# Correlation of measured and simulated inductances in laminated bus bars

Abdelilah Goulmane, Simon Dario, Thomas Fouet

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Abdelilah Goulmane, Simon Dario, Thomas Fouet. Correlation of measured and simulated inductances in laminated bus bars. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0035. hal-05075810

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# Correlation of measured and simulated inductances in laminated bus bars

Abdelilah GOULMANE¹, Simon DARIO², Thomas FOUET¹

¹MERSEN Pontarlier, 6 rue Claude Chappe, 25300 PONTARLIER, France

²MERSEN Angers, 2 rue du Déry, 49480 VERRIERES-EN-ANJOU, France

Email: abdelilah.goulmane@mersen.com, simon.dario@mersen.com, thomas.fouet@mersen.com

URL: https://www.mersen.com/en

## Acknowledgements

This work is part of the AdvanSiC project. This project has received funding from the European Union's Horizon Europe program under grant agreement No 101075709. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.

## Keywords

Bus bar, Power Modules, Converter, Inductance, Simulation, Measurement

## Abstract

Bus bars are key in the development and improvement of electrical systems that use power converters that are an essential part in smart grid systems. Bus bars, and especially laminated ones, can help reduce inductance and minimize switching losses, thereby enhancing the overall performance of converters. This highlights the critical need to accurately determine the inductance of bus bars, and then optimize their design accordingly. This publication focuses on different way to measure inductance of laminated bus bars and the correlation with simulation. The presented results demonstrate the sensitivity of inductance measurements, primarily influenced by the measurement setup. Well controlled, it is possible to obtain a strong correlation with numerical models, allowing fast design improvements before prototyping.

## I - Introduction

### A - Smart grids and power conversion

The traditional electrical grid, has long served as the backbone of our society, delivering reliable power from centralized plants to consumers.

However, the limitations of this conventional grid are becoming increasingly apparent. The growing demand for electricity, the integration of renewable energy sources, and the need for better performance has accelerated the development of a more advanced system: the smart grid.

A smart grid is an enhanced electrical grid that employs advanced technologies to improve the efficiency, reliability, and sustainability of electricity services, coming from various sources, including wind or solar plants. Medium voltage Direct Current (MVDC) or High voltage DC (HVDC) transmission could be part of this improvement, and in this case, converting stations are needed to manage variety of current profiles [1].

Power converters play a very important role in the field of energy conversion. Regarding solar applications, inverters are used to convert the energy of a direct current source (DC) (generated by photovoltaic panels) into alternating current (AC) energy, which can be fed into the network for use.

These inverters are generally composed of power modules (integrated semiconductor switches), capacitors and a bus bar, interconnecting these components. The latter plays a very important role as a low inductance filter by minimizing the switching losses that increase with the switching frequency of the active components and therefore affect the efficiency of power converters [2].

The work presented here is part of AdvanSiC project, where objective is to develop and characterize cost-effective inverters for MVDC grid applications, including wind and solar inverters for DC converter stations. These inverters are introducing high-voltage SiC MOSFET semiconductors with nominal voltage up to 2500 V. Since operating voltages are increasing, reduce inductance becomes critical to improve inverter efficiency, and it is essential to
have a high confidence in numerical models and inductance measures.

This work focuses the measurement of inductance using different methods, and the correlation with numeral models obtained by Finite element method (FEM).

# B – Importance of bus bars

## What is a bus bar?

Bus bars, made of conductive material (typically copper or aluminum), are used to distribute electrical power within a switchboard, panel, or electrical system. It serves as a central point for connecting multiple circuits and provides a common connection point for current to flow through [3].

