
RCLCircuit
RCLcircuits (also often called RLCcircuits) are often encounterd in the description of the discharge characteristic of a pulse generator, especially for capacitors and including Marxbank generators.
The challenge is to estimate peak current (and voltage) and how fast they could be achieved. Conversely, a comparison with measured outputs allows
in particular estimations on systeminductances. The tool will help to estimate current peaks and associated rise times. Voltage peaks can then be derived from the values of respective circuit elements
R and L.
Current waveforms for underdamped, critically damped and overdamped parameters:
Underdamped solution (δ < ω_{0}):



Critically damped solution (δ = ω_{0}):



Overdamped solution (δ > ω_{0}):



Solutions for the transient response when discharging a capacitance, C, through an inductance, L, and load resistance, R, can be found in respective textbooks.
In short: from Kirchof's law for the voltage along a closed loop, a second order differental equation can be derived. Depending on the solution of the characteristic
equation, the response can be described by either an underdamped, a critically damped or an overdamped solution.
Kirchhof's law for voltage loops:
Characteristic equation:
with resonance frequency (undamped natural frequency):
and damping attenuation:
and damped natural frequency:
or the "frequency" for the overdamped response:
Approach to Derive Solutions
Ansatz:
Boundary conditions:


