Capacitor in the AC circuit with Cobra3 LEP Related Topics Capacitance, Kirchhoff s laws, Maxwell s equations, AC impedance, Phase displacement Principle A capacitor is connected in a circuit with a variable-frequency voltage source. The impedance and phase displacement are determined as a function of frequency and of capacitance. Parallel and series impedances are measured. Equipment Resistor in plug-in box 47 Ω 3904.62 Resistor in plug-in box 00 Ω 3904.63 Resistor in plug-in box 220 Ω 3904.64 Capacitor (case 2) µf/250 V 393.0 Capacitor (case 2) 2.2 µf/250 V 393.02 Capacitor (case 2) 4.7 µf/250 V 393.03 Connection box 06030.23 Connecting cord, l = 500 mm, red 0736.0 2 Connecting cord, l = 500 mm, blue 0736.04 2 Cobra3 Basic Unit, USB 250.50 Measuring module function generator 2.00 PowerGraph Software 4525.6 Cobra3 Universal writer software 4504.6 Power supply, 2 V- 25.99 2 PC, Windows 95 or higher Tasks. Determine the impedance of a capacitor as a function of frequency. 2. Determine the total impedance of capacitors connected in series and in parallel. 3. Determine the phase displacement between current and voltage over a RC network as a function of frequency. Set-up and procedure. Impedance measurement Connect the Function Generator Module to the Cobra3 unit and set up the equipment according to Fig.. The Analog In 2 / S2 should be connected in a way that it measures the voltage drop over the capacitor. Connect the Cobra3 unit to your USB port. Connect both Cobra3 and Function Generator Module to their 2 V supplies. Start the measure program on your computer. Select the Gauge PowerGraph. Circuit for impedance measurement Fig. : Experimental set-up PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen P244055
LEP Capacitor in the AC circuit with Cobra3 On the Setup chart of PowerGraph click the Analog In 2 / S2 symbol and select the module Burst measurement with the following parameters to enable the Analog In 2 / S2 to perform ac measurements. The obtained values are ac amplitude values, i.e. the positive peak voltage. To obtain the effective voltage in case of sine waves the values have to be divided by 22. Add a Virtual device with two calculated channels like this: Fig. 4: Virtual device settings (channel ) Fig. 2: Analog In 2 / S2 settings for ac measurement Click the Function Generator symbol and set the parameters like this: Fig. 5: Virtual device settings (channel 2) Fig. 3: Function Generator settings Set the channels to be recorded like this (see Fig. 6) and configure a diagram to be seen during measurement like this (see Fig. 7) 2 P244055 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen
Capacitor in the AC circuit with Cobra3 LEP Plot the impedance against the inverse frequency exchange the set for the x-axis with Measurement > Channel manager. The linear dependance of the impedance from the inverse frequency can be seen. To put more curves into the same diagram use Measurement > Adopt channel (Fig. 9). The impedance of the capacitor is independent of the resistance value. Fig. 6: PowerGraph settings Fig. 9: Impedance dependence on the time scale auto range 2. Total impedance of parallel and series connection Also measure the impedance of capacitors in parallel and in series connection. auto range Fig. 7: Display settings Record curves for different values of resistance and capacitance. After clicking the Continue button the Start measurement button apperars. You may stop the measurement when the current does not rise much any longer, but for easy evaluation with Adopt channel... it is best to always record the same number of values the drop down menue under Stop condition you find a feature for this. If you select only current I and voltage U2 to be displayed, current and voltage curves plotted against frequency may look like this: Capacitors in parallel Fig. 8: Current/voltage dependence on the frequency Capacitors in series PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen P244055 3
LEP Capacitor in the AC circuit with Cobra3 The rule for adding capacitances in parallel is C tot = C + C 2. Capacitances in series sum up like Use the Survey function for phase shift evaluation. With a sample rate of 200 khz one channel corresponds to 5 µs. C tot C C 2 or C tot C C 2 C C 2 3. Phase shift measurement Connect the Analog In 2 / S2 terminals so as to measure the voltage drop over both capacitance and resistance. Fig. : Sample measurement for phase shift Here a curve obtained with 2,2 µf and 00 Ohm: Start the Gauge Cobra3 Universal Writer and select the Fast Measurement chart so that your Cobra3 can be used similar to an oscilloscope. Set the parameters like this: Fig. 2: Phase shift Fig. 0: Fast Measurement and Funcion Generator settings Record curves with different frequencies for each combination of resistance and capacitance and take down the phase shift to be plotted in a separate curve. For low frequencies it may be better to put the voltage higher to get lower current noise as the current is quite low for low frequencies. You may check the phase shift with a resistor (without capacitance) to be zero. Fig. 3: Tangent of the phase shift vs. frequency 4 P244055 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen
Capacitor in the AC circuit with Cobra3 LEP Theory and evaluation The voltage U C on a capacitance C with charge is t Q t2 It2 dt The voltage on the resistance R is with current U R t2 R I t2 R dq dt U t2 U C t2 U R t2 Q t2 C for a resistor and a capacitance in series connected to an ac voltage source. Differentiating this equation yields I di R C dt v U 0 sin v t2. This differential equation has the solution 0 U C t2 Q t2 C. dq R dt. I t2 dq dt U 0 cos v t2. The impedance is then U 0 and the impedance of the capacitance alone (R = 0) is In a plot of impedance vs. inverse frequency is the slope m hence with v = 2 p f m 2 R 2 a I 0 B v C b. v C Rˆ C. 2 p C and C 2 p m From Fig. 9 the slope values read: imprinted value / µf µf slope / Ohm/ms measured value / µf 4.7 36 ±2 4.42 ±0.2 2.2 72 ±4 2.2 ±0. 42 ±7.2 ±0.05 I t2 I 0 cos v t w2 with tan w2 v CR 7 0 i.e. the current is ahead of the voltage and ms m 36 Ohm>ms C 36 V 4.42 0 6 A 2 p ms m 72 Ohm>ms C 72 V 2.2 0 6 A 2 p As V As V 4.42 mf 2.2 mf I 0 U 0 2 B R2 a v C b. ms m 42 Ohm>ms C 42 V.2 0 6 A 2 p As V.2 mf PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH 37070 Göttingen, Germany P244055 5
LEP Capacitor in the AC circuit with Cobra3 6 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen P244055