## Problem

Charging or discharging a capacitor may cause energy loss even if no dissipative elements are apparent.

Figure 1.

Figure 1 (a) shows a capacitor C1 charged to voltage Vi and no voltage on capacitor C2 before switch closure. C1 is equal to C2 and the energy in the system is:

Energy = (C1*V1^2)/2

After switch closure (b), the charge and voltage is divided equally between the two capacitors (conservation of charge) and the total energy in the system is:

Energy = (C1*(V1/2)^2)/2 + (C2*(V2/2)^2)/2 = (C1*V1^2)/4

**Half the energy has disappeared. Where did it go?**

For a more sophisticated derivation, references to earlier papers, and an application see On Lossless Switched-Capacitor Power Converters.

For another interesting application see Recovered Energy Logic - A Highly Efficient Alternative to Today's Logic Circuits.

**Where the energy went.**

**Loss**. The energy lost in directly switching voltage to a capacitor at another potential is lost in parasitic resistance, and if the resistance is too low, in arcing or welding of the switch contacts or in radiation.

**Resistance**. The easiest loss mechanism to show analytically is the loss in parasitic resistance, such as the capacitor equivalent series resistance (ESR) or wiring resistance. Adding this resistance to the circuit and calculating the power dissipated shows the energy loss. The energy loss is independent of the value of the resistance.

**Arcing**. Most switches used in power supplies are solid state and arcing is not a problem, but if a capacitor is charged through a contact, arcing may be a problem.

**Radiation**. High rates of change of voltage or current result in radiation. Directly switching voltage to a capacitor at another potential is a source of radiation (EMI).

**Note**. The losses are the same if C1 is a voltage source instead of a capacitor.

## Relevance

The problem affects power-loss and efficiency, electromagnetic interference (EMI), contact arcing, power MOSFET drive circuits, snubbers, solid state relays, resonant and duty-cycle switching converters, clocks, and selection of circuit topology.

**Topology**. Occasionally one sees switched-capacitor circuits in the literature that are proposed as efficient power conversion circuits. This is usually not the case and energy loss calculation and measurements should be made to determine what the losses really are. These topologies are often used in power MOSFET drive circuits (gate capacitance), clocks, snubbers, etc. There are often "lossless" topologies that can be substituted for these circuits.

**Efficiency**. High-efficiency circuit design requires controlling all power loss mechanisms. The losses in charging and discharging circuit and parasitic capacitance needs to be considered along with other loss mechanisms.

**EMI**. If no dissipative elements are present in the circuit, energy loss from direct charge and discharge of capacitance is radiated as electromagnetic energy, usually undesirable.

**Arcing**. Contact arcing is a loss mechanism involving resistance and radiation and is usually undesirable.

## Solvability

By transferring energy through an inductor or by controlling the charging waveform, a capacitor can be charged without loss of energy (ideally).

**Approaches**. The loss mechanism in charging or discharging a capacitor comes from trying to instantaneously change the voltage across a capacitor. There are two common "lossless" circuit solutions to prevent these losses: switching energy through and inductor or switching at zero capacitor voltage.

**Inductor**. The inductor solution places an inductor in the circuit between the capacitor and switched source so that the instantaneous voltage appears across the inductor, not the capacitor. The familiar buck-converter is an example.

**Switch Matrix**. An example of zero-voltage switching are switch-matrix ac-to-ac (cycloconversion) and ac-to-dc converters (rectifier) where an alternating ac waveform is switched to the capacitor when it is equal to the capacitor voltage.

**Resonance**. Resonant converters are another example of zero voltage switching. In this class of converters, a resonant circuit is used to provide a waveform that can be switched when the voltage is zero across a capacitor.

## Solution

Transfer energy through an inductor or with a resonant or sinusoidal waveform with switching at zero voltage.

**Inductor**. Transferring energy through an inductor is the heart of most dc-to-dc converters. Rudy Severns has done an excellent job of categorizing this class of converter topologies.

*Switchmode Converter Topologies - Make Them Work for You**Modern DC-DC Switchmode Power Conversion Circuits*

**Resonance**. A 1988 collection of papers on resonant circuits edited by Kit Sum makes a good starting point for resonant converters. Especially recommended is the second paper, an invited paper by Steve Freeland, *An Introduction to the Principles and Features of Resonant Power
Conversion*.

**Switch Matrix**. A switching matrix is the simplest conceivable form of converter, consisting of input lines and output lines connected with switches. Peter Wood has a good introduction.

## Personal Anecdote

The problem in Figure 1 on what happens to the energy when a charged capacitor is switched into another capacitor at a different voltage was a homework problem in my first circuit course so I learned about this early. However, I am constantly amazed at how often such a power electronics "perpetual motion machine" circuit is proposed as an efficient converter.

The most flagrant example I know about was on a three-year government project to make a highly efficient hybrid dc-dc converter using only capacitors and no inductors. When the first paper was given at a respected conference, several from the audience commented on the problem -- suggesting that it was inductance in the banana-plug-wired breadboard that made the circuit work. Undaunted, they reported on circuit improvements the next year, still with their banana plug breadboard. Their plans were to make it a hybrid circuit next year. Next year there was no paper. We speculated that when they finally got rid of wiring inductance, they found the circuit did not work, more than two years after it was obvious to experienced power electronics designers. In all fairness, this was in the mid 1970's and even major power supply companies were having problems with switching-mode power supplies. Concurrently, the sponsoring activity was doing some outstanding work in the field. The lesson-to-be-learned is that when someone questions your work, you should seek a side meeting to explore their concerns - they may know something you need to know. In this case it would have saved going down a blind path.

## On the Web

I have not found anything on the web on this subject yet.