<figure><img src="image_2.png" /><figcaption>Fig. 1. Example of traditional copper bus bars</figcaption></figure>

### 1) Laminated bus bar

Different bus bars interconnexion technologies exist. One of these technologies is the laminated bus bar. Used for several decades in various industrial and railway application, as an alternative to cables, generating high inductance [2]. The principle consists of layering insulation and conductors, thanks to a lamination process, leading to improved electrical properties [4][5]. The laminated bus bars primary advantage is its ability to handle higher current densities than traditional cable systems, resulting in a more efficient and compact design. Picture of a typical laminated bus bar for power conversion is presented in Figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Laminated bus bar for power conversion</figcaption></figure>

The design of laminated bus bars helps reduce size and weight while preserving key physical properties, particularly electrical and thermal characteristics. In addition, conductive layers are face-to-face with an important overlapping, very flat, and separated by very thin insulant layers (from 0.1mm to 2 or 3mm), favorable to reduce inductance.

### 2) Inductance in bus bars

Electromagnetic induction is created when a flow of current I through a circuit generates a magnetic flux Φ, which is defined by two parameters according to the following equation:

$$ \Phi = LxI $$

with:

L: Inductance in H

I: Current passing through the circuit at A

Φ: Magnetic flux in (H/A)

The inductance, or stray inductance, is a characteristic of the conductor, or the circuit crossed by the electric current.

Each conductor or bus bar has its own inductance which positively or negatively influences the commutation loop.

The total inductance of a bus bar is composed of:

- Self-inductance, specific to a single conductor and depending on the geometry. This inductance is represented in Figure 3.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Self-inductance of 1 conductor</figcaption></figure>

- The mutual inductance, resulting of the influence between at least 2 closed conductors (Figure 4).
<figure><img src="image_5.png" /><figcaption>Fig. 4 Mutual inductance between 2 conductors</figcaption></figure>

Therefore, the total inductance of a laminated bus bar is calculated by the following equation:

$$L_{\text{Busbar}} = \sum L_{\text{P}} + 2 * \sum L_{\text{M}}$$

with:

* $\sum L_{\text{P}}$: Sum of all self-inductances (positive)

* $\sum L_{\text{M}}$: Sum of all mutual inductances (negative)

During the design of inverters, the objective is to reduce the value of the inductance to some nH. Since stray inductance is a challenge for electrical devices using semiconductors components (especially SiC), inverters are sensitive to overvoltage induced by the commutation current loop (CCL), presented in Figure 5 [6].

<figure><img src="image_6.png" /><figcaption>Fig. 5. Typical electrical diagram of a converter</figcaption></figure>

The switching devices operate at various frequencies inducing voltage peaks that are increased proportionally to the total stray series inductance of the CCL, following the equation:

$$V_{\text{peak}} = L_{\text{CCL}} \times \frac{di}{dt}$$

with:

* $V_{\text{peak}}$: Surge voltage, in volt

* $L_{\text{CCL}}$: Sum of inductances in CCL, in H

* $\frac{di}{dt}$: Current variation in function of time, in A.s⁻¹

To prevent surge voltage exceeding breakdown voltage, the stray inductance should be minimized. The inductance of each component should be strongly optimized and reduced, to avoid breakdown of the converter devices [9].

However, the shape and design of the bus bars, as well as the assembly of the conductors that compose it, are essential in the design processus. Calculate the inductance of a bus bar is possible through numerical simulation tools, and very instrumental to validate a design and optimize it easily. In any case this calculation should be validated by measuring experimentally the inductance.

### 3) Simulation and measurement

In this work, we propose for different bus bars, to compare simulation and experimental results, with the objective to correlate both. For both measure and simulation, different methods are proposed in the literature.

Calculation of inductance could be performed by numerical simulation, allowing estimation of the inductance applying the Finite Element Method (FEM), or Partial Element Equivalent Circuit (PEEC) to numerically solve partial differential equations [7].

About measurement, two ways of measurement are possible.

The direct measurement method, based on the use of high precision instruments and strong setup of the measurement circuit to directly measure the inductance. Using an impedance-meter (or RLC Bridge), this method consists of shunting the bus bar and measure the inductance of the concerned connection using measuring clamps. Also, another direct measurement method is the inductance voltage integral method [8].

