Linear Circuit Equations from Maxwell’s Equations

Abstract

In an elementary circuit with inductance, resistance, capacitance and applied voltage in series as shown in Figure 37 the Kirchhoff circuit equation is

$$L\ddot I + R\dot I + \frac{I}{C} = \frac{{d{E^e}}}{{dt}},$$

(14.1.1)

$$L\dot I + RI + \frac{Q}{C} = {E^e},$$

(14.1.2)

where E^e is the applied electromotive force or EMF, I is the current, Q is the charge on the capacitor plates, C is the capacitance, L is the inductance and R is the resistance. For two or more circuits inductively coupled the equations are

$${L_{11}}{\dot I^1} + {L_{12}}{\dot I^2} + {R^1}{I^1} + \frac{{{Q^1}}}{{{C_1}}} = {E^1},$$

(14.1.3)

$${L_{12}}{I^1} + {L_{22}}{I^2} + {R_2}{I^2} + \frac{{{Q^2}}}{{{C_2}}} = {E^2},$$

(14.1.4)

where L₁₂ is the mutual inductance and L₁₁ and L₂₂ are the self inductances. The question naturally arises as to just what is the relation of these equations to the more general Maxwell electromagnetic field equations which must govern the circuit behavior.