Resonance resistance

The resonance resistance is the impedance of an electrical circuit at a resonance frequency . Depending on the circuitry of the components, there can also be several resonance frequencies. A distinction is made between series resonance with particularly low and parallel resonance with particularly high resonance resistance .

Explanation

Electronic circuits always contain coils and capacitors as frequency-dependent reactances so that signals of different frequencies can be separated. Their values ​​are given as complex numbers in the context of the complex AC calculation , with the imaginary components of coils and capacitors having opposite signs . If these imaginary components compensate each other at certain frequencies, there is resonance.

Particularly simple conditions arise when the circuit is an oscillating circuit with only two components:

• In a parallel resonant circuit, the voltages on both components are the same. As the frequency increases, the current through the capacitor increases, while the current through the coil decreases. At a very specific frequency they have the same value but opposite direction. Then the currents compensate each other and no total current flows in the supply lines to the resonant circuit. The circuit isolates at this frequency, which corresponds to an infinitely large resonance resistance. Since the phase shift of real components is always less than 90 °, perfect compensation cannot be achieved with them, which is why the resonance resistance cannot exceed a maximum value of many thousands of ohms.

• In a series resonant circuit, the currents are the same and can assume any value. If you use ideal, i.e. loss-free components, the voltages on the coil and capacitor can compensate each other at a very specific frequency, the resonance frequency. Then the total voltage is zero, which corresponds to a frequency-selective short circuit. Perfect compensation cannot be achieved with real components, which is why the resonance resistance “only” reaches a minimum of a few ohms.

Sometimes it is assumed (incorrectly) that when the total voltage disappears, both partial voltages must also be very low. In fact, the opposite is true: the resonance converters in scanners and notebooks use this resonance increase to generate around 700 V alternating voltage from just 12 V at the connections of the coil and capacitor for operating the fluorescent tubes .

The assumption that no current flows at all in the parallel resonant circuit is just as wrong. In the case of resonance, a considerably higher current can flow through the coil than in the supply line to the resonant circuit. The quotient of both values ​​- the quality factor - can reach values ​​of around 100 at low frequencies and a low-loss coil, and even values ​​over 1000 for VHF frequencies and a silver-plated cup circle .

Basics

General

Equivalent circuit diagram for complex resistances / conductance values: 1. Impedance Z → 2. Admittance Y

As already mentioned above, when alternating current flows through a conductor , a coil or a capacitor, in addition to the ohmic resistance R, there is also a reactance X, the size of which depends on the frequency and the course of the alternating current. In order to be able to calculate this, one uses the complex alternating current calculation, in which one uses the equivalent impedance Z for the resistance R and the admittance Y for the conductance G. The impedance or admittance includes both the real / ohmic component and the reactive component of the resistance or conductance. As with real conductance and resistance, the impedance and admittance can be converted into one another at any time. Assuming ideal components, the real component (R or G) for the coil and the capacitor is zero and for the conductor / resistance the reactive component (X or B) is zero. For the sake of simplicity, this is also assumed for most calculations. However, this is not possible with the resonance resistance, since it is precisely these components that determine the resonance resistance.

The resonance resistance is therefore dependent on the ohmic components and the course of the current. The course specifically influences the resonance frequency and the reactance or reactance value and is therefore an elementary part of every calculation. Since the sinusoidal curve is of the greatest importance in electrical engineering, this curve will be examined in more detail below.

In order to avoid confusion, it should be clear beforehand that we always consider the impedance of the entire resonant circuit when calculating the resonance resistance. The consequence of this is that the real part R (effective resistance) of the entire resonant circuit can also be frequency-dependent, especially with parallel impedances. This can never happen when considering the impedances of individual components, since the real part there always and exclusively consists of ohmic components.

Mathematically

In general, the resonance resistance corresponds to the impedance at resonance . The impedance corresponds to the amount of the impedance. Since the reactance becomes zero in the case of resonance, the resonance resistance is the real part of the impedance. ${\displaystyle X}$ ${\displaystyle Z_{r}}$

Impedance:

${\displaystyle {\underline {Z}}=R+jX}$

${\displaystyle {\underline {Z}}=\operatorname {Re} \{{\underline {Z}}\}+j\operatorname {Im} \{{\underline {Z}}\}}$

Impedance:

${\displaystyle Z=|{\underline {Z}}|={\sqrt {R^{2}+X^{2}}}}$

Resonance resistance: (Is the impedance at resonance)

Resonance condition:

${\displaystyle \operatorname {Im} \{{\underline {Z}}\}=X=0}$so ( reactance of the resonant circuit is zero)

It follows

${\displaystyle Z_{r}=|R+j0|={\sqrt {R^{2}+0^{2}}}}$,

so

${\displaystyle Z_{r}=R=\operatorname {Re} \{{\underline {Z}}\}}$.