The indirect method using the Impedance Resonance method. This method uses the impedance resonance stray inductances in the bus bar by creating a resonant circuit with a capacitor and resistor. The resonant frequencies at different stages are measured. The inductance are derived by analysing the impedance at these frequencies. This method is well adapted to validate the design of a bus bar, as well as the simulated value.

Therefore, the presented work focuses on the correlation between the simulated and measured inductance values on bus bars. First part is dedicated to the inductance simulation of different bus bars: firstly, a single-conductor bus bar, and secondly a laminated bus bar with 2 conductors.
In a second part, experimental results associated to these bus bars will be presented, and discussed, with the objective to propose a correlation between simulation and measurement.

## II. Bus bars inductance simulation

The simulations were carried out using two different software:

- ANSYS for the inductance simulation of a bus bar composed of a single conductor.

- COMSOL Multiphysics for the simulation of the bus bar composed of 2 conductors.

Each software uses the finite element method (FEM). The principle of these tools is based on the use of Maxwell's differential equations which makes it possible to calculate the inductance of the bus bar. The FEM method requires only a 3D model of the bus bar, allowing fast iterative calculations.

The simulation informs about the magnetic field on a specific frequency range. This is particularly useful since the inductance depends on the frequency according to the following relationship:

$$L = \frac{X}{(2\pi f_c)}$$

With:

- L: the inductance in H

- $f_c$: the resonant frequency in Hz

- X: the norm of imaginary part of the complex impedance: $Z = R + iX$

Therefore, the inductance simulation allows to validate or optimize the design of the bus bar, considering the operating conditions of the inverter and more particularly the slope and the switching frequency as shown in the Figure 5, which represents an example of transition from the low voltage state to the high voltage state due to the MOSFET switching.

<figure><img src="image_7.png" /><figcaption>Fig. 6 MOSFET's switching transient – VDS and ID waveforms: turn-off [10]</figcaption></figure>

## A – Inductance simulation of a single-conductor bus bar

For this study, sample is a rectangular copper bar of 50mm² section. Representation of the bar is presented in Figure 7, and dimensioning information in Table 1.

<figure><img src="image_8.png" /><figcaption>Fig. 7 – Single-conductor bus bar</figcaption></figure>

<table><caption>Table 1 - Bus bar dimensions:</caption><thead><tr><td>Length (mm)</td><td>Width (mm)</td><td>Thickness (mm)</td></tr></thead><tbody><tr><td>200</td><td>25</td><td>2</td></tr></tbody></table>

Using Ansys software, it is possible to configure the current's path by setting the input and output conditions of the latter, as shown in the Figure 8, which facilitates the simulation of the inductance between two positions of the bus bar.

<figure><img src="image_9.png" /><figcaption>Figure 8 – Configuration of current's path on Ansys</figcaption></figure>

Simulation was performed on a frequency range from 1 Hz to 100 MHz, leading to inductance evolution as a function of frequency (Figure 9).

Simulation results show inductance values for DC and AC:

- DC inductance (red curve on Figure 8), where inductance is constant at **94.14 nH**.

- AC inductance (green curve on Figure 8), where values decrease between **94.14 nH** and **82.5 nH** respectively for frequencies of **1 Hz** and **100 MHz**.

Evolution of inductance as a function of frequency is explained by the current distribution into the bus bar (including skin effect). With DC or at very low frequencies, current distribution is
almost homogeneous into the bus bar section. When the frequency increases, current will be more localized at the surface of the conductor (skin and proximity effects), leading to an evolution of inductance.

