Find the dynamics of the circuit.
Find the dynamics of the circuit.
A. <p>R<sub>1</sub>I<sub>1</sub>(s) + LsI<sub>1 </sub>(s) + LsI<sub>2</sub> (s) = V(s)</p> <p><span class="math-tex">\(Ls{{I}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{cs{{I}_{2}}\left( s \right)}+Ls{{I}_{1}}\left( s \right)=~0~\)</span></p>
B. <p>R<sub>1</sub>I<sub>1</sub> (s) + LsI<sub>1</sub>(s) – LsI<sub>2</sub>(s) = V(s)</p> <p>LsI<sub>2</sub>(s) + R<sub>2</sub>I<sub>2</sub> (S) + Cs I<sub>2</sub> (s) + LsI<sub>1</sub>(s) = 0</p>
C. <p>R<sub>1</sub>I<sub>1</sub>(s) + LsI<sub>1</sub> (s) + LsI<sub>2</sub>(s) = V(s)</p> <p><span class="math-tex">\(L{{s}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0\)</span></p>
D. <p>R<sub>1</sub>I<sub>1</sub> (s) + LsI<sub>1</sub>(s) – LsI<sub>2</sub>(S) = V(s)</p> <p><span class="math-tex">\(Ls{{I}_{2}}\left( s \right)+{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0\)</span></p>
Please scroll down to see the correct answer and solution guide.
Right Answer is: D
SOLUTION
For the dynamics of the circuit, we require voltage and current equations associated with energy storing and dissipating elements in the Laplace form.
Now the circuit diagram in Laplace form is as shown below.
The voltage across the capacitor is,
\({{V}_{c}}\left( s \right)=\frac{1}{Cs}{{I}_{2}}\left( s \right)\) ---(i)
By applying the KVL in the input side,
∴ V(s) = I1(s) R1 + (I1(s) – I2(s)) sL
⇒ R1 I1 (s) + I1(s) sL – I2(s) sL = V(s)
By applying the KVL in the output side,
Vc(s) = I2(s) R2 + (I2(s) – I1(s)] sL
From equation (i)
\(sL~{{I}_{2}}\left( s \right)+{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-sL~{{I}_{1}}\left( s \right)=0\)
