Resistors

where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

A source-free RL circuit occurs when its DC source is suddenly disconnected. The energy already stored in the inductor is released to the resistors.

The energy that was initially stored in the inductor is eventually dissipated in the resistor.

A source-free RC circuit occurs when its DC source is suddenly disconnected. The energy already stored in the capacitor is released to the resistors.

The energy that was initially stored in the capacitor is eventually dissipated in the resistor.

where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

Typical examples of second-order circuits are RLC circuits, in which the three kinds of passive elements (resistors, inductors, and capacitors) are present. Other examples are RL and RC circuits.

A circuit comprising a resistor and capacitor and a circuit comprising a resistor and an inductor are called RC and RL circuits, respectively. The differential equations resulting from analyzing RC and RL circuits are of the first-order.

In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators.

The impedance \(Z(s)\) of the resistor is