Using Kirchhoff's Current Law (KCL), determine the voltage developed across the coil in the figure shown.

Given i₂ = 5e^−2t; i₄ = 3 sin t; and v₃ = 4e^−2t

Concept

The voltage across the inductor coil is given by:

\(V_L=L{di_L\over dt}\)

where, L = Inductor

\({di_L\over dt}=\) Rate of change of current across the inductor

The current across the capacitor is given by:

\(I_C=C{dV_C\over dt}\)

where, C = Capacitor

\({dV_C\over dt}=\) Rate of change of voltage across the capacitor

Calculation

Given i2 = 5e−2t; i4 = 3 sin t; and v3 = 4e−2t

KCL states that “The algebraic sum of all incoming current is equal to the algebraic sum of all outgoing current at a node.” 

Applying KCL at node A, we get:

Sum of incoming current = Sum of outgoing current

\(i_3+i_2+i_1=i_4\)

\(2{d\over dt}(4e^{-2t})+5e^{-2t}+i_1=3sint\)

\(i_1=3sint-5e^{-2t}+16e^{-2t}\)

\(i_1=i_L=3sint+11e^{-2t}\)

The voltage across the inductor coil is given by:

\(V_L=4{d\over dt}(3sint+11e^{-2t})\)

V_L = 12 cos t − 88e−2t