59 RC Decay Introduction: A capacitor is a device for storing charge and energy. It consists of two conductors insulated from each other. A typical capacitor is called a parallel-plate capacitor and is symbolized -| |-or -| (-. It consists of two conduction plates separated by a small distance. When the plates are con- nected to a charging device, e.g. a battery, charge is transferred from one conductor to the other until the potential difference between the conductors, due to their equal and opposite charges, equals the applied potential difference, e.g. the potential difference between the battery terminals. The amount of charge (Q) depends on the geometry of the capacitor and is directly proportional to the potential difference (V). The proportionality constant is called the capacitance (C). (The gap between plates is usually filled with a dielectric material to increase the capacitance.) Charge is measured in coulombs and voltage in volts, so capacitance has units of coulombs/volt that is called a farad. This experiment will investigate the discharge of a capacitor through a resistance in an electric circuit. The increase or decrease of charge on the capacitor is exponential in character. This is an important type of behavior in nature since it occurs whenever the instantaneous change in a quan- tity is only dependent on the amount present at that time. Population and bacterial growth, money in a savings account, and energy use may follow such a law. Written in electrical symbols the charge, as a function of time, on a discharging capacitor is given by: where Q is the instantaneous charge at time t Q o is the initial charge at the beginning of the time interval, t R is the resistance in the circuit C is the capacitance of the capacitor used. The product RC is called the time constant of the circuit. It has units of ohms-farads, or sec- onds, and is the time for the value of Q o to fall by 1/e, as can be seen by substituting time, . The value 1/e ? 0.37, so at , Q = 0.37 Q o . Since this contains an awkward, rounded con- stant value, t 1/2 or half time is usually determined experimentally instead of RC time directly. t 1/2 is defined as the time necessary for the Q to increase or decrease to ˝ of its maximum value. Two cases will be studied in this experiment. In the first, the exponential discharge or decay is slow enough that measurements can be made with a mechanical timer. In the second part, the decay is so fast that an oscilloscope must be used. QCV= Q Q ° e t? RC()? = tRC= tRC= 60 Procedure: Slow exponential decay The circuit shown in Figure 11.1. will be used to measure slow discharge of a capacitor. When the voltage has been applied to the capacitor for several time constants, it can be considered fully charged to . When the power supply is then disconnected, the capacitor will dis- charge through the circuit containing the voltmeter. (Be careful to remove the cables to the power supply and not just turn the power supply to zero output, as the components of the power supply must be completely removed from the circuit.) The voltmeter here is connected in series. This is an unusual use of a voltmeter. The discharging capacitor results in a current flow through this cir- cuit loop. The size of the current, however, is extremely small and would require a very sensitive meter to record. However, the large resistance of a multimeter used in the voltmeter mode results in a measurable voltage drop with even a small current flow. Using the voltmeter in series thus allows for a reasonable measure of the current change - and therefore the charge change - with time, since the current is proportional to the voltage according to Ohm's Law. Using a stopwatch and working in teams, readings from the voltmeter should be taken every 2 seconds after the power supply is disconnected. (One person times; one reads the voltmeter out loud; and one records the voltage.) Plotting voltage vs. time, the time for the voltage (and there- fore the current and charge) to decay to 1/2 of the original value can be determined. When the charge has decayed to 1/2 of the original charge (Q = (1/2)Q o ) . Solving for t ˝ in terms of the originally defined time constant (RC) or Now, looking at your graph. Where the voltage is ˝ the original voltage the half-life of the RC circuit. The half-life time (t 1/2 ) can be read directly from the graph at V 1/2 . With the t 1/2 value and Figure 11.1: Circuit used to measure slow discharge of a capacitor. R = 10k , C = 1.0µF, R v = 10M Q 0 CV 0 = Power Supply C P ? ? Q Q 0 ------- 1 2 --- e t 12? RC()? == 12?()ln t 1 2? ? RC ------------ = t 12? 0.69RC= 61 the equation above, the RC value can be calculated. Now, this value can be compared to the resis- tance and capacitance values of the circuit. Measure R directly with the ohmmeter and use the nominal value for C (or the measured value if a capacitance meter is available). Compare (RC) graph and (RC) measured Do the uncertainty values overlap? Thus, do you have agreement for the RC value? A graph can be ?fit? to data. Using Excel, fit your data taken to an exponential function of the form above, i.e. f(x) = exp(x/?). Using the solve functions of Excel, compare the ? value to the known RC value. How does the curve fitting value compare to the data graphed value at t 1/2 ? How does it compare to the component measured value? Why might this value be a better esti- mate of RC than the t 1/2 value of RC? Make sure to include your analysis of these three values of RC in your conclusion. Procedure: Rapid Exponential Decay When RC times are in the millisecond (msec) range, a stopwatch is useless. To measure such decay times an oscilloscope is used. The circuit to be used for the rapid exponential decay measurements is shown in below in fig- ure 11.2. The signal generator produces a voltage pattern as shown in Figure 11.3 when set on square wave. Here it will function like a battery, turned on for some time and then turned off for an equal time, in continuing cycles. T is the period, the time for one complete repeat of a varying signal. The frequency (f) of the signal is variable and f = 1/T. You want to adjust Figure 11.2: Rapid Decay Circuit Square Wave Generator Oscilloscope Vertical Inputs 62 the frequency so that for one cycle the voltage is "on" for a time sufficient to fully charge the capacitor and the time the capacitor has to discharge (the time the square wave output is zero) is sufficient for it to fully discharge. If both of those times, each equal to half a period (T/2), are chosen to be about 4 time constants (4RC), the signal on the oscilloscope screen should be ade- quate for accurate measurements of the RC value. The oscilloscope trace should look like Figure 11.4. Figure 11.3: Step Voltage of Signal Generator Figure 11.4: Waveform from oscilloscope T 2T time voltage voltage time T 2T mary RC Decay.fm