It can be seen that the impedance of the oscillating circuit at resonance, i.e. the resonance resistance , is only represented by the effective resistance . In the later calculation, however, you will see that the effective resistance of the parallel resonant circuit, unlike that of components, is frequency-dependent and thus the resonance frequency plays a decisive role in the calculation.

In the following means:

${\displaystyle X}$= Reactance

${\displaystyle R}$= Effective resistance

${\displaystyle j}$= imaginary unit

${\displaystyle Z_{C}}$= Impedance of the capacitor

${\displaystyle Z_{L}}$= Impedance of the coil

${\displaystyle X_{C}}$ = capacitive reactance

${\displaystyle X_{L}}$ = inductive reactance

${\displaystyle R_{Ohm}}$= ohmic resistance

${\displaystyle R_{C}}$ = capacitive loss resistance

${\displaystyle R_{L}}$ = inductive loss resistance

The loss resistance includes the contact resistance and line losses. A series-connected ohmic series resistor has an additive effect on the loss resistance, i.e. ${\displaystyle R_{L_{\rm {ges}}}=R_{\rm {Verlust}}+R_{\rm {Ohm}}}$

Series resonant circuit

Loss-free series resonant circuit

Resonance resistance of a series resonant circuit

The series resonant circuit (also called suction circuit ) consists of a coil and a capacitor that are connected in series. The resonance resistance would be zero for ideal components, but the loss resistances or an ohmic resistance increase the resonance resistance.

General formulas

Impedance

The impedance of the series resonant circuit results as follows:

Derivation:

${\displaystyle {\underline {Z}}={\underline {Z_{C}}}+{\underline {Z_{L}}}=R_{C}+jX_{C}+R_{L}+jX_{L}}$

Formula:

${\displaystyle {\underline {Z}}=(R_{C}+R_{L})+j(X_{L}+X_{C})}$

Resonance frequency

The following condition must be met for the resonance frequency of the series resonant circuit:

${\displaystyle \operatorname {Im} \{{\underline {Z}}\}=0}$

It results:

${\displaystyle \,X_{L}+X_{C}=0}$

Resonance resistance

The following then applies to the resonance resistance:

${\displaystyle Z_{r}=\operatorname {Re} \{{\underline {Z}}\}=R_{C}+R_{L}}$

For sinusoidal gradients

Resonance frequency

By inserting the formulas for the reactance one arrives at:

${\displaystyle \omega _{r}L-{\frac {1}{\omega _{r}C}}=0}$

${\displaystyle \omega _{r}=\omega _{0}={\frac {1}{\sqrt {LC}}}}$( Thomson's oscillation equation )

Resonance resistance

The general formula applies:

${\displaystyle Z_{r}=\operatorname {Re} \{{\underline {Z}}\}=R_{C}+R_{L}}$

annotation

It can be seen that the resonance resistance represents the lowest resistance of a series resonant circuit. The resonance resistance of a series resonant circuit is independent of the course of the alternating voltage. The use of an ohmic series resistor is essential for coils and capacitors with high quality, since the loss resistances are then so low in relation to the internal resistance of the voltage source that they practically represent a short circuit in the case of resonance.

Parallel resonant circuit

Lossless parallel resonant circuit

Resonance resistance of a parallel resonant circuit

The parallel resonant circuit (also called blocking circuit ) consists of a coil and a capacitor that are connected in parallel. With ideal components, the resonance resistance would be infinitely high, but the loss resistances reduce the value of the resonance resistance significantly.

General formulas

Impedance

The impedance of the parallel resonant circuit results as follows:

${\displaystyle {\frac {1}{\underline {Z}}}={\frac {1}{\underline {Z_{C}}}}+{\frac {1}{\underline {Z_{L}}}}}$

${\displaystyle {\underline {Z}}={\frac {{\underline {Z_{C}}}\cdot {\underline {Z_{L}}}}{{\underline {Z_{C}}}+{\underline {Z_{L}}}}}={\frac {(R_{C}+jX_{C})\cdot (R_{L}+jX_{L})}{(R_{C}+jX_{C})+(R_{L}+jX_{L})}}={\frac {(R_{C}R_{L}-X_{C}X_{L})+j(R_{C}X_{L}+R_{L}X_{C})}{(R_{C}+R_{L})+j(X_{L}+X_{C})}}}$

