To analyze the circuit depicted on the right panel of Figure 1, we apply Kirchhoff’s loop law. According to the well-known Kirchhoff’s laws 1) the algebraic sum of the emfs in a single loop circuit equals the sum of the ohmic voltage drops across the resistors. And 2) In a multi-mesh circuit at any node, the algebraic sum of the currents adds to a zero. There are three such loops, two are selected. The two squares are the referenced shown by clockwise directions 1 and 2. Knowing the voltages across the capacitor with the capacitance C, the self-inductor with the inductance of L, and the obvious voltage loss across the

resistor R are subject to ( Δ V ) C = q C and ( Δ V ) L = L d d t i , and ( Δ V ) R = R i , respectively we advance to the next step. The loops contain additional emf’s, ϵ 1 , ϵ 2 , and ϵ 3 . Kirchhoff’s loop law applied to loop 1 yield,

ϵ 1 − ϵ 2 − 1 C 1 q 1 − L 1 d d t i 1 + L 2 d d t i 2 + 1 C 2 q 2 = R 1 i 1 − R 2 i 2 , (1)

Repeating the same principle to loop 2, we have

ϵ 2 − ϵ 3 − 1 C 2 q 2 − L 2 d d t i 2 + L 3 d d t i 3 + 1 C 3 q 3 = R 2 i 2 − R 3 i 3 , (2)

and applying Kirchhoff’s node law at node A, we have

i 1 + i 2 + i 3 = 0 , (3)

In these equations, we replace the currents with their derivatives concerning time, namely, i 1 = d q 1 d t , i 2 = d q 2 d t , and i 3 = d q 3 d t . Substituting these in (1), (2) and (3) yield,

ϵ 1 − ϵ 2 − 1 C 1 q 1 − L 1 q ¨ 1 + L 2 q ¨ 2 + 1 C 2 q 2 = R 1 q ˙ 1 − R 2 q ˙ 2 , (4)

ϵ 2 − ϵ 3 − 1 C 2 q 2 − L 2 q ¨ 2 + L 3 q ¨ 3 + 1 C 3 q 3 = R 2 q ˙ 2 − R 3 q ˙ 3 , (5)

q ˙ 1 + q ˙ 2 + q ˙ 3 = 0 , (6)

where the over dots stand for the derivative concerning time. The set of Equations, (4), (5), and (6) are three coupled linear second-order inhomogeneous differential equations with constant coefficients. With the initial conditions, q 1 ( 0 ) = q 2 ( 0 ) = q 3 ( 0 ) = 0 and q ˙ 1 ( 0 ) = q ˙ 2 ( 0 ) = q ˙ 3 ( 0 ) = 0 we seek solutions. We fail to solve them analytically, claiming this is a suitable example to utilize with a CAS. By selecting a reasonable set of parameters using NDSolve command of Mathematic [5] we solve them numerically. Before doing so we adopt a strategy. 1) we set the parameters such that the double-looped circuit becomes symmetric, i.e. the parameters associated with loop 1 are the same as the parameters of loop; 2) Then by assigning small values to the embodied components, R, C, L, and ϵ and stronger ϵ for the mid common branch we anticipate a symmetric signal characteristics output. Upon justifying our intuitive point by breaking the symmetry we consider nonsymmetric elements leading the general features of the circuit. The Mathematica [5] codes running these scenarios are as follows.

For the symmetric case, we store the values of the components in values2b, units are SI. As mentioned, we assign small values to ϵ₁ and ϵ₂ and a stronger emf to the mid branch ϵ₂.

values2b={ϵ1->5.,ϵ2->50.,ϵ3->5.,R1->2.,R2->5.,R3->2.,L1->3.,L2->5.,L3->3., C1->4.,C2->10.,C3->4.}

