Capacitance & Inductance
While resistance characterizes the opposition to a steady electric current (DC), capacitance and inductance characterize how components react to alternating current (AC) and changing voltages or magnetic fields. Together, these form the broader concept of impedance. Metrology in this domain is crucial for the development of high-frequency communications, power distribution networks, and advanced semiconductor manufacturing.
Fundamental Definitions
Capacitance is the ability of a system to store an electric charge. It is defined as the ratio of the change in electric charge ($Q$) to the corresponding change in its electric potential ($V$).
The SI unit of capacitance is the farad (F), defined as one coulomb per volt. In practical electronics, a farad is exceptionally large, so microfarads ($\mu\text{F}$), nanofarads (nF), and picofarads (pF) are predominantly used.
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. A changing current creates a changing magnetic field, which in turn induces an electromotive force (EMF) that opposes the initial change in current (Lenz's Law).
The SI unit of inductance is the henry (H). One henry is defined as the inductance that produces an induced voltage of one volt when the current is changing at a rate of one ampere per second. Common sub-units include millihenries (mH) and microhenries ($\mu\text{H}$).
Realizing the Farad: The Calculable Capacitor
Historically, realizing a precise primary standard for impedance was extremely challenging. A major breakthrough occurred in 1956 with the Thompson-Lampard theorem. This theorem proved that the cross-capacitance of a specific arrangement of four cylindrical electrodes depends only on the length of the cylinders and the permittivity of vacuum ($\varepsilon_0$).
Because the permittivity of a vacuum was an exact defined constant (prior to the 2019 SI redefinition), the capacitance could be calculated directly from a single length measurement. By using an optical interferometer to precisely measure the length of the electrodes, National Metrology Institutes could create a Calculable Capacitor.
This remarkable instrument allowed the farad (and consequently the ohm, via complex AC bridge techniques) to be linked directly to the SI unit of length (the meter), providing a foundation for electrical metrology for decades.
Modern Realization: The Quantum Hall Effect
With the advent of the Quantum Hall Effect (QHE) and the 2019 redefinition of the SI units, the hierarchy of electrical standards was inverted. The von Klitzing constant ($R_K$) now provides an exact, primary standard for resistance.
Today, highly advanced metrology laboratories use purely electronic quadrature bridges to derive capacitance directly from the quantized Hall resistance. The relationship relies on precise angular frequency ($\omega = 2\pi f$), linking the farad fundamentally to quantum resistance and time (frequency). Once capacitance is established, inductance can also be derived using similar bridge resonance techniques.
Industrial Measurement Techniques
While NMIs use complex quadrature bridges, industrial laboratories and manufacturing floors rely on robust, highly accurate working standards and measurement instruments to quantify reactive components.
- LCR Meters: The workhorse of impedance metrology. These digital instruments measure Inductance (L), Capacitance (C), and Resistance (R). They operate by applying an AC voltage (at a selectable frequency) across the device under test (DUT) and measuring the resulting current and phase angle. From the voltage-to-current ratio and the phase difference, the instrument's microprocessor calculates the complex impedance and extracts the L, C, and R values.
- Maxwell-Wien Bridges: A classical AC bridge circuit used to measure unknown inductances in terms of a known capacitance and known resistances. While largely replaced by digital LCR meters in routine calibration, the fundamental principles of balancing the real (resistive) and imaginary (reactive) components of an AC bridge remain essential in high-echelon metrology.
- Vector Network Analyzers (VNAs): At high frequencies (RF and microwave), lumped element models of L and C break down. VNAs are used to measure the scattering parameters (S-parameters) of networks, evaluating how electrical signals are reflected and transmitted. This is critical for characterizing the high-frequency parasitic inductance and capacitance of components.