Lab 4 - Guide

Lab 4 - Guide - Experiment Guide for RC Circuits I....

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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 μ F = 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: Since Q is the integration of current over time, we can write: Differentiating this equation, we obtain the I-V characteristic equation for a capacitor: (Eq. 1) 2.RC Circuits An RC (resistor + capacitor) circuit will have an exponential voltage response of the form v (t) = A + B e -t/RC where constant A is the final voltage and constant B is the difference between the initial and the final voltages. ( e x 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 B e - τ /RC = B e -1 = 0.368 * B 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 characteristic equation for a capacitor, and Equation 3 is the defining I-V characteristic equation for a resistor (Ohm’s Law). C dt t i C t Q t v = = ) ( ) ( ) ( V C Q = dt t dV C t i ) ( ) ( = 1 of 10
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(Eq. 2) (Eq. 3) Combining Equations 1 and 3 into 2, we obtain the following first-order linear differential equation: (Eq. 4) If V IN is a step function at time t =0, then V C and V R are of the form: (Eq. 5) RC t R e B A V / ' ' + = If a voltage difference exists across the resistor ( i.e. V R < 0 or V R > 0), then current will flow (Eq. 3). This current flows through the capacitor and causes V C to change (Eq. 1). V C will increase (if I > 0) or decrease (if I < 0) exponentially with time, until it reaches the value of V IN at which time the current goes to zero (since V R = 0). For the square-wave function V IN as shown in Figure 2(a), the responses V C
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Lab 4 - Guide - Experiment Guide for RC Circuits I....

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