Electric Capacitance

Lecture

Electric capacitance — a characteristic of a conductor, a measure of its ability to accumulate electric charge. In circuit theory, capacitance refers to the mutual capacitance between two conductors; it is the parameter of a capacitive element of an electric circuit (a capacitor), represented as a two-terminal device.

In the International System of Units (SI), capacitance is measured in farads, the commonly accepted symbol for capacitance being C.

Capacitance is calculated as the ratio of the electric charge to the potential difference between the conductor and infinity, or between conductors

C=Qφ−φref Electric Capacitance ,

where Q — charge, φ — potential of the conductor, φref — potential of the other conductor or the potential at infinity (usually taken as zero).

Capacitance depends on the geometry and shape of the conductors and on the electrical properties of the surrounding medium (its permittivity).

Definition. Some formulas

For a single conductor, capacitance equals the ratio of the conductor's charge to its potential, assuming all other conductors are infinitely far away and that the potential at an infinitely distant point is taken as zero. In mathematical form, this definition is written as

Electric Capacitance ,

where Q — charge, φ — potential of the conductor. For example, the capacitance of a conducting ball (or sphere) of radius R is (in the SI system):

Electric Capacitance

where ε0 — the electric constant (8.854⋅10−12 F/m), Electric Capacitance — the relative permittivity.

Derivation of the formula

It is known that Electric Capacitance

Since Electric Capacitance , substituting the found φ here, we obtain .

For a system of two conductors, separated by a dielectric or vacuum and carrying equal but opposite charges ±Q, the capacitance (mutual capacitance) is defined as the ratio of the charge magnitude to the potential difference between the conductors. If the potential of one of the conductors is taken as zero, the formula C=Q/φ remains valid for this case as well.

A discrete circuit element based on the system described above, possessing significant capacitance, is called a capacitor. The two conductors in this case are called plates.

For a parallel-plate capacitor, the capacitance equals:

Electric Capacitance ,

where S — the plate area (the plates are assumed identical), d — the distance between the plates.

The electrical energy stored by a capacitor is

2 Electric Capacitance ,

where U — the voltage between the plates.

Notation and units of measurement

Capacitance is conventionally denoted by the capital Latin letter C (from the Latin capacitas — capacity, volume).

In the SI system, capacitance is expressed in farads[ , abbreviated «F». A conductor has a capacitance of one farad if, at a surface potential of one volt, it carries a charge of one coulomb. One farad is a very large capacitance; real conductors have capacitances on the order of nano- or microfarads. The «farad» is named after the English physicist Michael Faraday.

The unit of capacitance in the CGS system is the centimeter. Relation: 1 cm of capacitance ≈ 1.1126 pF; 1 F = 8.988×10¹¹ cm of capacitance.

Properties of capacitance

Capacitance is always positive, except in the case of certain structures with ferroelectrics.
Capacitance depends only on the geometric dimensions of the conductor and the dielectric properties of the medium (for a capacitor — the insulating material filling it).
Capacitance depends indirectly on temperature and signal frequency (through the dependence of the medium's permittivity εr on these quantities).
In the case of a medium with constant εr values, capacitance is a constant, but in the case of a nonlinear medium, where εr depends on the electric field strength, the capacitance will change with voltage.
When applied to a sinusoidal current circuit with frequency ω, the «capacitance» element can be assigned a reactance .
The voltage across a capacitance cannot change abruptly.

Differential capacitance

The differential (small-signal) capacitance is defined as the derivative of the conductor's charge with respect to potential

Electric Capacitance ,

which is determined for selected conditions Electric Capacitance . This quantity characterizes the response of the conductor to a small change in potential. If the dependence of charge on potential is linear, then , but more complex cases occur in practice.

Measurements of the so-called capacitance-voltage characteristics of metal-dielectric-semiconductor structures — the dependences Electric Capacitance at various frequencies ω of the potential's variation over time t according to the law φ=φ0+Δφsin⁡(ωt) — have become widespread. Such measurements provide valuable information about the quality of the dielectric.

Negative electric capacitance

Physicists have found the first example of a substance with ferroelectric properties that can exhibit negative electric capacitance. The possibility of such an effect was theoretically predicted by the Soviet physicist Lev Landau about 70 years ago, but its experimental observation had previously been considered practically impossible. The authors call the results important both for fundamental science and for technological applications. The paper was published in the journal Nature.

Ferroelectrics are crystalline substances characterized by the spontaneous emergence of polarization at certain temperatures in the absence of an external field. This phenomenon is analogous to ferromagnetism. These substances were discovered about 100 years ago. The first complete description of the emergence and destruction of the polar phase in ferroelectrics was given by Lev Landau through his theory of second-order phase transitions. This theory also predicted the existence of negative capacitance, but it had not previously been observed in such substances. Today the phenomenon of ferroelectricity has found numerous applications in electronics, nonlinear optics, memory devices, temperature sensors, and even in the field of neuromorphic computing.

In a new study, scientists from Germany and Romania proved that negative capacitance is characteristic of hafnium-zirconium oxide, and that it Electric Capacitance

is explained precisely by Landau's theory. In the simplest case, the capacitance of a parallel-plate capacitor is always a positive coefficient between the charge on the plates and the applied voltage. That is, the greater the applied voltage, the greater the charge that can be accumulated in the capacitor. If a capacitor had negative capacitance, the charge on it would increase as the voltage decreased and would decrease as the voltage increased. Landau's theory predicted that negative capacitance would be observed if the thermodynamic Helmholtz free energy corresponding to a given ferroelectric, as a function of polarization, had two minima. It was precisely this characteristic that was measured in the new experiment.

Comparison of the experimental results with Landau's theory

The authors note that the new material may find application in the near future, since negative capacitance can be used to amplify the applied potential, thereby reducing power losses to a level unattainable by present-day electronics.

«Most ferroelectrics are very difficult to integrate into modern semiconductor manufacturing methods, but Electric Capacitance is already used in these processes,» explains the paper's lead author Michael Hoffmann of the Laboratory of Nanoelectronic Materials (Germany). — This means that products using this effect could soon appear. Another advantage of is that it retains its ferroelectric properties even at film thicknesses of less than 10 nanometers, which is important for future miniaturization».