Resonance transformer

Coil L and capacitor C form a resonance transformer

A resonance transformer , also known as a Boucherot circuit , is a circuit similar to an oscillating circuit consisting of a capacitor and a coil in order to achieve power matching between components or assemblies at a given frequency . For low-frequency applications, ferrite cores can also be used in the coil to increase the inductance. In the case of high-frequency applications , however, the iron core of the coil is generally not required , as its physical properties distort the transformed AC voltage and limit the power.

General

Like a transformer, a resonance transformer can transform both voltage and current, but (like an autotransformer ) it has no galvanic isolation and moreover only works in a narrow frequency band. It is therefore only used if the frequency does not change significantly.

Its leakage inductance is often increased by the application, for example if a large distance is given (e.g. air for mobile energy transmission) or is necessary (e.g. because of the insulation). Resonant operated transformers are a way of transferring energy with high efficiency. See also resonance converter.

A particular advantage of the resonance transformer is the low-pass effect , which reduces the harmonic content of the transmitted signal.

calculation

To determine the values ​​of the coil L and capacitor C for the purpose of power matching, the impedances of the resonance transformer on both sides must correspond to the values ​​of the two external resistors R ₁ and R ₂ .

Power adjustment means that in the above circuit, with exemplary resistance values ​​for R ₁ and R ₂ , either

• a source (left) with the internal resistance R ₂  = 30 Ω should  deliver as much power as possible to the consumer R ₁ = 140 Ω or
• a source (right) with the internal resistance R ₁  = 140 Ω should  deliver as much power as possible to the consumer R ₂ = 30 Ω.

In both cases, the value of the consumer appears to be increased or decreased by a certain factor with respect to the value on the other side of the resonance transformer framed in red in the figure.

The dimensioning of the resonance transformer can be done either graphically with a Smith diagram or, as in the following, mathematically within the framework of the complex AC calculation. The kirchhoff rules and the laws for series connection and parallel connection apply . The inductive resistance Z _{L of} the coil L with the angular frequency ω  = 2 · π · f is given by

${\displaystyle Z_{L}=\mathrm {j} \omega L}$

and for the capacitive resistance Z _{C of} the capacitor C.

${\displaystyle Z_{C}={\frac {1}{\mathrm {j} \omega C}}=-{\frac {\mathrm {j} }{\omega C}}}$

The following generally applies to the equivalent resistance Z _{total of} two resistors connected in parallel

${\displaystyle Z_{\mathrm {gesamt} }={\frac {Z_{1}\cdot Z_{2}}{Z_{1}+Z_{2}}}}$

If you apply this formula to the parallel connection of R ₁ and C , the result is

${\displaystyle Z_{R_{1}\|C}=Z_{a}={\frac {R_{1}\cdot {\frac {-\mathrm {j} }{\omega C}}}{R_{1}+{\frac {-\mathrm {j} }{\omega C}}}}={\frac {R_{1}-\mathrm {j} \omega CR_{1}^{2}}{(\omega CR_{1})^{2}+1}}=R_{1}{\frac {1-\mathrm {j} Q}{Q^{2}+1}}}$

with the auxiliary quantity Q  =  R ₁ ωC , the quality factor . For the equivalent resistance of a series connection , the individual resistances must be added; in this case it results

${\displaystyle Z_{Z_{a}+Z_{L}}=Z_{a}+\mathrm {j} \omega L}$

The following applies to power adjustment for this circuit (shown above)

${\displaystyle \,R_{2}=Z_{a}+\mathrm {j} \omega L}$

Given known values ​​of ω, R ₁ and R _2, this complex equation breaks down into two real determining equations for L and C , since the imaginary part of the right-hand side of the equation must be zero. The solutions are:

${\displaystyle Q={\sqrt {{\frac {R_{1}}{R_{2}}}-1}};\qquad C={\frac {Q}{R_{1}\omega }};\qquad L={\frac {R_{1}Q}{\omega (Q^{2}+1)}}={\frac {QR_{2}}{\omega }}}$

Example: The power adjustment in the picture is to be calculated for the frequency 100 MHz. So Q  = 1.915; C  = 22 pF and L  = 91 nH.

Example: Adaptation of a dipole antenna

Dipole with matching circuit for coaxial cable

The input resistance of a dipole antenna strongly depends on the location of the feed. If you split the dipole in the middle and connect a symmetrical cable there, you have to set its impedance to about 74 Ω in order to achieve power matching . If the dipole is not interrupted in the middle (as in the adjacent picture), the power can be fed in asymmetrically at one end. With thin wire antennas, an impedance of around 2200 Ω is measured at this point. As a rule, the radio station is connected to the antenna via an unbalanced coaxial cable with an impedance of 75 Ω or 50 Ω, which is why a low-loss transformer must be interposed to avoid a strong mismatch . Since a dipole antenna only has a relatively small bandwidth of a few percent of the center frequency, a narrow-band resonance transformer is very well suited for resistance adjustment.

