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CH4_Problems - 2 04 C hapter 4 Transients Note You can...

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204 Chapter 4 Transients Note: You can check the answers to many of the problems in this chapter by using a computer- aided circuit-analysis program such as Multisim Problems Section 4.1: First-Order RC Circuits P4.1. Suppose we have a capacitance C discharging through a resistance R. Define and give an expression for the time constant. To attain a long time constant, do we need large or small values for R? For C? *P4.2. The dielectric materials used in real capac- itors are not perfect insulators. A resistance called a leakage resistance in parallel with the capacitance can model this imperfection. A 100-~tF capacitor is initially charged to 100V. We want 90 percent of the initial energy to remain after one minute. What is the limit on the leakage resistance for this capacitor? *P4.3. The initial voltage across the capacitor shown in Figure P4.3 is vc(O+) = -10V. Find an expression for the voltage across the capaci- tor as a function of time . Also, determine the time to at which the voltage crosses zero. t = 0 R =50 kQ + + vc(t) Figure P4.3 *P4.4. A 100-~tF capacitance is initia ll y charged to 1000 V. At t = 0, it is connected to a 1-kn resistance. At what time t2 has 50 percent of the initial energy stored in the capacitance been dissipated in the resistance? *P4 .5. At t = 0, a charged 10-~tF capacitance is con- nected to a voltmeter, as shown in Figure P4.5. The meter can be modeled as a resistance. At t = 0, the meter reads SOV. At t = 30s, from National Instruments or OrCAD Capture from Cadence Inc. See Appendix D for more information. the reading is 25 V. Find the resistance of the voltmeter. )I t t = 0 , ' VM Figure P4.5 *P4.6. At time t1, a capacitance Cis charged to a volt- age of vl. Then, the capacitance discharges through a resistance R. Write an expression for the voltage across the capacitance as a function of time for t > t1 in terms of R, C, V1,andt1. P4.7. Given an initially charged capacitance that begins to discharge through a resistance at t = 0, what percentage of the initial voltage remains at two time constants? What percent- age of the initial stored energy remains? P4 .8. The initial vo ltage across the capacitor shown in Figure P4.3 is vc(O+ ) = 0. Find an expres- sion for the voltage across the capacitor as a function of time, and sketch to scale versus time. P4.9. In physics, the half-life is often used to characterize exponential decay of physical quantities such as radioactive substances. The half -life is the time required for the quantity to decay to half of its initial value. The time constant for the voltage on a capacitance dis- charging through a resistance is r = RC. Find an expression for the half-life of the voltage in terms of R and C. P4 .10. We know that a 50-~tF capacitance is charged to an unknown voltage Vi at t = 0. The capac- itance is in parallel with a 3-kn resistance.
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