Chapter 3.2, Problem 27P

Applications.

Electric Circuits. The theory of electric circuits, such as that shown in Figure 3.2.6, consisting of inductors, resistors, and capacitors, is based on Kirchhoff’s laws: (1) At any node (or junction), the sum of currents flowing into that node is equal to the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, and (2) the net voltage drop around each closed loop is zero. In addition to Kirchhoff’s laws, we also have the relation between the current $i\left(t\right)$ in amperes through each circuit element and the voltage drop $v\left(t\right)$ in volts across the element:

$v-Ri$, $R=\text{resistance}\text{in}\text{ohms}$;

$C\frac{dv}{dt}=i$, $C=\text{capacitance}\text{in}{\text{farads}}^{\text{4}}$;

$L\frac{di}{dt}=v$, $L=\text{inductance}\text{in}\text{henries}$.

Kirchhoff’s laws and the current-voltage relation for each circuit element provide a system of algebraic and differential equations from which the voltage and current throughout the circuit can be determined. Problems 27 through 29 illustrate the procedure just described.

Consider the circuit shown in the Figure 3.2.6. Let $i_1,i_2$ and $i_3$ be the currents through the capacitor, resistor, and inductor, respectively. Likewise, let $v_1,v_2$ and $v_3$ be the corresponding voltage drops. The arrows denote the arbitrarily chosen directions’ in which currents and voltage drops will be taken to be positive.

(a) Applying Kirchhoff’s second law to the upper loop in the circuit, show that

$v_1-v_2=0$. (i)

In similar way, show that

$v_2-v_3=0$. (ii)

(b) Applying Kirchhoff’s first law to either node in the circuit, show that

$i_1+i_2+i_3=0$. (iii)

(c) Use the current-voltage relation through each element in the circuit to obtain the equations

$C{v}_1\text{'}=i_1,v_2=R{i}_2,L{i}_3\text{'}=v_3$. (iv)

(d) Eliminate $v_2,v_3,i_1$ and $i_2$ among Eqs.(i) through (iv) to obtain

$C{v}_1\text{'}=-i_3-\frac{v_1}{R},L{i}_3\text{'}=v_1$. (v)

These equations form a system of two equations for the variables $v_1$ and $i_3$.