The following values ​​result for a frequency of 3.6 MHz and a 50 Ω cable:

${\displaystyle {\begin{aligned}Q&={\sqrt {{\frac {2200}{50}}-1}}=6{,}56\\C&={\frac {6{,}56}{2200\,\Omega \cdot 2\pi \cdot 3{,}6\,\mathrm {MHz} }}=132\,\mathrm {pF} \\L&={\frac {6{,}56\cdot 50\,\Omega }{2\pi \cdot 3{,}6\,\mathrm {MHz} }}=14{,}5\,\mu \mathrm {H} \end{aligned}}}$

This circuit has a number of advantages over the otherwise common feed at the "current belly" in the middle of the dipole:

• The resonance frequency of the antenna can be shifted by about 10% from the calculated values by a slight deviation from C or L without the standing wave ratio assuming inadmissibly large values. This corresponds to an increased bandwidth of the antenna.
• The resonance transformer is easily accessible at the end of the antenna.
• With long wire antennas, there is no heavy coaxial cable with a balun hanging in the center of the dipole .

It can be seen as a disadvantage that with the final feed at the higher-impedance input the feed voltage is higher (in the example by a factor of 6.56). At P = 100 W, the peak voltage then increases from 100 V to 656 V. This must be taken into account when dimensioning components.

Pi filter

Pi filter for resistance transformation

In high-frequency technology, power transistors and electron tubes are preferably operated as switches (C mode) in order to avoid unnecessary heat loss. According to the laws of Fourier analysis , abrupt switching on and off of a voltage creates many harmonics that are emitted and can interfere with the function of other devices. To prevent this, low-pass filters , resonant circuits or resonance transformers of a sufficiently high quality Q must be installed. A rule of thumb states that from Q  ≥ 8, the harmonics of the alternating voltage are sufficiently suppressed.

In the simple resonance transformers just described, Q depends exclusively on the ratio of the resistances at the input and output. If the resistances have approximately the same value, Q is too low to ensure an appreciable filter effect. This can be changed by combining two resonance transformers. The circuit is reminiscent of the Greek letter π , which is why the name pi-filter became established . Sometimes the term Collins filter is also used because it became known for its good properties in radio equipment from the Rockwell Collins company of the same name .

The components are calculated in two stages: Resistance R ₂ is transformed by C ₂ and L ₂ to a very low intermediate  value R ₃ ≈ 1 Ω, which can be imagined from the connection of the two red rectangles. However, R ₃ is not present as a component, but this fictitious intermediate value is stepped up to the desired resistance R ₁ by C ₁ and L ₁ . Since both resonance transformers have high quality factors Q , the desired filter effect is achieved.

A change in the calculated number of turns is necessary if L ₁ and L _{2 are} usually combined into a single coil - both coils with the number of turns n are then magnetically coupled and the total number of turns has a value kn where 2> k> for resonance, depending on the coil shape 2 ^0.5 , resulting in a total inductance L ₁  +  L ₂ .

Applications

The resonance transformer is used in different areas. Some application areas are listed below as examples.

• In radio equipment and high frequency technology, resonance transformers are used, which can also serve as band filters :
  • Adaptation of the wave impedance of a cable to an antenna ( impedance transformation ).
  • Matching between two successive amplifier stages ( transistors ).
  • In both of the above cases, equivalent circuits can also be used.
• To transfer electrical power:
  • For operating cold cathode tubes in flat screens or in electronic ballasts for compact fluorescent lamps and energy saving lamps to generate the necessary operating voltage for the tubes. It is typical that these resonance converters automatically provide the required high ignition voltage when the cold cathode tube is not yet ignited due to the high output impedance. In this case, the resonance transformer is formed with a real transformer for operation on low voltage and for galvanic isolation .
  • In television receivers with cathode ray tubes , the line transformer required to feed the line deflection coils works as a resonance converter during line return and then generates the anode voltage of the picture tube.
  • In quasi-resonant switched-mode power supplies, the switching transitions take place as a resonant half-wave; this can drastically reduce switching losses .
  • The Tesla transformer generates voltages well over 100 kV through resonance exaggeration in an air-core coil due to its natural resonance.
  • Mobile devices with wireless charging are equipped with a resonant secondary winding and are brought closer to a transmitter coil (primary coil) for charging or communication ( wireless charging , wireless energy transfer or passive RFID )

literature

• Heinz M. Hiersig (Hrsg.): VDI-Lexikon Energietechnik . Springer-Verlag Berlin-Heidelberg GmbH, Berlin 1994, ISBN 3-642-95749-8 .
• Hans Rein, K. Wirtz (Hrsg.): Textbook of wireless telegraphy . Springer Verlag, Berlin 1917.
• Alfred Fraenckel: Theory of alternating currents. 3rd edition, Springer Verlag, Berlin 1930.
• Johann Siegl: Circuit technology - analog and mixed analog-digital . 4th edition, Springer Verlag, Berlin / Heidelberg 2010, ISBN 978-3-642-13303-9 .
• Richard Marenbach, Dieter Nelles, Christian Tuttas: Electrical energy technology . Basics, energy supply, drives and power electronics, Springer Fachmedien, Wiesbaden 2013, ISBN 978-3-8348-1740-2 .
• Fritz Schröter, N. von Korshenewsky, WT Runge (Hrsg.): Textbook of wireless communications technology . Television technology part two, Springer Verlag, Berlin 1963.
• Otto Zinke and Heinrich Brunswig: Textbook of high frequency technology . Volume 1, High Frequency Filters - Lines - Antennas, 4th edition, Springer Verlag, Berlin / Heidelberg 1990, ISBN 978-3-540-51421-3 .

See also

• Line theory # λ / 4 line

Individual evidence

1. ^ Resonance converter by Jörg Rehrmann: The power supply and converter manual

Web links

• Basics of the resonance transformation (accessed on November 3, 2017)
• Resonance transformation (accessed November 3, 2017)