${\displaystyle {\underline {Z}}={\frac {(R_{C}^{2}R_{L}+R_{C}R_{L}^{2}+R_{C}X_{L}^{2}+R_{L}X_{C}^{2})+j(R_{C}^{2}X_{L}+R_{L}^{2}X_{C}+X_{C}X_{L}(X_{L}+X_{C}))}{(R_{C}+R_{L})^{2}+(X_{L}+X_{C})^{2}}}}$

Since the ratio is too often very small, it is usually neglected for the sake of simplicity, so that a simplified formula for the impedance results: ${\displaystyle R_{C}}$ ${\displaystyle R_{L}}$

${\displaystyle {\underline {Z}}={\frac {(R_{L}X_{C}^{2})+j(R_{L}^{2}X_{C}+X_{C}X_{L}(X_{L}+X_{C}))}{R_{L}^{2}+(X_{L}+X_{C})^{2}}}}$

Resonance frequency

At the resonance frequency of the parallel resonant circuit, the following condition must again be met:

${\displaystyle \operatorname {Im} \{{\underline {Z}}\}={\frac {R_{C}^{2}X_{L}+R_{L}^{2}X_{C}+X_{C}X_{L}(X_{L}+X_{C})}{(R_{C}+R_{L})^{2}+(X_{L}+X_{C})^{2}}}=0}$

This is the case when the numerator is zero and the denominator is different from zero: it follows:

${\displaystyle R_{C}^{2}X_{L}+R_{L}^{2}X_{C}+X_{C}X_{L}(X_{L}+X_{C})=0}$ and ${\displaystyle (R_{C}+R_{L})^{2}+(X_{L}+X_{C})^{2}\not =0}$

Resonance resistance

The following then applies to the resonance resistance:

${\displaystyle \operatorname {Re} \{{\underline {Z}}\}=Z_{r}={\frac {R_{C}^{2}R_{L}+R_{C}R_{L}^{2}+R_{C}X_{L}^{2}+R_{L}X_{C}^{2}}{(R_{C}+R_{L})^{2}+(X_{L}+X_{C})^{2}}}}$

For sinusoidal gradients

Resonance frequency

(For formulas see reactance )

${\displaystyle R_{C}^{2}\cdot (\omega _{r}L)+R_{L}^{2}\cdot \left(-{\frac {1}{\omega _{r}C}}\right)-{\frac {L}{C}}(\omega _{r}L-{\frac {1}{\omega _{r}C}})=\omega _{r}L\cdot \left(R_{C}^{2}-{\frac {L}{C}}\right)-{\frac {1}{\omega _{r}C}}\cdot \left(R_{L}^{2}-{\frac {L}{C}}\right)=0}$

The formula for the resonance angular frequency of a parallel oscillating circuit is thus:

${\displaystyle \omega _{r}={\frac {1}{\sqrt {LC}}}\cdot {\sqrt {\frac {\left(R_{L}^{2}-{\frac {L}{C}}\right)}{\left(R_{C}^{2}-{\frac {L}{C}}\right)}}}}$

If is, Thomson's oscillation equation applies . ${\displaystyle \,R_{C}=R_{L}}$

If is, the losses of the components slightly influence the resonance frequency, the resonance frequency is attenuated. ${\displaystyle R_{C}\not =R_{L}}$

If and is accepted, the equation simplifies to: ${\displaystyle \,R_{C}=0}$ ${\displaystyle R_{L}\not =0}$

${\displaystyle \omega _{r}={\sqrt {{\frac {1}{LC}}-\left({\frac {R_{L}}{L}}\right)^{2}}}}$

In the case that and is, i.e. assuming only ideal components, this equation does not apply, since in this case the denominator of the fraction would also be zero, the resonance frequency is not reached in this case the resonant circuit blocks completely beforehand. ${\displaystyle \,R_{C}=0}$ ${\displaystyle \,R_{L}=0}$

Resonance resistance

${\displaystyle Z_{r}={\frac {R_{C}^{2}R_{L}+R_{C}R_{L}^{2}+R_{C}\cdot \omega _{r}^{2}L^{2}+R_{L}\cdot {\frac {1}{\omega _{r}^{2}C^{2}}}}{(R_{C}+R_{L})^{2}+\left(\omega _{r}L-{\frac {1}{\omega _{r}C}}\right)^{2}}}}$

As a simplification one can now assume that ( Thomson's oscillation equation ) and thus: ${\displaystyle \omega _{r}\approx \omega _{0}}$

${\displaystyle \,X_{L}+X_{C}=0}$ and ${\displaystyle X_{L}^{2}=X_{C}^{2}={\frac {L}{C}}}$

This follows the formula:

${\displaystyle Z_{r}={\frac {R_{C}^{2}R_{L}+R_{C}R_{L}^{2}+{\frac {L}{C}}(R_{C}+R_{L})}{(R_{C}+R_{L})^{2}}}}$

