PHYS-­‐102 LAB-­‐04 CAPACITORS and RC-­‐CIRCUITS 1. Objective The objective of this experiment is to measure the capacitance of a single capacitor in an RC circuit and to examine the effective capacitance of two capacitors when connected in a series, and a parallel configuration. 2. Theory Figure 1.A series RC circuit. In this lab the capacitance will be experimentally determined by measuring the time-­‐dependent voltage across a capacitor in an RC circuit driven by a periodic square voltage. The theoretical aspects of an RC circuit and how the capacitors behave in serial and parallel connections have been discussed in your textbook and only a brief review is presented here. Consider an RC circuit as shown in the diagram. If switch S is pushed down so as to start charging the initially uncharged capacitor C, then at some time t, the magnitude of the charge on one of the capacitor plates is given by Q(t) = Qo(t)[ 1-­‐ exp(-­‐t/RC)] [1] where Qo(t) = VbattC. The potential difference across C is given by V(t) = Vbatt [ 1-­‐ exp(-­‐t/RC)] [2] If switch S is thrown in the ’up’ position after the capacitor has been fully charged, the charge on the capacitor will exponentially decay with time and is given by Q(t) = Qo(t)exp(-­‐t/RC) [3] [4] Q(t) = Qo(t)exp(-­‐ 1) = Qo(t)/e = 0.37 Qo(t) [5] And the voltage across C is given by V(t) = Vbatt exp(-­‐t/RC) When t = RC, according to [3] or the charge has fallen to 37% of its original value. The time when t = RC is called the time constant τ of the RC circuit. [Note: during the charging process, the charge on C would have risen to 63% of its final value in one time constant. Similar statements can be made for V(t).] Experimentally, it is easier and more convenient to measure t1/2, the time it takes for the voltage across the capacitor (or the charge on it) to rise (during charging from an initial Q = 0 state) or fall (from an initial Q = Q0 state) to half its maximum value. Thus ½ = exp[ -­‐ t1/2 / RC] = exp[ -­‐ t1/2 / τ]
or
τ = RC = t1/2 /ln 2
[6]
3. Experimental Procedure
The experimental procedure for this laboratory is very simple. An RC circuit is driven by a periodic square voltage and the resulting time-­‐dependent voltage across the capacitor(s) is displayed on an oscilloscope screen from which t1/2 is
measured followed by the use of eq.[6] to determine C.
3.1 Apparatus
[i]. AT-­‐700 Portable Analog/Digital Laboratory. [ii] An assortment of connector wires, capacitors and resistors.
You will be working with the AT-­‐700 Portable Analog/Digital Laboratory. It consists of two main parts. As shown in the picture below there is a breadboard surrounded by a variety of functional input and output circuits. In this experiment we are only concerned with just one device – the function generator (FG). The FG is a device that can produce periodic voltage signals – sinusoidal, triangular and square wave. Here, we will only utilize the square wave signal. The solderless breadboard allows electrical components to be easily plugged in, removed and interconnected without cumbersome soldering and desoldering. Here is a brief description of various terms and how a breadboard works. The holes that you see in a breadboard are called tie points. You connect two tie points using a connector wire. The tie points are organized in two distinct arrangements called the distribution strip and the terminal strip (see diagram below). Figure 2a. The AT-­‐700 Portable Analog/Digital Laboratory. The distribution strip is a series of two parallel rows. In a given distribution strip, all tie points in a given row are connected by a copper base strip under the tie points. This means you can connect two wires by simply sticking two wires anywhere in a given row. In a terminal strip all five vertical tie points in a vertical column are connected by a base conducting strip. Thus you can connect two wires by sticking them in the same column in a terminal strip. Figure 2b. Close up of the Bread Board. 3.2 Experimental Procedure A. Resistance Measurement. You will be given two capacitors of capacitance in the 5.0 to 10.0nF range and a resistor, R of nominal resistance of 10.0 – 20.0 kΩ. Measure the actual value of R using a digital multimeter and enter this value in Table-­‐I. The capacitance values will be imprinted on the capacitors, however, we will determine the capacitance using an RC circuit and then compare these values to the imprinted values. B. Single Capacitor Using one of the capacitors and the resistor R, construct the circuit as shown below. OSC
Ch-1
V1/2
R
t1/2
c
O
FG
C
OSC
V1/2
a
b
t1/2
time
Ch-2
Figure 3a. Single capacitor circuit charging and discharging with response curve. Adjust the FG frequency in the 10 – 20kHz range (this can be later adjusted to optimize the reading precision). The voltage across the capacitor will be displayed on CH-­‐02 of the oscilloscope. CH-­‐01 will show the driving voltage being applied to the RC circuit. (Note: The amplitude of the sinusoidal and the triangular voltage signals can be adjusted with the Amplitude knob. However, the amplitude of the square wave is not adjustable and is fixed at 5.0 V). With the Vertical Position knob adjust the vertical position of the CH-­‐