We write,

eq1=ϵ1-ϵ2-1/C1q1[t]-L1q1’’[t]+1/C2q2[t]+L2q2’’[t]==R1q1’[t]-R2q2’[t]/.values

eq2=ϵ2-ϵ3-1/C2q2[t]-L2q2’’[t]+1/C3q3[t]+L3q3’’[t]==R2q2’[t]-R3q3’[t]/.values

eq3=q1’[t]+q2’[t]+q3’[t]==0/.values

initialCond=q1[0]==q2[0]==q3[0]==q1’[0]==q2’[0]==q3’[0]==0

with these codes, we solve the equations numerically displaying the solutions. I.e. charge on the capacitors q(t), voltages across the capacitors, currents through the loops, and the voltages across the self-inductors vs. time, respectively. These are depicted in Figure 2

solq=NDSolve [{eq1,eq2,eq3,initialCond}/.values,{q1[t],q2[t],q3[t]},{t,0,300}]

As mentioned in the figure caption, although each graph appears to contain only two signals in reality there are three. The thickened blue signals completely overlapped on the magenta acting as one signal as expected.

The interpretation of shown grouped signals of each plate is straightforward. The upper left graph is the charging characteristics of the capacitors. It follows the routine charging trends as pointed out initially they begin with no charge q ( t = 0 ) = 0 and gradually grow to their plateau saturation. The second plot on the first row is the scaled version of the first, this is because q ( t ) = ( Δ V ) C . It depicts the exponential growth of the voltage across the capacitors. The first and the second graphs of the second row are the classic current jumps and their exponential decay of the voltages across the current and the self-conductors, respectively.

With confidence in the accuracy of the crafted formulations and codes next we assign asymmetric values to the elements. These are stored in values2c, units are SI as well.

values2c={ϵ1→5.,ϵ2→50.,ϵ3→10.0,R1→2.,R2→5.,R3→5.0,L1->3.,L2->5.0,L3->5.0,C1->4.,C2->4.,C3->5.0};

We run the same mentioned code with the values2c. The description of the graphs depicted in Figure 3 is the same as in Figure 2. The impact of the asymmetric parameters is evident. The blue of the magenta signals is separated as expected. For the signals displayed on the first row, we used the same period as we used in Figure 2. However, for the signals on the second row because of the values of the chosen elements, we used a shorter period enabling displaying the signal characters.

By comparing the signals displayed in Figure 2 and Figure 3 we realize the impact of the symmetric vs asymmetric parameters. Here again, the general behavior of the signals is what one expected. These are the mimics of the classic singles with no surprises.

Here we show how easily one may modify the circuit dramatically by replacing one of the DC power supplies with an AC driver. Although there are three DC sources as shown in Figure 1, here for the sake of demonstrating the procedure we replace only one driver, ϵ₂. The interested reader may wish to try replacing ϵ₁ and/or ϵ₃ or ultimately all three. As mentioned, we have complete control over setting the parameters. Just for the sack of demonstration, we set the parameters as,

values2d={ϵ1->5.0,ϵ2->50.0 Sin[π t],ϵ3->5.0,R1->2.0,R2->5.0,R3->2.0,L1->3.0,L2->3.0,L3->3.0,C1->4.0,C2->4.0,C3->4.0}

The second driver is a sinusoidal source, ϵ₂(t) = 50 sin[πt]. Its voltage amplitude is 50.0 V and its low frequency is f = 1/2 Hz. Replacing the ϵ₂ in (1) and (2) with the sinusoidal driver easily is accomplished via the driver’s character given in value2d. The circuit is shown in Figure 4.

Here the voltages of the left and the right loops as well as the values of the resistances, capacitances, and the self-inductors are set the same. The character of

the emf of the central branch is replaced with an AC source. The values of the other elements of the circuit are set differently as well. This procedure shows how easy it is to switch from DC to an AC source. As expected, the effective value of the power supply and its oscillating character dominates the time dependency behavior of the signals of interest. Eyeballed view of the characters of Figure 2 and Figure 3 reveals the impact of the DC vs AC sources, as shown in Figure 5. Utilizing the codes makes it easy to either place the source in the left and or right loop. Or even better design a fresh circuit either based on one’s curiosity and need.