Lab #4 Capacitors and Inductors. Capacitor Transient and Steady State Response


 George Sparks
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1 Capacitor Transient and Steady State Response 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 2 W = CV (1) 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 dv i(t)= C dt (2) and t 1 v(t) = idt v(t0). C (3) t 0 i c (t) Figure 1: Typical Capacitor Circuit Symbol. 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. 1
2 t=0 R V s C v c (t)  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: t/ τ v (t) = V (1 e ) (4) c s where, τ, is the time constant of the circuit. The time constant is given by: i c (t) Figure 2: Capacitor Charging Circuit. 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. 1 ~.63. It is customary to measure this point on the charge or discharge curve to determine τ 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: Vs t/τ ic(t) = e. (6) R When the initial voltage on the capacitor is nonzero the voltage across the capacitor over time is given by: V C t =V s (V 0 V s )e t τ (7) Where: V 0 is the initial voltage across the capacitor and V S is the source voltage at time 0. i c t = (V 0V S ) e t τ (8) R Equation 7 can be written in terms of the initial and final voltage across the cap. 2
3 V C t =V [V 0 V ]e t τ (9) Where: V 0 is the initial voltage across the capacitor at time 0 and V is the final or steady state value of the source voltage. We can determine the current running through a capacitor by measuring the voltage across a resistor in series with the device under test (DUT). The current going through a set of components that are in series is always the same phase in each component. Therefor measuring the current through any one of the components will be the current through any of the components. If the component is a resistor then we can measure the voltage across the resistor and divide it by the resistance to give the current. This is easy as long as there is a resistor in the right place in the circuit. You are lucky because there is a resistor in the right place for you to measure current with it. Note this only works with a resistor because the current through it and voltage across it are always in phase. You can actually use the scope to measure the current through a device and the voltage across it to see what the phase relationship is between the current and voltage. You actually measure the current through a different device which is in series and the voltage across the actual device. Here is how to do it. VPULSE 20mS 10mS Vin R1 1K Vout C1 0.1uF A2 A2 A1 A1 A2 SUBV A1 SUBV Vr Vc Math1 ADDV Vsource Figure 3: Current measurement with the Analog Discovery. You measure V R1 using the A1 input. The A1 voltage gets divided by the resistor to scale the measured voltage to current in Amps. WARNING: MEASURING PHASE ANGLE IS NOT A TRIVIAL TASK. Measuring phase angle was covered in the first lab it will not be repeated here. 3
4 Instructional Objectives Determine the time constant of a simple RC circuit. Measure the phase angle between a voltage and a current. Measure the transient response of an RC. Measure the steady state phase response of a C. Procedure Parts needed for this lab: 0.1uF film capacitor (Red one), A 1K (1.05K) and 10K resistor. That s it. For all experiments in this lab you will be using a breadboard and the Analog Discovery measurement system. 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 Analog Discovery has an input impedance is 1MΩ. The A1 and A1 or A2 and A2 get connected across the resistor and across the capacitor so the impedance will always be in parallel with the resistor or capacitor. 1. What is the input impedance of the Analog Discovery A1 to A1 and the impedance of R1? Analog Discovery input impedance Ω. Resistor value R1 Ω. 4
5 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. Do you need to worry about the input impedance when determining Ƭ? 2. Measure the charging of a capacitor to determine τ charge : Build the circuit shown below. A2 A2 A2 Vr Vin R1 1K Vout SUBV Math1 Vsource VPULSE C1 0.1uF A1 A1 A1 SUBV Vc ADDV Figure 4: RC circuit. Setup the source to put out a 0 to 4V (2V 2V OFFSET ) square wave at 200Hz. Set triggering to C2 Rising edge at about 2V Set the time base to 500uS/Div. Measure the initial and final voltages, V INIT, V FINAL across the capacitor. 3. Use the cursors to measure the time constant τ charge. Put cursor at the most negative across V C, (V INIT ) right where the voltage starts rising. Change the Horizontal Time Base to 20uS or 50uS/Div. Set the other cursor to the voltage which is 1 ~0.63 the way to V FINAL. This is 63% from V INIT to V FINAL = V INIT 0.63(V FINAL V INIT ) V. From Eq. 9 above. Capture the resulting display for your report. τ charge. Figure 5 shows the display I captured. 5
