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The following simulation sequence describes and explains the use of a negative inductance for a simple filter example.

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The depicted schematic illustrates a measurement setup with a capacitor.

In addition to the intentional element of the capacitor – the capacitance (C1) – the parasitic elements of the capacitor are included within the circuit – the parasitic resistance of the capacitor (RC1) – and the parasitic inductance of the capacitor (LC1). These parasitic components mostly are termed as ESR and ESL inside the data sheet. The chosen values are typical for a foil-capacitor in the range between 100 and 200 V – and are somewhat adjusted to get “convenient” figures.

The circuit is according to common test setups in a 50 Ω system.

V1 together with R1 approximates the output of a signal generator, R2 represents the input of a power meter or an analyzer.

The open-circuit voltage (2 V) is such, as to give 1 V across a load of 50 Ω .

This is also the reference of the simulator for 0 dB, meaning neither attenuation nor amplification of the input signal compared to the voltage across R2.

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x/dB = 20 • log ( U(R2) / 1 V )

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Thus the simulation provides the same values as a real measurement setup, without further conversion.

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The circuit shows the typical frequency response:

– for low frequencies the reactance of the capacitor dominates

– at the resonant frequency the ESR can be read – here RC1 with 0.01 Ω

20log(0.01Ω/50Ω) + 6dB = -68dB (+6dB due to Uo = 2V)

– above the resonance frequency the parasitic series inductance dominates and

deteriorates the effect of the filter capacitor with increasing frequency

at 100 MHz the attenuation is only about 12 dB

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Inserting a negative inductance of equal value in series with the parasitic inductance of the capacitor (simulations in the frequency domain allow to specify negative values), results in the mutual cancellation of both inductances – and we get this ideal response – without dropping off towards high frequencies. The attenuation remains constant at a value that results from the voltage division between R1 and RC1.

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Unfortunately, in reality we cannot fall back to negative components just like that. Active circuits for generating negative components are known, however, such circuits are less suitable for high frequencies and high currents and voltages .

Inductors are natural ingredients of filter assemblies, be it as explicit filter elements, or as implicit components in the form of lines. With magnetic coupling of inductors, comes a number of complex effects, we want to utilize here.

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The following table presents the relationship between two coupled coils, a transformer and the T-equivalent circuit of the transformer.

Looking at the conversion formulas between the coupled coils and the T-equivalent circuit in detail, we find L1 and L2 are larger than Lprim and Lsek and – ** L3 becomes negative ! ** (for positive values of k)

**In reverse, this means that at the connection point of the real coupled coils (or the center tap of the real transformer respectively) the negative inductance appears as real effect !**

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The coupling coefficient k indicates what percentage of the generated magnetic field is flowing through both coils in common.

k = 1 means 100% of the field of Lprim goes through Lsek (and vice versa) – both coils are perfectly coupled – a leakage field does not exist – hence, there is no leakage inductance.

When k = 0 no field portion of a coil goes to the other – the inductors are completely decoupled.

Negative k stands for an inverted sense of winding or cross connections with one of the coils.

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To generate an appropriate negative inductance, it is not required to have the coupled coils symmetrically divided, also the coupling coefficient will not be 1 in reality. For example, can the relatively small negative inductance be set by using a tap approaching the end of a larger coil .

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The simulation examples on the other hand, shall depict the background as graphic as possible.

In case of symmetrical coils, there is

Lprim = Lsek = Lcoupled

and the conversion formulas simplify to

L1 = L2 = 2•Lcoupled and L3 = -Lcoupled = -Lprim = -Lsek for k = 1

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Accordingly, should the circuit just below have the required negative inductance of -10nH

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The result validates our assumption – we have virtually the same response as before. Only at very high frequencies a slight effect can be seen due to the inserted small coils.

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Now the cross-check of the converted T-equivalent circuit. If everything is correct, exactly the same frequency response shall occur. Please notice – the coils in the T-equivalent circuit are **not **coupled, i.e. completely independent of each other!

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et voilà !

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For better understanding, here is again the circuit with the two real coils, however without coupling, i.e. k = 0

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Apart from the very highest frequencies, we have the same response as in the simulation run with capacitor alone.

Without coupling there is not a negative inductance. The small additional inductance in the longitudinal branches has little effect.

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It should be noted in circuits with coupled coils (or transformers) how to handle the “orientation” of the coil relative to the magnetic circuit. The small circle (or dot) marks the end with the same electrical polarity for coupled coils (whether achieved through sense of winding or by interchanging / crossing of lines). There is no magnet circuit or magnetic core drawn into the schematics for the simulation. Without the mark , neither simulation nor simulator , would actually know, which way round the coupled coils were penetrated by the magnetic field – and at what end there would be plus or minus.

In this example, Lsek is magnetically reversed (which corresponds to a common-mode choke) – therefore k = -1

This is also a very vivid demonstration of how easily confusion in coupled coils happens, generally – and in simulations in particular. Although in the simulation, k is positive, k has to be taken negative for the calculation according to formulas in the T-model , because Lsek related to the T-model has been inverted. In the simulation both the magnetic orientation (to the magnetic circuit), as well as the electric orientation (in the electric circuit), as well as the direction of k (sgn (k), positive/negative) relative to the original schematic has to be considered.

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Thus the inductors in the longitudinal branch cancel each other – and the attenuation decreases with increasing frequency up to 0 dB.

The effective inductance at the connection point (L3) is positive, due to k = -1 and is adding to the parasitic inductance of the capacitor LC1. So the resonant frequency is somewhat reduced.

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In the simulation, you can also simply set a negative k-value, in order to produce a magnetically inversed coupled coil (which is not possible in reality – you will have to either change the sense of winding or cross the connections)

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Simulations with LTspice IV – a free Spice III simulator, schematic capture and waveform viewer (Linear Technology)

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