RC Circuits. 1 Introduction. 2 Capacitors


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1 1 RC Circuits Equipment DataStudio with 750 interface, RLC circuit board, 2 voltage sensors (no alligator clips), 2x35 in. leads, 12 in. lead Reading Review operation of DataStudio oscilloscope. Review operation of the DataStudio signal generator. 1 Introduction The 3 basic linear circuits elements are the resistor, the capacitor, and the inductor. The voltage across a resistor is proportional to the current through the resistor. The voltage across a capacitor is proportional to the time integral of the current into the capacitor. The voltage across an inductor is proportional to the time derivative of the current through the inductor. This lab is concerned with the characteristics of capacitors and circuits consisting of a resistor and a capacitor in series (RC circuits). The primary focus will be on the response of an RC circuit to a step voltage.in this case will be using an AC square wave as a step voltage. (A step voltage is a constant voltage which instantaneously changes to a different constant voltage.) Also, you will be varying the frequency of a square waveform. The frequency will vary the time of the period. A period is the overall time it takes to complete one cycle. In the lab write up the period will be labeled as capital T. 2 Capacitors A capacitor is a 2 terminal circuit element that stores energy in its electric field. In its simplest form a capacitor consists of 2 parallel plates separated by a small gap which is usually filled with a nonconducting dielectric material. In practice many capacitors are made by sandwiching a dielectric sheet such as mylar between 2 pieces of metal foil and rolling this assmebly up into a cylinder. If a positive current I enters one plate of a capacitor and exits the other plate a positive charge Q builds up on the first plate and a negative charge Q builds up on the second plate. A voltage V C develops across the capacitor. The charge Q and voltage V C are linearly related by the equation Q = CV C, (1) where C is a constant called the capacitance. Eq. 1 can also be written as V C = 1 C Idt. (2) The unit of capacitance in S.I. units is the farad (F). One farad is a huge capacitance and more usual units are the micro farad (µf) and the pico farad (pf). The constant C depends on the geometry of the capacitor and the dielectric properties of the insulator between the plates. The most important specification for a capacitor of a given value is its maximum voltage rating.
2 2 3 RC Circuits A series RC circuit with a voltage source V(t) connected across it is shown in Fig. 1. The voltage across the resistor and capacitor are designated by V R and V C, and the current around the loop by I. The signs are chosen in the conventional way. I is positive if it is in the direction of the arrow. Kirchoff s law, which says that the voltage changes around the loop are zero, may be written Note: I =change in charge(q) overtime. Letting V R = IR = ( d Q)R and V dt C = 1 Q C Eq.(3) may be written as V R + V C = V. (3) R d dt Q + 1 Q = V. (4) C Of particular interest in this lab is the solution of Eq.(4) when V is a constant voltage. The solution is Q = (Q 0 CV )e t RC + CV, (V = constant) (5) where Q 0 is the charge on the capacitor at time t=0. The quantity RC is the time constant of the circuit. Time constant references the time it takes for the current to flow in out of the capacitor and resistor. Energy is stored in capacitor and the flow of charge and discharge will be controlled by the resistor. The time dependence of the circuit is exponential when V=constant. Beginning at time t = 0 the parameters of this circuit exponentially approach constant values. V C approaches V and I approaches 0. In a time 3RC those parameters are within 95% of the final values. As a function of time the voltage across the capacitor is 1 C Q, the current in the circuit is I = dq dt, and the voltage across the resistor is V R = IR. 3.1 A time varying voltage to an RC circuit. We can use this result to understand what happens when a constant voltage is applied to a RC series circuit, or if a constant voltage is instantaneously changed from one constant value to another constant value. A key point is that when the voltage is switched, the charge on and the voltage across the capacitor do not change instantaneously. V C and Q are continous but not differentiable at the switching point. The voltage across the resistor and the current in the circuit do change instantaneously. V R and I are neither continuous or differentiable at the switching point. Consider the situation shown in Fig.(2). For times t < 0 the applied voltage is 0, and we assume that the applied voltage has been zero long enough so that V C = Q = V R = I = 0. At t=0 a constant voltage V is applied to the circuit for a time T/2 RC. For the time interval 0 < t < T/2 the voltage across the capacitor rises exponentially from 0 toward V and the voltage across the resistor decays exponentially for V toward zero. As shown in Fig.(2) at time T/2 the applied voltage is returned to 0. As T RC at time T/2 the voltage across the capacitor is essentially V. For times t > T/2, V C decays exponentially to 0. Let ɛ be a very small increment of time. At time t = (T/2 ɛ) V R is
3 3 almost 0. At time t = (T/2 + ɛ) V R has jumped to V. Can you explain why? For t > T/2, V R goes exponentially to 0. The above response of an RC circuit to a constant voltage which is first zero, then V, and then zero again can be observed by applying a square wave to the circuit where half of the square wave has zero voltage and the other half voltage V. If T is the period of the square wave, we continue to assume that T RC. Make a single plot that shows the square wave, V C, and V R, but read the next paragraph first and check that your plot agrees with the statements made. Regardless of the relative magnitudes of T and RC, if a periodic voltage is applied to a series RC circuit, V C, V R, and I must obey the following conditions. 1. The time average value of V C must equal the time average value of the applied voltage. You can prove this by integrating Eq.(3) over one period. 2. The time average value of the I (and V R ) must be zero. If they were not, what would happen to the voltage across the capacitor? Exercise. If T RC, graph the response of an RC circuit to an applied symmetric square wave where the voltage oscillates between +V and V? Your graph should show the applied square wave, V C, and V R. Hint. At the instant the square wave switches the magnitude of the voltage across the resistor is either 2V or 2V. If a high frequency square wave, such that T RC, is applied to an RC circuit, the voltages changes across the capacitor are minimal. The capacitor does not have time to charge much before the voltage is reversed. Most of the voltage changes occur across the resistor. As mentioned previously, the average voltage of the capacitor will be the time average voltage of the square wave, and the average V R will be zero. Fig. 3 shows the voltages for a square wave that oscillates between the constant voltage V and 0 (ground), and Fig. 4 shows the voltages for a square wave that oscillates between +V and V. In both these figures, the exponential dependences of V C and V R are approximated as straight lines. Neither V C or V R approach steady state values. 3.2 Charging and discharging of an RC circuit The capacitor does not charge instantaneously. The charge is proportional dq/dt. In time when the capacitor does become charged the current will go to zero. This occurs at a certain time constant which is related to the capacitance of the capacitor and the resitance value. 4 Experiments Charge Current Voltage Discharging Q 0 e t RC I 0 e t RC V 0 e t RC Charging Q 0 (1 e t I 0 (1 e t V 0 (1 e t RC ) RC ) RC ) In the following sections you will be observing the charging and discharging of the capacitor. This is related to the time constant of the RC circuit. You will be applying voltage square waves that have various periods. Depending on the time constant of the circuit and period of the square being applied you will see various response curves across the capacitor and resistor.
