Kirchoff’s Current Law (KCL) states that the sum of all currents coming out of a node in an electric circuit must sum to zero. This is a consequence of the conservation of charge, any that charge that flows must be matched by a flow of charge in.
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Kirchoff’s Voltage Law (KVL) states that the sum of all voltage drops around a closed loop must equal zero. When one tracks changes in electric potential around a loop, one must end up back at the original electric potential, so the voltage drops must sum to zero.
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Capacitors consist of separated surfaces which have the capacity to store charge, $q_c$, by the voltage difference between the surfaces. This charge is proportional to the voltage across the capacitor, $e_{12}$, according to the capacitance, $c$. Differentiating this equation with respect to time indicates that the current into a capacitor is related to the time derivative of the voltage across it.
$$ q_c=ce_{12}\Rarr i_c=c\dot e_{12} $$
Related to its ability to store charge, a capacitor stores energy. This makes the voltage across a capacitor a very suitable state variable. Application of Kirchoff’s current law to a node adjacent to a capacitor allows us to relate the rate of change of the voltage across the capacitor to other currents in the circuit, by the above equation. This gives a differential equation perfect for use in state-space form.
$$ {\large\boldsymbol\varepsilon}c=\frac 12ce{12}^2 $$
We may also note that, analogous to the velocity of a mass, we cannot instantaneously change the voltage across an inductor without infinite power. However, analogous to an instantaneous change in the acceleration of a mass, we can instantaneously change the current into a capacitor.
Inductors generate a magnetic flux through the core of their coiled wires, and thus operate on a linkage between this flux and the current passing through them. The time derivative of this flux linkage, $\lambda_{12}$, is the voltage drop, $e_{12}$, induced across the inductor. Thus, the voltage across an inductor is proportional to the time derivative of the current through it according to its characteristic inductance, $L$.
$$ \lambda_{12}=Li_L\Rarr e_{12}=L\dot i_L $$
Related to its capacity to induce a voltage, an inductor stores energy. This makes the current through an inductor a very suitable state variable. Application of Kirchoff’s voltage law to a loop containing the inductor allows us to relate the rate of change of the current through the inductor to other voltage drops in the circuit, by the above equation. This gives a differential equation perfect for use in state-space form.
$$ {\large\boldsymbol\varepsilon}_L=\frac 12Li_L^2 $$
We may also note that, analogous to a the force on a spring, we cannot instantaneously change the current through an inductor. However, we can instantaneously change the voltage across an inductor.
Inductors consist essentially of two inductors whose magnetic fluxes are coupled. This coupling ensures that the ratio of voltages across the inductors is equivalent to the ratio between the number of turns in their coils. Additionally, it ensures that the ratio of currents through the inductors is equivalent to the inverse of the ratio between the number of turns in their coils.