Lumped electrical circuits can be viewed naturally as dynamical systems governed by ordinary differential equations. The state of a circuit is determined by its energy-storing elements, while resistive elements introduce dissipation but no additional state.
In particular, capacitors store energy in electric fields and introduce state variables associated with voltage, while inductors store energy in magnetic fields and introduce state variables associated with current. Once these state variables are identified, the remaining circuit equations follow from Kirchhoff’s laws and element constitutive relations.
State Variables in Circuits
Each independent capacitor voltage and inductor current corresponds to a state variable. A circuit with no capacitors or inductors is purely algebraic, while a circuit containing energy storage elements gives rise to differential equations.
The number of independent state variables determines the order of the resulting dynamical system. RC circuits produce first-order dynamics, while RLC circuits produce second-order dynamics. More complex circuits give rise to higher-dimensional systems.
Series Connections
When elements are connected in series, they share the same current. From a dynamical systems perspective, series connections typically modify existing state equations rather than introducing new state variables.
For example, adding a resistor in series with an RLC circuit increases dissipation, while adding an inductor in series increases effective inertia. The structure of the governing equations remains the same, but the functional form or parameters may change.
In this sense, series connections act by adding terms or modifying relations within an existing dynamical system rather than increasing its dimension.
Capacitors connected in series do not generally introduce additional state variables, since their charges are constrained to be equal and their voltages are algebraically related. As a result, a series combination of capacitors can be represented by a single effective capacitance with one associated state variable.
Parallel Connections
Parallel connections share the same voltage but allow current to split into multiple paths. When energy-storing elements are connected in parallel, each branch may carry its own independent state variable, leading to additional differential equations.
For example, inductors connected in parallel each introduce their own current state, requiring additional equations to describe the system. In this case, the system dimension increases, and the circuit must be modeled using a vector-valued state.
Parallel capacitors, by contrast, share a common voltage and therefore combine into a single effective capacitance without increasing the number of state variables.
Circuits as ODE Systems
Once state variables are chosen, circuit equations can be written in standard dynamical systems form. Kirchhoff’s laws provide algebraic constraints, while capacitor and inductor relations introduce time derivatives.
If all circuit elements obey linear constitutive relations, the resulting system is a linear system of ordinary differential equations. When nonlinear elements are present, the same modeling procedure leads instead to nonlinear differential equations. In both cases, the underlying state-space structure remains unchanged.
From this perspective, circuit analysis becomes an exercise in identifying state variables, writing conservation laws, and assembling a system of ordinary differential equations. This viewpoint makes explicit the close connection between electrical circuits, mechanical systems, and control theory.
Summary: When Does a Circuit Element Add a State?
In lumped electrical circuits, state variables arise from independent energy storage. Capacitors store energy in electric fields and introduce state variables associated with voltage, while inductors store energy in magnetic fields and introduce state variables associated with current.
An energy-storing element contributes a new state variable only if its voltage or current is not algebraically constrained by Kirchhoff’s laws and the circuit topology. If the voltage or current of an element can be eliminated using algebraic relations alone, then no additional state is introduced.
- Capacitors in series share the same charge and do not introduce independent voltage states.
- Capacitors in parallel share the same voltage and do not introduce independent voltage states.
- Inductors in series share the same current and do not introduce independent current states.
- Inductors in parallel carry independent currents and therefore introduce additional state variables.
More generally, new state variables appear whenever energy-storing elements are separated by circuit configurations that break algebraic constraints, such as inductors between capacitors, capacitors between inductors, nonlinear elements, or switching elements.
In summary, series and parallel connections are useful heuristics, but the true criterion for adding state is the presence of independent energy storage modes.