**L.2. Equivalent Circuits and Simulation Models**

Modern measuring equipment, such as the HP4195A impedance analyzer and similar instruments, allow computer-aided derivation of equivalent circuits and their optimization. Constant parameters for inductance L, capacitance C and resistance R are required for the simulation of electronic circuits. An incorrect approach in defining an equivalent circuit leads to flawed or incorrect simulation results.

Equivalent circuit parameters are in no way legally binding, exactly observable characteristics, but show the user the approximate positions of the main tolerances. The fact that this data was previously unavailable has much to do with test engineering and marketing traditions, which should be replaced by efforts towards user-friendliness in the future.

Electronic components, resistors, capacitors, inductors, conductors, approximations to ferrite materials and insulation materials can be adequately characterized with AC parameters. The condition is that their properties do not alter appreciably with different high voltage and current levels or that they are operated in the small signal range. One then speaks of linear components as opposed to non-linear components such as varistors, diodes, transistors etc.

As a consequence of spatial size in the construction of components, partial capacitances exist between each point on the surface of a conductor and every other. Conductors generate magnetic fields through current flow and have a partial inductance by virtue of the energy stored in these magnetic fields. The finite conductivity of copper, silver etc. or the conductivity of layered resistors

generated locally lead to partial resistances. Parallel conductors may also be seen as sections of high-frequency lines at high frequencies, possessing a wave resistance, signal traveling time and perhaps signal attenuation.

Reality is generally extremely complicated. If one wished to plot or even calculate a microscopically fine circuit with all partial resistances, capacitances or inductances, one would be destined for failure. Fortunately for the user, accurate equivalent circuits may easily be found for capacitors and inductors without iron or ferrite cores. Approximations of other inductive components may also be well described, whose averaged parameters are much better than unknown characteristics. Real measurement values as curves are compared to the equivalent circuit models to demonstrate the derivation of these values to the user. This particularly applies to EMC ferrites, with which interference is suppressed, but no precise circuitry is formed.

**2.1 The most important types of equivalent circuits**

**Capacitors โ series resonance**

*Fig. 1.39: Series resonance circuit and impedance against frequency*

The ideal capacitor C_{s} is influenced by the lead inductance L_{s} (of the order of several nH) and the track resistance R_{s} (typ. of the order of 20 mฮฉ โฆ 100 mฮฉ, for cold electrolyte capacitors up to 1 ฮฉ). At low frequencies the capacitive component predominates, at the self-resonant frequency

the track resistance is measurable. Above the series resonance, the inductive component predominates, which may to some extent be influenced by having short connecting lengths. The phase curve changes from โ โ90ยฐ to โ +90ยฐ in the region of resonance. The phase point 0ยฐ therefore precisely defines the point of resonant frequency, often much more precisely than is possible by means of amplitude measurement. The phase curve also serves the computer-aided determination of the equivalent circuit (1st approximation).

**Inductance โ parallel resonance**

*Fig. 1.40: Parallel circuit and impedance against frequency*

The ideal inductance L_{p} is influenced by the unavoidable winding capacitance C_{p} and the assimilated R_{p} losses (core material, winding losses). An infinitely large parallel resistance R_{p} is desirable for inductors, for EMC ferrites on the other hand, a very low and broadband resistance R_{p} is the desired parameter. The inductance L_{p} plays a lesser role and is the key to transforming the resistive loss into the line.

This means that in the model, in the place of the EMC ferrite, the line is cut open and the equivalent circuit is interposed. A component is only exactly described by all three parameters L, C and R. A single parameter is insufficient, although other properties may be predicted for known configurations.

Imagine a three-dimensional coordinate system with the axes R_{p}, L_{p}, C_{p}; the component is a point in this space and its tolerance field is a cuboid body. Various components are located in this tolerance space as different points.

*Fig. 1.41: Tolerance space*

This mathematical treatment of series and parallel resonance may be summarized as follows:

Unified (normalized) resonance curves for series and parallel resonant circuits as a function of the normalized imbalance ฮฉ in all the values of L, C and R are included. Standardized detuning unifies the parameters of quality factor and resonant frequency f_{RES}. This makes it easier to describe the behaviour of resonant systems in their frequency range

(ฮf โฆ frequency deviation from the resonant frequency)

Series resonant circuits:

*Fig. 1.42: 1 + jฮฉ (magnitude and phase) as a function of ฮฉ*

Parallel resonant circuits:

*Fig. 1.43: 1 / 1 + jฮฉ (magnitude and phase) as a function of ฮฉ*

**Series resonance**

**Parallel resonance**

Resonant condition:

for series resonance:

for parallel resonance:

Normalized off-resonance:

for series resonance:

for parallel resonance:

The detuning v denotes how far the resonant circuit is detuned compared with the imposed oscillation. In short: Measure of the deviation of the (circuit) frequency from the resonant (circuit) frequency.

The normalized curves are plotted in Figures 1.42 and 1.43 for R = 1 (top magnitude, bottom phase).

Selecting the suitable operands (normalization), it is possible to illustrate all characteristics as two standard curves. They serve to provide a basic understanding, whereas in applications, the measured resonance curves are evaluated or in computer simulations, the numerical values of L, R and C.

**Derivation of simple equivalent circuits**

In the following examples, measurement curves derived with the HP 4195 are analyzed. It is however in principle irrelevant with which measuring device or configuration the curves are derived.

The analysis is based on fundamental theoretical modeling concepts.

*Fig. 1.44: Impedance curve of a coil with 6 winding turns on a ceramic former โ
6 mm*

Figure 1.44 shows an impedance of a small air coil of 6 mm diameter and 6 winding turns. A simple estimate of inductance from the values at 10 MHz and 100 MHz gave an L of 63.7 nH. As the phase ฮ is between 85ยฐโ90ยฐ, one may assume a pure inductance and obtains from |Z| = 4 ฮฉ at 10 MHz and |Z| = 40 ฮฉ at 100 MHz.

