The Capacitor Paradox
This is much more than a theoretical paradox; it is what happens in circuits in the real world and has practical applications in circuit design. Two ideal capacitors have the same capacitance C; one charged to voltage V and charge Q and the other uncharged. The energy stored in the charged capacitor is ½(capacitance)(voltage2) = ½CV2.
2C farads, ½V volts, charge Q
total energy ¼ CV2
The initial voltage = charge/capacitance, V = Q/C.
When the capacitors are connected in parallel by an ideal switch, the resulting compound capacitor has:
- charge conserved = Q
- capacitance doubled = 2C
- voltage = charge/capacitance = Q/2C = ½V
The stored energy ½(capacitance)(voltage2) = ½(2C) (½V)2 = ¼ CV2.
So, where is the other half of the initial energy?
Learned folk have proposed these solutions:
If the switch closes in a finite time, the energy is lost in the switch.
If there is series resistance then half the energy is dissipated in this, regardless of the value of the resistance.
If the capacitors are connected with finite leads then they act as a loop antenna and energy is lost by electromagnetic radiation.
Series inductance (ideal) causes the circuit to oscillate continuously and no energy is lost. If the inductor is not ideal the oscillation will decay until only half the original energy remains in the charged capacitors, energy being lost by electromagnetic radiation and/or heat depending on whether the inductor radiates or has resistance or both.
All of these solutions require a particular change in the given conditions of the paradox which specifies ideal components. The switch is ideal and closes in zero time. The capacitors are ideal lossless components with zero series impedance, zero dielectric loss and zero lead length. Such ideal capacitors are not rare; every time you design a capacitor circuit and ignore ESR and other parasitic effects, you are assuming an ideal capacitor. Of course in practical circuits the loss mechanisms can be any of the suggested solutions depending on the ways that the circuit departs from the ideal.
Another suggestion is that energy is required to move electrons across the finite plates of the charged capacitor to and across the finite plates of the uncharged one. It follows from this that, in order to satisfy the law of conservation of mass-energy, this energy must be either:
- stored in the circuit and retrievable in some way other than in the charge on the capacitors and/or
- lost to the circuit as electrical, magnetic, electromagnetic, mechanical, thermal, chemical energy etc.
In my opinion, no energy is required for the transport of electrons through and across these lossless capacitors and the required storage or loss mechanisms are just not there.
To my simple way of thinking if all these solutions require changes to the initial conditions to avoid an impossible outcome then the initial conditions must be impossible. Another example of the impossible constructed from ideal components is 2 ideal voltage sources of different value connected in parallel. An infinite current flows for as long as you like but no energy is involved. This is like the irresistible force acting on the immovable object. The problem is in the given conditions.
For circuit designers this is the really interesting bit. We have seen that charging one capacitor from another is inefficient. It can be shown that the same thing happens if you charge from a voltage source. 50% of the energy from the source is dissipated in series resistance, be it a physical resistor or the incidental parasitic resistance in capacitors, switches, semiconductors, inductors or the radiation resistance of antenna-like leads. So a capacitor or voltage source is no way to charge a capacitor unless you are willing to wear the inefficiency and stress on other components.
Ignoring the capacitor paradox can lead to some very inefficient or unreliable designs. Early capacitor discharge ignitions were typically push pull transistor oscillators charging the capacitor via a step-up transformer and rectifier. In effect the discharged capacitor was being charged from a voltage source, although with some series leakage inductance from the transformer. Something like 50% of the input energy would have been dissipated in the transistors, transformer and rectifier. Failure of the transistors was not rare and one published circuit provided for automatic switching back to conventional ignition on failure. Rail inverters, switched capacitor converters, charge pumps and Cockcroft-Walton voltage multipliers can be disappointingly inefficient if driven by square waves.
What's the alternative? Use a current source. Later capacitor discharge ignition designs used flyback converters which are much more efficient and reliable in this application because they are current sources. Similar converters are favoured for charging photoflash capacitors. Buck converters can be extremely efficient because by interposing an inductor and diode in the charging path they act as current sources. Most PC power supplies use push pull forward converters, which are just dual transformer coupled buck converters.
The practical implications of the capacitor paradox are seen in other areas. The power supply current of CMOS logic is largely due to the charging and discharging of parasitic capacitance. SPICE simulation of power MOSFETS driven by a stiff voltage source shows that peak gate currents of more than an amp are possible. This is because of the MOSFET's large input capacitance multiplied by the Miller Effect. Adding a series gate resistor to suppress parasitics also reduces the peak gate current but does nothing for the efficiency. You may need more robust drivers than you expect.