Space charge

Space charge is a localized region of excess negative charge that occurs near a metal object when heated to incandescence in a vacuum. This effect was first observed by Thomas Edison in light bulb filaments, where it is sometimes called the Edison Effect.

Cause

Simply

When a metal object is placed in a vacuum and is heated to incandescence, the energy is sufficient to cause electrons to "boil" away from the surface atoms and surround the metal object in a cloud of free electrons. This is called thermionic emission. The resulting cloud is negatively charged, and can be attracted to any nearby positively charged object, thus producing an electrical current which passes through the vacuum.

Theoretical

If the vacuum is 10^-6mm pressure or less, the main vehicle of conduction is electrons. Equation $i=(1-{\tilde {r}})A_{0}T^{2}\exp \left({\frac {-e\phi }{kT}}\right)$A/cm² gives the current per cm² of cathode as a function(mathematics)function of cathode temperature. The emitted electrons do not all have the same but have a Maxwellian distribution of speeds, viz., proportional to $\exp \left(-{\frac {{\frac {1}{2}}mv^{2}}{kT}}\right)dv$ where v is the speed.

A₀ = ${\frac {4\pi mek^{2}}{h}}$ = 120 A/cm²K²
k = Boltzmann's constant = 1.38 x 10^-23J/K
h = Planck's constant = 6.55 x 10^-34J sec
φ = work function of the cathode (electron-volt)
T = absolute temperature
ř = a reflection coefficient

The reflection coefficient can be as low as 0.105 but is usually near 0.5. For Tungsten, (1 - ř)A₀ = 60 to 100 and φ = 4.52. At 2500°C, the emission is 300mA/cm²

The emission current is many times greater than that normally collected by the electrodes, except in some pulsed valves such as the cavity magnetron used in radar: the cathode would suffer rapid exhaustion if the full emission were drawn off. Most of the electrons emitted by the cathode are driven back to it by the repulsion of the cloud of electrons in its neighbourhood. This is called space charge effect, and it determines the current-voltage characteristic of tube.

Occurrence

Space charge is an inherent property of all vacuum tubes. This has at times has made life harder or easier for electrical engineers who used tubes in their designs. For example, space charge significantly limited the practical application of triode amplifiers because it impedes the flow of electrons from cathode to anode, thus reducing the level of gain that could be achieved in such tubes.

On the other hand, space charge came in quite handy in some tube applications because it generates a negative EMF within the tube's envelope, which could be used to create a negative bias on the tube's grid. This could improve the engineer's control and fidelity of amplification.

Space charges can also occur within a solid, liquid, or gas dielectric. For example, when gas near a high voltage electrode begins to undergo dielectric breakdown, electrical charges are injected into the region near the electrode, forming space charge regions in the surrounding gas. Space charges can also occur within solid or liquid dielectrics that are stressed by high electric fields. Trapped space charges within solid dielectrics are often a contributing factor leading to dielectric failure within high voltage power cables and capacitors.

Child's Law

Also known as the Child-Langmuir Law or the Three-Halves Power Law, Child's Law states that the space charge-limited current (SCLC) in a plane-parallel diode varies directly as the three-halves power of the anode voltage ${V_{a}}$ and inversely as the square of the distance $d$ separating the cathode and the anode.

Mathematically: $I_{a}=JS=2.33\times 10^{-6}{\frac {S\nu {V_{a}}^{3/2}}{d^{2}}}$.

Where $I_{a}$ is the anode current, J the current density, and S the area. This assumes the following:

1. The electrodes are planar, parallel, equipotential surfaces of infinite dimensions.
2. The electrons have zero velocity at the cathode surface.
3. In the interelectrode region, only electrons are present.
4. The current is space-charge limited.
5. The anode voltage remains constant for a sufficiently long time so that the anode current is steady.

Mott's steady-state space-charge-limited conduction model

The steady-state space-charge-limited conduction-current density $J$ in a plane-parallel dielectric sample with electrode separation $L$ is proportional to the square of the applied voltage $V$.

Mathematically: $J={\frac {9{\epsilon }{\mu }{V}^{2}}{8{L}^{3}}}$

This assumes the following:

1. There is only one type of charge carrier present.
2. The material has no intrinsic conductivity, but charges are injected into it from one electrode and captured by the other.
3. The carrier mobility $\mu$ and the dielectric permittivity $\epsilon$ are constant throughout the sample.
4. The electric field at the charge-injecting cathode is zero.

Shot noise

Space charge tends to reduce shot noise. Electrons (and positive charge carriers) come with their own built-in negative feedback.

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See also

• Thermionic emission
• Vacuum tubes
• Grid leak

References

Telecommunications by A. T. Starr, Second Ed., Sir Isaac Pitman & Sons, Ltd, London, 1958

Physics of Dielectrics for the Engineer by R. Coelho, Elsevier Scientific Pub. Co., Amsterdam, 1979