Since it is usually much smaller than , you can set for further simplification , so that only the following remains: ${\displaystyle R_{C}}$ ${\displaystyle R_{L}}$ ${\displaystyle R_{C}=0}$

${\displaystyle Z_{r}={\frac {{\frac {L}{C}}(R_{L})}{(R_{L})^{2}}}={\frac {L}{R_{L}\cdot C}}}$

This formula clearly shows that the loss resistance of the coil reduces the resonance resistance. If it were zero, the resonance resistance would be infinitely large. ${\displaystyle R_{L}}$

Note to R _L

The low resistance connected in series to the coil can also be described as high resistance connected in parallel . ${\displaystyle R_{L}}$ ${\displaystyle R_{p}}$

${\displaystyle R_{p}={\frac {1}{G_{L}}}={\frac {R_{L}^{2}+X_{L}^{2}}{R_{L}}}}$   

The following then applies to sinusoidal alternating voltages:

${\displaystyle R_{p}={\frac {R_{L}^{2}+\omega ^{2}L^{2}}{R_{L}}}}$   

In the case of resonance ( ) is: ${\displaystyle \,R_{C}=0}$

${\displaystyle \omega =\omega _{r}={\sqrt {{\frac {1}{LC}}-\left({\frac {R_{L}}{L}}\right)^{2}}}}$

which was derived a little further above. Since the actual resonance circuit from C and at resonance has an infinite resistance, the impedance is alone . After insertion, the resistance in the case of resonance is: ${\displaystyle L_{p}}$ ${\displaystyle Z_{r}}$ ${\displaystyle R_{p}}$

${\displaystyle R_{p}={\frac {R_{L}^{2}+\left[{\frac {1}{LC}}-\left({\frac {R_{L}}{L}}\right)^{2}\right]L^{2}}{R_{L}}}}$   

${\displaystyle Z_{r}=R_{p}={\frac {L}{R_{L}\cdot C}}}$

The expression derived above with the prerequisite therefore also applies to without this restriction. ${\displaystyle \omega _{r}\approx \omega _{0}}$ ${\displaystyle R_{C}=0}$

Note on R _C

In the case of capacitors, the loss is often given as the loss conductance instead of the loss resistance . In this case, it is not the impedance that is used for illustration , but the admittance , i.e. the conductance loss parallel to the susceptance . ${\displaystyle G_{C}}$ ${\displaystyle R_{C}}$

Derivation

${\displaystyle {\underline {Y_{C}}}={\frac {1}{\underline {Z_{C}}}}={\frac {1}{R_{C}+jX_{C}}}}$

Forming leads to:

${\displaystyle G_{C}+jB_{C}={\frac {R_{C}}{R_{C}^{2}+X_{C}^{2}}}-j{\frac {X_{C}}{R_{C}^{2}+X_{C}^{2}}}}$

formula

${\displaystyle G_{C}={\frac {R_{C}}{R_{C}^{2}+X_{C}^{2}}}}$ or. ${\displaystyle R_{C}={\frac {G_{C}}{G_{C}^{2}+B_{C}^{2}}}}$

Other resonance resistances

All of the above formulas only apply to wavelengths that are much larger than the dimensions of the components. If the wavelength falls below a few centimeters, coils and capacitors can either no longer be implemented or they show additional properties that drastically modify the electrical behavior. In addition, components are used that contain neither coils nor capacitors and still show frequency-dependent impedances with resonance effects.

Quartz crystal

The reactance of a quartz oscillator changes very strongly between f _r and f _a

If the impedance of the standard quartz crystal component is measured over a large frequency range, a peculiar law results: closely spaced pairs of series resonance f _r and parallel resonance f _a can be measured at regular intervals , the frequencies of which differ by less than one percent. The exact values ​​are defined by the dimensions of the crystal and can hardly be changed. For example, if f _r1  = 5 MHz and f _a1  = 5.003 MHz forms the first pair, the second follows at 15 MHz and the third at 25 MHz.

Resonant lines

Suction circle effect of a copper strip on an insulator (gray)

Circuits in the range of radar frequencies are often built as strip lines , whereby the resonance resistances of particular line lengths are used. If high-frequency energy is passed from left to right in the adjacent figure, the λ / 4 strip shows series resonance in a narrow frequency range and acts like a suction circuit .

literature

• Edwin Wagner, Heinz-Ulrich Seidel: General electrical engineering: alternating current technology - compensation processes - lines . 3. Edition. Carl Hanser Verlag, 2005, ISBN 978-3-446-40018-4 .