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Series RLC Circuit
Figure 113 shows the schematic diagram of a series RLC circuit to which is applied a constant dc voltage V. In this circuit the applied voltage satisfies the following equation
where L = constant selfinductance in henries
Equation 173 is of exactly the same form as Eq. 157, which applies to the mechanical system with mass, friction, and elasticity. The integral in the lefthand side of Eq. 171 is eliminated by differentiation with respect to time. Hence
If Eq. 158 is divided by M, the result is
A comparison of Eq. 175 with Eq. 174 shows these equations to be identical except for the values of the constant coefficients. This comparison then shows that solution of Eq. 172 must be of the same form as that of Eq. 175, the only difference being in the constant coefficients. The solution of Eq. 175 is given by Eq. 162 as follows
On that basis the solution of Eq. 174 is
Because of the selfinductance the current must be zero at t = 0, the instant when the dc voltage is applied; so from this boundary condition when t = 0, then i = 0, and
Equation 176 establishes the relationship between the unknown constants B_{1} and B_{2} but does not evaluate them in terms of known constants. Either of these constants, B_{1} or B_{2} can now be evaluated, however, in terms of the known constants R, L, and C and the applied voltage V for the same boundary condition as follows. The current is zero at t = 0. This means that the term Ri in Eq. 173 must be zero at t = 0 also. Furthermore, since at t = 0 no current has as yet been applied to the capacitor, the time integral of the current must also equal zero. Substitution of
in Eq. 171 yields
and
From Eqs. 177 and 178 there results
The process involved in the solution of Eqs. 158, 159, and 160 when applied to this case yields
and
Equations 179 and 180, when substituted in Eq. 176 yield
where a = R/2L and
From Eq. 181 it is evident that the current i may undergo two different kinds of variation with respect to time. The current may (1) oscillate or it may (2) rise to a maximum and then gradually die out without reversing its direction. Current oscillatory When the constant b in Eq. 181 is real, i.e., if
the current i oscillates, its successive amplitudes decreasing at a rate determined by the constant a. The current and energy relationships of a series RLC circuit are shown graphically as functions of time in Fig. 114. When the current in Fig. 114 is positive the circuit absorbs energy from the source of voltage and when the current is negative the circuit returns energy to the source. If all of the energy in the circuit were reversible, i.e., if there were no heat losses, the negative current peaks would equal the positive current peaks and the circuit would return as much energy to the source when the current is negative as it received during the half cycle when the current was positive. To satisfy the condition of no heat losses the resistance R of the circuit must be zero. When R = 0, the constant a in Eq. 181 is also zero, and the equation then defines a steady alternating current. Current nonoscillatory When the constant b in Eq. 181 is imaginary, i.e., if
then the trigonometric sine in Eq. 181 becomes a hyperbolic sine and the current flows in one direction only, building up to a maximum value and then gradually dying out as the capacitor becomes charged to a potential difference that approaches the value of the applied voltage V. This is a case in which the reversible energy is so great in relation to the energy stored in the selfinductance L and in the capacitance C that there is no energy returned to the source while the capacitor is charged.
The case that serves as a dividing line between an oscillating current and a nonoscillating current is a critically damped circuit where the constant b = 0, i.e.
If b is made zero in Eq. 181 the righthand side of that equation becomes an indeterminate that can be evaluated on the basis that
Substitution of Eq. 180 in Eq. 179 yields
Whether the current oscillates or not, the irreversible energy, i.e., the energy converted into heat is expressed by
The total energy input to the circuit is
and since the applied voltage V is a constant it can be taken outside the integral, thus
The final voltage across the capacitor determines the final energy stored in the capacitor and, in this case, this final value of voltage is the constant applied voltage V. The final energy stored in the capacitor is therefore given by Eq. 155 as
It therefore follows that, with a constant applied dc voltage, the irreversible energy in the series RLC circuit equals the stored energy so that, in this case also, onehalf of the energy input is irreversible. Since Eq. 167 for the case of the mechanical system is similar to Eq. 181, it follows that conditions analogous to the cases of the oscillating and nonoscillating current must exist. Thus, if in Eq. 167 the constant b is real, the motion of the mass M reverses its direction periodically, and the mass gradually comes to rest. If on the other hand the constant b is imaginary, the mass M moves only in the direction of the applied force F, and gradually comes to rest without any reversal in the direction of motion. It is significant that when a constant force is applied to a system in which the energy is finally stored in the deformation of a body such as a spring, onehalf of the energy input to the system is converted into heat. This is true also in the case of an electric circuit in which the energy is finally stored in an ideal capacitor. Thus, under the condition of a constant applied force, a spring is capable of storing, at best, only 50 percent of the applied energy in reversible form. This is also true of a capacitor under the condition of constant applied voltage.


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