# Resonant (CLC) Charging

Matched (C2=C1)
Ring Up (C2<<C1)
Ring Down (C2>>C1)
The equations, diagrams, and waveforms on this page describe Resonant (CLC) Charging of one capacitor from another capacitor through a charging inductor. This form of charging is frequently used in pulsed power and power conditioning circuits in order to efficiently transfer energy from one stage to another. The following equations are used to describe the critical parameters of resonant charging, including the resonant frequency (omega), the charging current, the voltage on the load capacitor, the maximum possible load capacitor voltage, and the efficiency of the energy transfer.

Each of the waveform plots shown below can be clicked on to open up the full size version of the graph in a separate window.

 Equation for Resonant Charging Circuit Resonant Frequency Omega L is the circuit inductance (H) C1 is the source capacitance (F) C2 is the load capacitance (F) Equation for the Current in a CLC Resonant Charging Circuit Current V0 is the initial voltage on the C1 capacitor (V) Equation for the Load Capacitor Voltage in a CLC Resonant Charging Circuit C2 Voltage Same as above Equation for the Maximum Possible Load Capacitor Voltage in a CLC Resonant Charging Circuit Maximum C2 Voltage Same as above Equation for the Energy Transfer Efficiency in a CLC Resonant Charging Circuit Energy Transfer Efficiency Same as above

The following graph illustrates the relationship between these last two parameters (Maximum C2 Voltage and Energy Transfer Efficiency) and the ratio of capacitances (C2 / C1).One can see from the graph that maximum energy transfer is obtained when the two capacitances are equal. A second condition of interest is that where the load capacitance is much smaller than the source capacitance. In this case, the maximum load capacitor voltage approaches 2 times the initial voltage on the source capacitor.

A voltage gain of ~2X is therefore possible during the resonant charging. This can be used in many cases to reduce the operating voltage of the front end power system.

Graph of the Maximum Load Capacitor Voltage and Energy Transfer Ratio as a Function of the Ratio of Capacitances in a CLC Resonant Charging Circuit

Matched Capacitance Resonant Charging (C2=C1)

As noted above, this case allows maximum energy transfer from the source capacitor to the load capacitor. All of the energy from C1 can be transferred to C2.  Since the capacitors are equal, this implies that the voltage on C2 will equal the initial voltage on C1 (as shown in the waveforms). The voltage on the source capacitor also reaches zero volts at the same time that the load capacitor reaches its maximum voltage. In real life, resistive losses in the circuit will reduce the maximum possible voltage on C2 somewhat. One can also see that the charging current in this case is a simple half-sinusoid shape.

The viewer should also note that the simulation has assumed that the switch has opened at the end of the half-period (this has been done in all three cases). If this were not done, the energy would merely resonantly transfer back to the source capacitor. In real life, this can be done by several techniques including the addition of a diode to the circuit to only allow energy transfer in the forward direction.

Schematic Diagram of a CLC Resonant Charging Circuit With Equal Capacitors

Circuit Waveforms for a CLC Resonant Charging Circuit With Equal Capacitors

Ring-Up Resonant Charging (C2<<C1)

This is the second case of interest since as noted above, the voltage on the load capacitor can be doubled providing a voltage gain in the system of 2X. In this case, the load capacitor voltage is approximately doubled (~1.8X) as the source capacitor voltage only drops by ~10%.

Circuit Schematic for a CLC Resonant Charging Circuit Where the Initial Capacitor is Much Larger Than the Load

Circuit Waveforms for a CLC Resonant Charging Circuit Where the Initial Capacitor is Much Larger Than the Load

Ring-Down Resonant Charging (C2>>C1)

Circuit Schematic for a CLC Resonant Charging Circuit Where the Initial Capacitor is Much Smaller Than the Load

Circuit Waveforms for a CLC Resonant Charging Circuit Where the Initial Capacitor is Much Smaller Than the Load

Send consulting inquiries, comments, and suggestions to richard.ness@nessengr.com