Question 6.4: Circuit Modeling Using the Node Method: One Node Consider th...
Circuit Modeling Using the Node Method: One Node
Consider the circuit shown in Figure 6.21. Derive the differential equation relating the output voltage v_{\mathrm{o}}(t) to the input voltage v_{\mathrm{a}}(t).
Note that the voltage at node 1 is unknown and we denote it as v_{1}. All currents entering or leaving node 1 are labeled as shown in Figure 6.21. Applying Kirchhoff’s current law to node 1 gives
i_{\mathrm{L}}-i_{\mathrm{R}}-i_{\mathrm{C}}=0.
Expressing the current through each element in terms of the node voltage, we have
\frac{1}{L} \int\left(v_{\mathrm{a}}-v_{1}\right) \mathrm{d} t-\frac{v_{1}-0}{R}-C \frac{\mathrm{d}}{\mathrm{d} t}\left(v_{1}-0\right)=0 .
Differentiating the above equation with respect to time results in
\frac{v_{\mathrm{a}}-v_{1}}{L}-\frac{\dot{v}_{1}}{R}-C \ddot{v}_{1}=0
Because the node voltage v_{1} is essentially the output voltage v_{\mathrm{o}}, the above equation can be rewritten as
R L C \ddot{v}_{\mathrm{o}}+L \ddot{v}_{\mathrm{o}}+R v_{\mathrm{o}}=R v_{\mathrm{a}},
which is the input-output equation relating the applied voltage v_{\mathrm{a}} and the output voltage v_{\mathrm{o}}.