<figure><img src="image_10.png" /><figcaption>Fig. 9 - Inductance simulation results on Ansys</figcaption></figure>

## B - Inductance simulation of a 2-conductors laminated bus bar

Simulation of a two-conductors laminated bus bar is also possible. A representative bus bar with 2 copper conductors is used for this study (Figure 10). Polyester insulation is used, allowing a very good overlapping between conductive layers (around 0.25mm insulation)

<figure><img src="image_11.png" /><figcaption>Fig. 10 - Bus bar with 2 layers conductors</figcaption></figure>

Using COMSOL Multiphysics, it is possible to simulate the CCL. All capacitors connections were shunted, replaced by short circuits (Figure 11) to allow current to flow through the bus bar. Even for the power module like the IGBT, is replaced by a short circuit and on which the power supply will be applied as well as the inductance measurement. The inductance simulations were performed considering only one module connection without short circuiting other module connections.

<figure><img src="image_12.png" /><figcaption>Fig. 11 - Bus bar's inductance simulation circuit</figcaption></figure>

The short circuits used to shunt the capacitor connections are magnetically isolated, so they do not influence the inductance simulation circuit.

Simulation was performed on a frequency range from 1 Hz to 100 MHz, leading to inductance evolution as a function of frequency (Figure 12).

<figure><img src="image_13.png" /><figcaption>Fig. 12 - Inductance simulation results on COMSOL Multiphysics</figcaption></figure>

The simulation results show inductance values between **12.51 nH** and **9.06 nH** respectively for frequencies of **1 kHz** and **1 MHz**.

## III. Bus bars inductance measurement

With a view to compare simulation and measures, next step was to measure the inductance of the bus bars previously simulated. For inductance measurement, 2 measurement equipment were used, as well as 2 different measurement method.

First, a dedicated inductance test bench was used, based on the impulse response method. Principle is to send a calibrated voltage pulse into the circuit loop concerned by the measurement. The measured response is a voltage and current curve, thanks to a current sensor and a voltage sensor.

The inductance is calculated using the following equation:
$$ \Delta V = L \times \frac{\Delta I}{\Delta t} $$

With:

ΔV: Voltage average in a time interval Δt in V

L: Inductance in H

Δt: Time interval in s

ΔI: Current difference in a time interval Δt in A

This measurement method is also known as the **ΔV method**, which is part of the category of direct inductance measurements.

Figure 13 below describes the inductance measurement circuit using the test bench.

<figure><img src="image_14.png" /><figcaption>Fig. 13 inductance measurement test bench circuit</figcaption></figure>

From Figure 13, it can be seen that in order to construct the inductance measurement circuit, the measured element must be shunted to close the measurement circuit.

This measuring equipment measures the total parasitic inductance of the busbar, including the self and mutual parts.

Another method is to use an impedance meter (or RLC Bridge). This equipment is used to precisely measure the impedance of components such as resistance, capacitance and inductance in different range of frequencies. For tests

<figure><img src="image_15.png" /><figcaption>Fig. 14 Impedance meter RLC Bridge</figcaption></figure>

## A - Inductance measurement of single-conductor bus bar

The bus bar concerned by the measurement has the same characteristics mentioned in the previous paragraph.

The figure below describes the inductance measurement circuit of the bus bar using the test bench.

<figure><img src="image_16.png" /><figcaption>Fig. 15 Bus bar inductance measurement circuit</figcaption></figure>

The measurement results are visualized using an oscilloscope, recording voltage and current during the test (Figure 16).

<figure><img src="image_17.png" /><figcaption>Fig. 16 Voltage and current curves during inductance measurement test</figcaption></figure>

According to the curves above, time interval, voltage and current values are:

- Δt= 5.84 µs

- ΔV= 12 V

- ΔI= 720 A

So, the value of the inductance is: L = 97.3 nH.

## B - Inductance measurement of laminated bus bar composed of 2 conductors

The bus bar concerned by the measurement has the same characteristics mentioned in the simulation paragraph, (Figure 17).
<figure><img src="image_18.png" /><figcaption>Fig. 17 - Laminated bus bar with 2 layers conductors</figcaption></figure>

Inductance is measured according to the two methods previously presented.