02 signal so that the signal trace is cut in half by the horizontal axis. Adjust the vertical scale of CH-­‐02 so that the voltage trace is large enough to fill as much of the oscilloscope screen as possible. This will improve the accuracy for locating the vertical halfway level. Adjust the time scale so that you can see at least one charging and one discharging scale well on the oscilloscope screen. Using the time markers, measure the t1/2 shown as Oa or bc in the diagram above. Make three measurements for each t1/2 .Enter these values in Table-­‐I and calculate C1 as indicated in the Table. Repeat the procedure outlined above for the second capacitor. C. Capacitors in Series Using the two given capacitors and the resistor, make the series circuit as shown in the diagram. Repeat the procedure outlined above for a single capacitor to measure the equivalent capacitance of C1 and C2. OSC
Ch-1
R
FG
C1
C2
OSC
Ch-2
Figure 4. Charging circuit with capacitors in series. D. Capacitors in Parallel Using the two given capacitors and the resistor, make the parallel circuit as shown in the diagram. Repeat the procedure outlined above for a single capacitor to measure the equivalent capacitance of C1 and C2. OSC
Ch-1
R
FG
OSC
Ch-2
C1
C2
Figure 5. Charging circuit with capacitors in parallel. LAB-­‐04 Capacitors and RC-­‐Circuits Name:_______________________ Sec./Group__________ Date:_____________ 4. Prelab Suppose, you are given three capacitors C1 = 2.0µF, C2 = 4.0µF, and C3 = 6.0µF. 1. What is the minimum and the maximum capacitance you can obtain using all three capacitors in a circuit? 2. If you apply a voltage Vab = 12.0V to the terminals of the series and parallel capacitor circuits of part [a] above to charge the capacitors, what is the ratio of the energy that can be stored in the 2.0µF and the 4.0µF capacitors when they are parts of [a] the minimum capacitance? [b] the maximum capacitance configuration? LAB-­‐04 Capacitors and RC-­‐Circuits Name:_______________________ Sec./Group__________ Date:_____________ 5. Data 5.1 Single Capacitors Resistance, R (kΩ) = _________ TABLE-­‐I Capacitor Average Capacitance t1/2 t1/2 Cexp = t1/2/(Rln2) (μsec) (μsec) (nF) C1 C2 (Below C1 and C2 are the values printed on the capacitor cases) %difference = ( C1 − C1,exp / C1 ) ×100 = %difference = ( C2 − C2,exp / C2 ) ×100 = 5.2 Capacitors in series TABLE-­‐II Average Equivalent Capacitance t1/2 t1/2 Cexp = t1/2/(Rln2) (μsec) (μsec) (nF) Ctheo. = C1C2/ (C1 + C2) = %difference = ( Ctheo − Cexp / Ctheo ) ×100 = LAB-­‐04 Capacitors and RC-­‐Circuits Name:_______________________ Sec./Group__________ Date:_____________ 5.3 Capacitors in parallel TABLE-­‐III Average Equivalent Capacitance t1/2 t1/2 Cexp = t1/2/(Rln2) (μsec) (μsec) (nF) Ctheo. = C1 + C2 = %difference = ( Ctheo − Cexp / Ctheo ) ×100 = 6. Analysis There is no extra analysis for this lab. 6. JUST FOR THE FUN OF IT. 6.1 When Ohm’s law is not obeyed. An LED is an example of a non-­‐Ohmic device – it I
does not obey the relation V = IR. An I-­‐V plot of an LED is shown in Fig.6. To demonstrate this behavior, connect two LED’s in parallel across the For positive voltage
I increases non-linearly
with V.
function generator of the AT-­‐700. Use a square wave signal in the 1 – 5 Hz range. If you don’t see V
the two LED’s blink (glow and dim) alternately, pull one of the LED’s, flip its leads and reinsert it in the distribution strip. Now the two LED’s will For negative voltage
almost no current
flows.
blink alternately – LED1 glows (no current O
Figure 6
through LED2) in the first half of the square wave when the current flows from a to b while in the other half of the square pulse the current flows from c to d and LED2 glows (no current through LED1). Figure 7 6.2 Connecting Ammeters and Voltmeters In this experiment we will use the brightness of an LED as an approximate indicator of the current flowing through it. Keep just one LED in the circuit, remove the other one you used in the experiment above. Take a resistor R1 in the range of 20 – 100Ω. Connect it in parallel with the LED. You will notice a sharp reduction in the brightness of the LED. Now connect the same resistance in series with the LED. Now what do you observe? Repeat the experiment with a large resistor R2 (10-­‐ 20kΩ) resistor. Note the change in the brightness of the LED when R2 is connected in parallel and then in series with the LED. Use your observations to explain why an ammeter (a low resistance device) in connected in series with a circuit element to measure the current through it, and a voltmeter (a very high resistance device) is connected across a circuit element to measure the voltage drop across it.