across V C1, (V INIT ) right where the voltage starts rising. Change the Horizontal Time Base to xs/div to cover most of the screen with the initial to final event (Fast one). Time base = xs/div. Set the other cursor to the voltage which is ( ) 0.63 the way to V FINAL. This is 63% from V INIT to V FINAL = V INIT (V FINAL -V INIT ) V. From Eq. 9 above. Capture the resulting display for your report.. Figure 5 shows the display I captured. 8

9 Figure 5: Captured RC transient measurement. 4. Use the cursors to measure the time constant. Change the Horizontal time base to the (Slow one). Put one cursor at the most positive voltage across V C1, (V FINAL) where the voltage starts falling. Change the Horizontal Time Base back to the (Fast one). Set the other cursor to the voltage which is ( ) 0.63 the way to V INIT. This is 63% from V INIT to V FINAL = V INIT (V FINAL -V INIT ) V. From Eq. 9 above. Capture the resulting display for your report. =. 5. Measure the peak current values during charge and discharge. First measure V R. Pos, Neg. What is R Ω. Calulate I CHARGE I DISCHARGE. 6. Change R1 to 30.0K. The following steps are similar to steps 2-3. Setup the W1 source to put out a 0 to 4V (2V 2V OFFSET ) square wave. Set the frequency of the square wave such that the voltage across the capacitor has sufficient time to reach steady state 5 (Note that the frequency will have to be much lower than the frequency used in previous steps, because of the larger value of ) Set triggering to Ch2 Rising edge at about 2V. Set to the time base to cover most of the screen with the entire initial to final event. Measure the initial and final voltages, V INIT, V FINAL across the capacitor. 9

10 7. Use the cursors to measure the time constant. Change the Time Base to cover the initial to final event to sufficiently measure. Put one cursor at the most negative across V C, (V INIT ) right where the voltage starts rising. Set the other cursor to the voltage which is 0.63 the way to V FINAL as in step 3 above. Capture the resulting display for your report.. Part 2 Measuring the transient response of an RL network. 8. Measure the charging of an inductor to determine : Build the circuit shown below. Figure 5: RC circuit. Setup the W1 source to put out a 0 to 4V (2V P with 2V OFFSET ) square wave at a frequency that has about 5 time at 4V and 5 time at 0. Set triggering to C2 Rising edge at about 2V Set the time base to a xs/div. which allows you to see a charging and discharging waveform on the same trace and which seems to show the initial and final voltages (Slow one L). Measure the initial and final voltages, V INIT, V FINAL across the resistor. (Note that we are measuring resistor voltage since inductor resists change in current. Voltage across inductor changes instantaneously). 9. Use the cursors to measure the time constant. Put cursor at the most negative across V R1, (V INIT ) right where the voltage starts rising. 10

11 Change the Horizontal Time Base to cover most of the rising event on the screen. (Fast one L). Set the other cursor to the voltage which is ( ) 0.63 the way to V FINAL. This is 63% from V INIT to V FINAL = V INIT (V FINAL -V INIT ) V. From Eq. 9 above. Capture the resulting display for your report Use the cursors to measure the time constant. Change the Horizontal time base to (Slow one L). Put one cursor at the most positive voltage across V L1, (V FINAL) where the voltage starts falling. Change the Horizontal Time Base to (Fast one L). Set the other cursor to the voltage which is ( ) 0.63 the way to V INIT. This is 63% from V INIT to V FINAL = V INIT (V FINAL -V INIT ) V. From Eq. 9 above. Capture the resulting display for your report Measure the peak inductor voltage values during charge and discharge. First measure V L1. Pos, Neg. 12. Are the values of and close to the expected (theoratical) value? Why or why not? 11

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