6 Figure 5: Captured RC transient measurement. Calculate the frequency that the time constant Ƭ represents. 1 ω Ƭ rads/sec, f Ƭ Hz. 4. Use the cursors to measure the time constant τ discharge. Change the Horizontal time base to 500uS/Div. Put one cursor at the most positive voltage across V C, (V FINAL) where the voltage starts falling. Change the Horizontal Time Base to 20uS or 50uS/Div. Set the other cursor to the voltage which is 1 ~0.63 the way to V INIT. This is 63% from V INIT to V FINAL = V INIT 0.63(V FINAL V INIT ) V. From Eq. 9 above. Capture the resulting display for your report. τ discharge. 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. Measure V OUT and V IN to determine the transfer function of this circuit. Replace with AWG2. Set AWG2 to a 2.0V P sine wave to the same frequency (Hz) calculated from Ƭ above. Change the time base to display at least 3 cycles. Measure V IN and V OUT. See Fig. 6. V IN, V OUT. Calculate the transfer function gain A. A = V OUT /V IN. Convert the gain to db. A db = 20log10(A). 6
7 7. Calculate the phase difference between the voltage across the capacitor and the current through the capacitor. Measure the time difference between the positive to negative zero crossings. Hint the time difference will be less than or equal to Ts/4. T S = 1/F S. t. Does the voltage or current waveform occur first?. Calculate the phase angle between the voltage and current with the correct sign on the angle. f Ƭ, θ = t*f Ƭ * (2) Change R1 from 1.0XK to 10.0K. The following steps are similar to steps 27. Replace AWG2 with. Setup the source to put out a 0 to 4V (2V 2V OFFSET ) square wave at 20Hz. Set triggering to Ch2 Rising edge at about 3V Set to the time base to 5mS/Div. Measure the initial and final voltages, V INIT, V FINAL across the capacitor. 9. (3) Use the cursors to measure the time constant τ charge. Change the Time Base to 200uS or 500uS/Div. 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. τ charge. Calculate the frequency that the time constant Ƭ represents. 1 ω Ƭ1 rads/sec, f Ƭ1 Hz. 10. (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. 11. (6) Measure the V OUT and V IN to determine the transfer function of this circuit. Replace with AWG2. Change AWG2 to a 2.0V P sine wave of the same frequency calculated from Ƭ. Change the time base to display at least 3 cycles. Measure V IN and V OUT. V IN, V OUT. Calculate the transfer function gain A. A = V OUT /V IN. Convert the gain to db. A db = 20log10(A). 7
8 12. (7) Calculate the phase difference between the voltage across the capacitor and the current through the capacitor. Measure the time difference between the positive to negative zero crossings. t. Does the voltage or current waveform occur first?. Calculate the phase angle between the voltage and current with the correct sign on the angle. f Ƭ1, θ = t*f Ƭ1 * Set up the circuit and Analog Discovery as shown in Figure 6 below. Connect 1 from the Analog Discovery to the protoboard. Remember is in the Analog Discovery so you only need to connect W1 to R1. A1 R1 10K A2 VIN 20Hz20KHz 5.000Vpp C1 0.1uF   VOUT A1 A2 Figure 6: RC Bode plot measurement setup. 14. Measure V C /V S : Run the Bode (Network Analyzer) application. The Help menu is quite good for this app. Use it if you want to play with the app. The suggested Bode Analyzer settings are listed here and shown in the figure below: Use as the signal source as shown in Fig. 6. Connect A1 to and A1 to. Connect A2 to C1 and A2 to. Set the waveform source to 5.0 V. Set the Start frequency to 20Hz. Set the Stop frequency to 20KHz. Use 200 Steps. Use AWG offset of 0V Set MaxGain = 1X 8
9 Adjust the Bode Scale to give a good looking plot which allows you to see the whole curve. Perhaps: Amplitude 0 to 50dB. Phase 0 to 90. Leave the Scope Channels set to the default values. What is the 3dB frequency in Hz? Zoom in if you have to. What kind of transfer function is this? High Pass, Low Pass or Band Pass. 15. Swap the capacitor and the resistor. The circuit should now look like the one shown in Figure 7. A1 A2 VIN 20Hz20KHz 5.000Vpp C1 0.1uF R1 10K   VOUT A1 A2 Figure 7: CR Bode plot measurement setup. Measure the Bode plot again. Capture this plot for your post lab report. What kind of filter is this? What is the 3dB frequency? Hz. How does the 3dB frequency compare to the frequency calculated from τ? <, >, = 9
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