The analyzer measures L = 58 nH as a result of its more accurately stored values than those displayed. Two curves are just distinguishable in Figure 1.44, a thick measurement curve and one, which the analyzer plotted as the result of its equivalent circuit measurement. It actually includes an equivalent circuit calculator. It calculates the equivalent circuit plotted in Figure 1.44 and writes the calculated values of |Z| and ฮ alongside the measurement curve. Without a computer, one would have to be content with just a rough calculation.

Multi-dimensional measures are required in EMC matters; precision is not of paramount importance.

*Fig. 1.45: Impedance curve of a 10 nF ceramic capacitor with approx. 5 mm long connecting legs*

Figure 1.45 shows the impedance curve of a ceramic disc capacitor of 10 nF capacitance rating with approx. 5 mm long connecting legs. By reading the measurement curves and calculation, C = 11.37 nF and L = 9.55 nH could be found. From the |Z| curve at ฮ = โ90ยฐ, the capacitance

and at ฮ = + 90ยฐ the inductive component

can be calculated.

|Z|_{200 kHz} = 70 ฮฉ, |Z|_{4 MHz} โ 3.5 ฮฉ, C = 11.37 nF, |Z|_{40 MHz} โ 2 ฮฉ,

|Z|_{200 MHz} = 12 ฮฉ and L = 9.55 nH.

The series resistance was 33.8 mฮฉ according to the measurement. With the correctly selected equivalent circuit (Figure 1.45), the analyzer measured R = 33.7 mฮฉ, C = 11.2 nF and L = 9 nH and the its coinciding curve showed that there were no further parasitic equivalent circuit parameters of any significance. Whereas the coil in Figure 1.45 is still useful at 200 MHz, the capacitor

is only applicable up to 30 MHz.

One recognizes from this, how terribly important it is to consider line inductances in association with blocking processes. An example in Figure 1.46 illustrates this.

*Fig. 1.46: The effect of a triangular pulse, as arises during the switching edge in logic circuits, on a capacitor with line inductances*

If a triangular pulse reaches a โnakedโ capacitor without inductance, a voltage reduction of 25 mV occurs in 10 ns with rounded falling edge. The line inductance of 9 nH alone adds a bipolar pulse with steep edge of ยฑ90 mV, an effect that everyone who has looked at their power supply line with an oscilloscope has observed. It is of little use to substitute 10 nF with 1 ฮผF; if the inductance remains, the noise voltage remains. The pure change in charge in a 10 nF capacitor due to the triangular pulse leads to a change in voltage ฮU at the capacitor.

The current edges ฮI/ฮt at just 9 nH generate a bipolar voltage shift of

*Fig. 1.47: Impedance curve and equivalent circuit of a resonant circuit*

As an example, Figure 1.47 shows a resonance circuit comprising coil and capacitor in parallel with a 1 kฮฉ resistor. The analyzer with the structure A selected wins the equivalent circuit hands down. Other structures give rise to poor or worthless approximations. The analyzer primarily attempts to approach the phase curve. One can manually improve the parameters and arrive at new approximations.

The inductance may be determined from the measurement curve in this case. Capacitance may only be found to a limited extent. |Z|_{3 MHz} = 4 ฮฉ, |Z|_{57.6 MHz} = 969 ฮฉ, |Z|_{200 MHz}= 24 ฮฉ, L_{p} = 207 nH, R_{p} = 989 ฮฉ, C_{p} = 36.4 pF.

The resonance condition then helps:

The analyzer results confirm the calculated values.

*Fig. 1.48: Equivalent circuit types from HP4195A*

*Fig. 1.49: Impedance curves for the parallel connection of two dissimilar capacitors (Legend of effective HF blocking)*

Figure 1.49 shows the impedance curve for the parallel connection of 2 dissimilar capacitors. The larger 150 nF capacitor was supplemented with a small 150 pF capacitor to improve HF blocking. It should โbalanceโ the disadvantage of the capacitorโs longer line to the load. The result is a resonant peak at 61.57 MHz with a Z of 590 ฮฉ (with no load attached). Every steep pulse edge would definitely cause a long damped oscillation and the significantly reduce the usefulness of blocking.

Derivation of the equivalent circuit from the |Z| curves takes place in several steps. The phase curve Q signalizes a capacitance of up to 2 MHz (zone I). The series resonance at 2 MHz is a consequence of the approx. 4 cm long connection legs of the 150 pF capacitor. The circuit is inductive in zone II to generate a parallel resonance with 590 ฮฉ at 60 MHz, where one can no longer

speak of blocking. From 30 MHZโ100 MHz (zone III) the impedance is greater than 100 ฮฉ and is extremely detrimental for internal signal coupling. The 150 pF capacitor starts to play a role beyond the parallel resonance and affects a series resonance at 154 MHz. The phase curve Q is observed in zone IV.

Calculated values are

in zone I |Z|_{600 kHz} = 2 ฮฉย โฉ 133 nF,

in zone II |Z|_{20 MHz} = 6.5 ฮฉย โฉ 52 nH,

in zone III resonance 61.6 MHz C_{b} = 1/ฯ^{2} ยท L = 128.6 pF,

in zone IV resonance 154.6 MHz L โ 1/ฯ^{2} ยท C_{b} = 8.25 nH

The equivalent circuit is obtained

*Fig. 1.50: Equivalent circuit of two parallel connected capacitors*

**ABC of CLR: Chapter L Inductors**

**Equivalent Circuits and Simulation Models – Circuit Types**

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