Resonant inductive coupling

Two resonances as a pair are observed

This process occurs in a resonant transformer, an electrical component which transformer consists of high Q coil wound on the same core with a capacitor connected across a coil to make a coupled LC circuit.

The most basic resonant inductive coupling consists of one drive coil on the primary side and one resonance circuit on the secondary side.[34][5][2] In this case, when the resonant state on the secondary side is observed from the primary side, two resonances as a pair are observed.[35][5] One of them is called the antiresonant frequency (parallel resonant frequency 1), and the other is called the resonant frequency (serial resonant frequency 1').[5] The short-circuit inductance and resonant capacitor of the secondary coil are combined into a resonant circuit.[36][5] When the primary coil is driven with a resonant frequency (serial resonant frequency) of the secondary side, the phases of the magnetic fields of the primary coil and the secondary coil are synchronized.[5] As a result, the maximum voltage is generated on the secondary coil due to the increase of the mutual flux, and the copper loss of the primary coil is reduced, the heat generation is reduced, and the efficiency is relatively improved.[2] The resonant inductive coupling is the near field wireless transmission of electrical energy between magnetically coupled coils, which is part of a resonant circuit tuned to resonate at the same frequency as the driving frequency.

In the transformer, only part of the flux generated by current through the primary coil is coupled to the secondary coil and vice versa. The part that couples is called mutual flux and the part that does not couple is called leakage flux.[37] When the system is not in the resonance state, this leads to the open-circuit voltage appearing at the secondary being less than predicted by the turns ratio of the coils. The degree of coupling is captured by a parameter called coupling coefficient. The coupling coefficient, k, is defined as the ratio of transformer open-circuit voltage ratio to the ratio that would be obtained if all the flux coupled from one coil to the other. However, if it is not open circuit, the flux ratio will change. The value of k lies between 0 and ±1. Each coil inductance can be notionally divided into two parts in the proportions k:(1−k). These are respectively an inductance producing the mutual flux and an inductance producing the leakage flux.

Coupling coefficient is a function of the geometry of the system. It is fixed by the positional relationship between the two coils. The coupling coefficient does not change between when the system is in the resonance state and when it is not in the resonance state, or even if the system is in resonance state and a secondary voltage larger than the turns ratio is generated. However, in the resonance case, the flux ratio changes and the mutual flux increases.

Resonant systems are said to be tightly coupled, loosely coupled, critically coupled or overcoupled. Tight coupling is when the coupling coefficient is around 1 as with conventional iron-core transformers. Overcoupling is when the secondary coil is so close and the formation of mutual flux is hindered by the effect of antiresonance, and critical coupling is when the transfer in the passband is optimal. Loose coupling is when the coils are distant from each other, so that most of the flux misses the secondary. In Tesla coils around 0.2 is used, and at greater distances, for example for inductive wireless power transmission, it may be lower than 0.01.

Generally the voltage gain of non resonantly coupled coils is directly proportional to the square root of the ratio of secondary and primary inductances.

${\displaystyle A=k{\sqrt {\frac {L_{2}}{L_{1}}}}\,}$

However, if in the state of resonant coupling, higher voltage is generated. The short-circuit inductance L_sc2 on the secondary side can be obtained by the following formula.

${\displaystyle L_{sc2}=(1-k^{2})\cdot {L_{2}}}$

The short-circuit inductance L_sc2 and the resonance capacitor Cr on the secondary side resonate. The resonance frequency ω₂ is as follows.

${\displaystyle \omega _{2}={1 \over {\sqrt {L_{sc2}C_{r}}}}={1 \over {\sqrt {(1-k^{2})\cdot {L_{2}}C_{r}}}}}$

Assuming that the load resistance is Rl, the Q value of the secondary resonance circuit is as follows.

${\displaystyle Q_{2}=R_{l}{\sqrt {\frac {C_{r}}{L_{sc2}}}}\,}$

The voltage generated in the resonance capacitor Cr at the peak of the resonance frequency is proportional to the Q value. Therefore, the voltage gain Ar of the secondary coil with respect to the primary coil when the system is resonating,

${\displaystyle A_{r}=kQ_{2}{\sqrt {\frac {L_{2}}{L_{1}}}}\,}$

In the case of the Type P-P, Q1 does not contribute to the voltage gain.

