Solved: Electric Circuits. The theory of electric circuits, such as that shown in Figure

Chapter 7, Problem 19

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Electric Circuits. The theory of electric circuits, such as that shown in Figure 7.1.2, consistingof inductors, resistors, and capacitors, is based on Kirchhoffs laws: (1) The net flow of currentinto each node (or junction) is zero, and (2) the net voltage drop around each closed loopis zero. In addition to Kirchhoffs laws, we also have the relation between the current I inamperes through each circuit element and the voltage drop V in volts across the element:V = RI, R = resistance in ohms;C dVdt = I, C = capacitance in farads1;LdIdt = V, L = inductance in henrys.Kirchhoffs laws and the currentvoltage relation for each circuit element provide a system ofalgebraic and differential equations from which the voltage and current throughout the circuitcan be determined. 19 through 21 illustrate the procedure just described.Consider the circuit shown in Figure 7.1.2. Let I1, I2, and I3 be the currents through thecapacitor, resistor, and inductor, respectively. Likewise, let V1, V2, and V3 be the correspondingvoltage drops. The arrows denote the arbitrarily chosen directions in whichcurrents and voltage drops will be taken to be positive.(a) Applying Kirchhoffs second law to the upper loop in the circuit, show thatV1 V2 = 0. (i)In a similar way, show thatV2 V3 = 0. (ii)(b) Applying Kirchhoffs first law to either node in the circuit, show thatI1 + I2 + I3 = 0. (iii)(c) Use the currentvoltage relation through each element in the circuit to obtain theequationsCV1 = I1, V2 = RI2, LI3 = V3. (iv)(d) Eliminate V2, V3, I1, and I2 among Eqs. (i) through (iv) to obtainCV1 = I3 V1R , LI3 = V1. (v)Observe that if we omit the subscripts in Eqs. (v), then we have the system (2) of thissection.