4 4 The period of the square wave is T. Both the situation T RC and T RC will be examined. For the former case, due to limitations of the equipment, the peak values of V R will be less than the theoretical values. One equipment limitation that you can easily see on the scope is the finite rise time of the square wave. 4.1 T RC, Positive Only Square wave If necessary, remove the Power Amplifier II connector from input channel C of the 750 interface. Hook up the circuit shown in Fig.(5) taking care with the polarities. On the RC circuit board connect one lead across the induction coil to short it out, choose R = 10 Ω and C = 100 µf. See Fig.(6) for an overview of the setup. Keep in mind the after you re done wiring the circuit you need to plug in the analog voltage sensors. Plug the voltage sensor that is across the capacitor to channel A and the voltage sensor that is across the resistor to channel B. Open DataStudio and click on the output terminals of the 750 interface. A signal generator window will pop up. Set the sample rate to 2000Hz. This is done in the measurements and sample rate tab of the signal generator. In the signal generator window, choose the positive only square wave, a voltage of 1.5 V, and a frequency of 50Hz. Open the scope display by dragging the scope icon to the voltage ouput terminals icon of the setup window. Set up the voltage sensors on analog channels A and B. Make the input to channel 2 of the scope voltage sensor A and the input to channel 3 of the scope voltage sensor B. This is done by draging the Scope icon from the Displays window to the Output Voltage icon in the data window. Next, drag Scope 1 icon from the Displays window to the Voltage ChA icon in the Data window. Last, again drag the Scope 1 icon form the displays winodw to the Voltage ChB icon in the Data window. The input to channel 1 of the scope should be the voltage ouput of the signal generator. For the parameters you are using, calculate T and RC. Is the assumption that T RC valid? Some remarks. When using the oscilloscope after you click Stop, the last traces are stored. The stored trace is usually better for examination than the live trace. The live trace may have jitter. You can you the smart cursor on the stored trace. When measuring scope trace parameters, judicious use of the sweep speed, vertical gain, and vertical trace position help. You can use many of the scope functions on the stored trace. You can print out the stored trace. Label your printed out traces immediately by V, V C, and V R as the printer is not a color printer. If you use same vertical sensivity for all 3 scope traces you can easily compare the trace values. The voltage of 1.5 V for the signal genertor is satisfactory for all the experiments. Higher values may saturate V R. For reliable triggering with a positive only square wave, make the trigger voltage a little positive.
5 5 Click Start and examine V C, V R and V. Are the results what you expect? Use the smart cursor to measure the time constant of the exponentials and compare your measurements to RC (1/e = 0.37). Note that V C is fairly similar in form and magnitude to the input square wave, and that V R looks a lot like the derivative of the square wave. Referring to the left hand side of Eq.(4), dq will be of the order of Q. With the assumption that T RC this dt T term can be neglected compared to the term Q. We now have, to some approximation, that C V = Q = V C C. The result of differentiating this expression is dv = I = V R. You have dt C RC constructed an electronic differentiator! 4.2 T RC, Symmetric Square Wave Use the same parameters as in section 4.1 except use a square wave that is symmetric with respect to ground Oscillates between (+V and V ). Compare your results to what you expect. 4.3 T RC, Positive Only Square Wave Apply a 100 Hz positive only square to an RC circuit with R=100 Ω and C=330 µf. What is the time constant and how does it compare to T? Compare your results to Fig. 3. Note that V R is fairly similar in form to the input square wave and that V C is similar in form to its integral. Using Eq.(4) can you show that this is plausible? In Fig. 3 is approximating the exponentials by straight lines reasonable? 4.4 T RC, Symmetric Square Wave Use the same parameters as in Section 4.3 except use a square wave that is symmetric with respect to ground. Compare to Fig. 4. Is approximating the exponentials by straight lines reasonable? 4.5 T RC, Symmetric Square Wave Use the same RC components as in Section 4.3. Apply a symmetric square wave with a frequency of 30 Hz. Are the results what you expect? If you have time try 10 Hz and 5 Hz. 5 Finishing up and additional questions 1. What would occur on the time to discharge the capacitor if you increased the size of the capacitor? Explain. 2. What real world applications can an RC circuit be used in? Please leave your bench in a state of maximum order. Thank you.
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