The WiTricity type magnetic resonance is characterized in that the resonant coils on the primary side and the resonant coils on the secondary side are paired. The primary resonant coil increases the primary driving coil current and increases the generated magnetic flux around the primary resonator. This is equivalent to driving the primary coil at high voltage. In the case of the type on the left figure, the general principle is that if a given oscillating amount of energy (for example a pulse or a series of pulses) is placed into a primary coil which is capacitively loaded, the coil will 'ring', and form an oscillating magnetic field.

Resonant transfer works by making a coil ring with an oscillating current. This generates an oscillating magnetic field. Because the coil is highly resonant, any energy placed in the coil dies away relatively slowly over very many cycles; but if a second coil is brought near it, the coil can pick up most of the energy before it is lost, even if it is some distance away. The fields used are predominantly non-radiative, near fields (sometimes called evanescent waves), as all hardware is kept well within the 1/4 wavelength distance they radiate little energy from the transmitter to infinity.

The energy will transfer back and forth between the magnetic field in the inductor and the electric field across the capacitor at the resonant frequency. This oscillation will die away at a rate determined by the gain-bandwidth (Q factor), mainly due to resistive and radiative losses. However, provided the secondary coil cuts enough of the field that it absorbs more energy than is lost in each cycle of the primary, then most of the energy can still be transferred.

Because the Q factor can be very high, (experimentally around a thousand has been demonstrated[38] with air cored coils) only a small percentage of the field has to be coupled from one coil to the other to achieve high efficiency, even though the field dies quickly with distance from a coil, the primary and secondary can be several diameters apart.

It can be shown that a figure of merit for the efficiency is:[39]

${\displaystyle U=k{\sqrt {Q_{1}Q_{2}}}}$

Where Q₁ and Q₂ are the Q factors of the source and receiver coils respectively, and k is the coupling coefficient described above.

And the maximum achievable efficiency is:[39]

${\displaystyle \eta _{opt}={\frac {U^{2}}{(1+{\sqrt {1+U^{2}}})^{2}}}}$

Because the Q can be very high, even when low power is fed into the transmitter coil, a relatively intense field builds up over multiple cycles, which increases the power that can be received—at resonance far more power is in the oscillating field than is being fed into the coil, and the receiver coil receives a percentage of that.

Unlike the multiple-layer secondary of a non-resonant transformer, coils for this purpose are often single layer solenoids (to minimise skin effect and give improved Q) in parallel with a suitable capacitor. Alternative resonator geometries include wave-wound Litz wire and loop-gap resonators (LGRs). In the Litz wire-based resonators, insulation is either absent or low permittivity and low-loss materials such as silk are used to minimise dielectric losses. The LGR geometries have the advantage that electric fields outside the resonator structure are very weak which minimizes human exposure to electric fields and makes the power transfer efficiency insensitive to nearby dielectrics.[40]

To progressively feed energy into the primary coil with each cycle, different circuits can be used. One circuit employs a Colpitts oscillator.[38]

In Tesla coils an intermittent switching system, a "circuit controller" or "break," is used to inject an impulsive signal into the primary coil; the secondary coil then rings and decays.

The receiver of a smart card has a coil connected to a chip which provides capacitance to give resonance as well as regulators to provide a suitable voltage

The secondary receiver coils are similar designs to the primary sending coils. Running the secondary at the same resonant frequency as the primary ensures that the secondary has a low impedance at the transmitter's frequency and that the energy is optimally absorbed.

Example receiver coil. The coil is loaded with a capacitor and two LEDs. The coil and the capacitor form a series LC circuit which is tuned to a resonant frequency that matches the transmission coil located inside of the brown mat. Power is transmitted over a distance of 13 inches (33 cm).

To remove energy from the secondary coil, different methods can be used, the AC can be used directly or rectified and a regulator circuit can be used to generate DC voltage.