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EE40 Lab 4-Lab Guide-P1 Experiment Guide for RC Circuits I. Introduction 1. Capacitors A capacitor is a passive electronic component that stores energy in the form of an electrostatic field. The unit of capacitance is the farad (coulomb/volt). Practical capacitor values usually lie in the picofarad (1 pF = 10-12 F) to microfarad (1 µF = 10-6 F) range. Recall that a current is a flow of charges. When current flows into one plate of a capacitor, the charges don't pass through (although to maintain local charge balance, an equal number of the same polarity charges leave the other plate of the device) but instead accumulate on that plate, increasing the voltage across the capacitor. The voltage V across the capacitor (capacitance C) is directly proportional to the charge Q stored on the plates: Q CV= Since Q is the integration of current over time, we can write: v tQ tCi t dtC( )( )( )= =! Differentiating this equation, we obtain the I-V characteristic equation for a capacitor: i t CdV tdt( )( )= (Eq. 1) 2. RC Circuits An RC (resistor + capacitor) circuit will have an exponential voltage response of the form v t A BetRC( ) = +! where constant A is the final voltage and constant B is the difference between the initial and the final voltages. (ex is e to the x power, where e =2.718, the base of the natural logarithm) The product RC is called the time constant (whose units are seconds) and is usually represented by the Greek letter τ. When the time has reached a value equal to the time constant, τ, then the voltage is BBeBelRC*368.0==!!" volts away from the final value A, or about 5/8 of the way from the initial value (A+B) to the final value (A). The characteristic “exponential decay” associated with an RC circuit is important to understand, because complicated circuits can oftentimes be modeled simply as resistor and a capacitor. This is especially true in integrated circuits (ICs). A simple RC circuit is drawn in Figure 1 with currents and voltages defined as shown. Equation 2 is obtained from Kirchhoff’s Voltage Law, which states that the algebraic sum of voltage drops around a closed loop is zero. Equation 1 (above) is the defining I-V characteristicEE40 Lab 4-Lab Guide-P2 equation for a capacitor, and Equation 3 is the defining I-V characteristic equation for a resistor (Ohm’s Law). RCinVVV += (Eq. 2) dtdVRCIRVCR== (Eq. 3) Combining Equations 1 and 3 into 2, we obtain the following first-order linear differential equation: dtdVRCVVCCin+= (Eq. 4) If VIN is a step function at time t=0, then VC and VR are of the form: RCtCBeAV!+= (Eq. 5) RCtReBAV!+= '' If a voltage difference exists across the resistor (i.e. VR < 0 or VR > 0), then current will flow (Eq. 3). This current flows through the capacitor and causes VC to change (Eq. 1). VC will increase (if I > 0) or decrease (if I < 0) exponentially with time, until it reaches the value of VIN at which time the current goes to zero (since VR = 0). For the square-wave function VIN as shown in Figure 2(a), the responses VC and VR are shown in Figure 2(b) and Figure 2(c), respectively.EE40 Lab 4-Lab Guide-P3 Note that if the frequency of the square wave VIN is too high (i.e. if f >> 1/RC), then VC and VR will not have enough time to reach their asymptotic values. If the frequency is too low (i.e. if f << 1/RC), the decay time will be very short relative to the period of the waveform and thus the exponential decay will be difficult to observe. As a rough guideline, the period of the square wave should be chosen such that it is approximately equal to 10RC, in order for the responses shown in Figure 2b-c to be readily observed on an oscilloscope. 3. Inductors An inductor is a passive electronic device that stores energy in a magnetic field. The unit of inductance is the henry (volt-second/ampere). Practical values of inductance range from one microhenry ( 1µH = 1 x 10-6 H) to one henry (1 H). Inductors are usually made by highly coiled wires, in which changing current generate a magnetic field. By Lenz’s Law, the changing magnetic flux produces a back-EMF (electromotive force), or a potential in the opposite direction of current flow and magnetic flux. While capacitors act to oppose changes in voltage, inductors oppose changes in current. The voltage v(t) across an inductor (with inductance L) is equal to the inductance multiplied by the change in current, di(t)/dt through the inductor: dttdiLtv)()( = (Eq. 6) II. Hands On 1. Determining the RC Circuit Configuration In this part of the experiment, you will make ohmmeter measurements to see if you can discover a method to determine if a resistor and capacitor are connected in series or in parallel.EE40 Lab 4-Lab Guide-P4 Get a 5.1 kΩ resistor and 100 µF capacitor from your TA. <> Recall that an ohmmeter has a built-in current source that sends a small current into the circuit under test. The ohmmeter reads the voltage across the circuit under test and determines the resistance of the circuit using Ohm’s Law. Build the circuit shown in Figure 3. Note that the ohmmeter’s current source keeps on charging up the capacitor. (For small values of capacitance, the capacitor will be fully charged almost instantly.) Question 1: Are you able to measure the value of the resistor? If not, explain the reason why you cannot make the measurement. Build the circuit shown in Figure 4. Note that the capacitor stops charging when the current through the resistor is equal to the current from the ohmmeter. Question 2: Explain how you got your ohmmeter reading for the circuit in Figure 4. Why does it take some time before the ohmmeter’s reading stabilizes? Question 3: Given a black box with either a series or parallel RC circuit, can you determine the RC configuration using an ohmmeter? If so, how? Identifying Physical Values in a Series RC Circuit Black Box and a Parallel RC Circuit Black Box The TA will give you two “black boxes” (if available). One contains a series RC circuit and the other contains a parallel RC circuit. Determine the basic resistorcapacitor configuration in


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Berkeley ELENG 40 - Experiment Guide for RC Circuits

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