**ΔV method :**

Measurement circuit is the same the simulation one. The Figure 18 below represents the measurement circuit 1. The capacitor connections are short-circuited with a copper wire (soldered) with the same dimensions for each one, very similar to the simulated ones.

The power supply is applied on the same module connection which its inductance has been simulated.

<figure><img src="image_19.png" /><figcaption>Fig. 18 - Bus bar inductance measurement circuit 1 using the test bench.</figcaption></figure>

The total inductance measured is L = 10.6 nH.

This value is the addition of the inductance of the bus bar and inductance of the capacitor's shunts. Unlike the simulation, these shunts are not magnetically isolated and therefore they influence the final measurement of the inductance.

To obtain only inductance of the bus bar, it is necessary to subtract the inductance value of the shunts. The measure of the shunt's inductance is done using an RLC bridge.

The Figure 19 shows the measurement of 1 shunt:

<figure><img src="image_20.png" /><figcaption>Fig. 19 - Shunt inductance measurement circuit using RLC Bridge</figcaption></figure>

The measured value, at the frequency 164 kHz is: **42.3 nH**. Since 54 shunts are connected in parallel, the total inductance of the shunts is: **L<sub>Shunts</sub> = 42.3/54=0.78 nH**.

The inductance value of the bus bar is L = 10.6 - 0.78 = 9.82 nH

**RLC Bridge method:**

An impedance meter (RLC Meter) RLC E4980A from Keysight is used. This equipment takes in consideration the inductances created by all of the components connected to the bus bar, including capacitors.

First, the measurement was done using the same bus bar with shunts (Figure 20).

<figure><img src="image_21.png" /><figcaption>Fig. 20 - (Bus bar + capacitors shunt) inductance measurement circuit using RLC Bridge</figcaption></figure>

The inductance measurement of the whole assembly (bus bar and capacitors shunt) is done at f = 164 kHz.

The measured inductance value is L= 21.95 nH. Compared to the previous value measured with ΔV method, it presents almost the double. Main hypothesis to explain that is that for ΔV method, bus bar is powered before sending the electrical pulse. The capacitors shunt, since they are connected in parallel with opposite current and
are close to each other, combined mutual inductances, reducing the total inductance. With RLC Bridge, shunts are not crossed by the current, so the equipment takes in consideration the inductance of each shunt, and this will increase the total inductance.

An alternative to measure the bus bar's inductance was to replace shunts by real capacitors. 27 capacitors (11 µF each) were welded to the bus bar. To carry out the measurement correctly, a dedicated junction fixture was used (Setup presented in Figure 21).

<figure><img src="image_22.png" /><figcaption>Fig. 21 - Bus bar inductance measurement circuit using impedance meter RLC Bridge</figcaption></figure>

The whole assembly (bus bar + capacitors) will be measured at f = 164 kHz. The measured inductance value is L = 10.2 nH.

To determine the inductance of the bus bar, it is necessary to subtract the inductance value of the capacitors. As the same way than for shunts before, the measure of the capacitor inductance was done using the RLC bridge, but at f = 1MHz, because the equipment measures the inductance value of the capacitor at this frequency. The Figure 22 shows the inductance measurement of 1 capacitor.

<figure><img src="image_23.png" /><figcaption>Fig. 22 - Capacitor inductance measurement circuit using RLC Bridge</figcaption></figure>

The measured value of the capacitor inductance is: 14.95 nH. Considering 27 capacitors connected in parallel, the total inductance of the capacitors is L<sub>Capacitors</sub> = 14.95/27 = 0.55 nH.

The inductance value of the bus bar is L = 10.2 - 0.55 = 9.65 nH.

## IV. Correlation between measured and simulated inductances

It is now possible to compare simulated and measured values of each tested bus bar.

### A - Single-conductor bus bar

Positioning of the voltage measuring lugs strongly affects the measurement results. It is necessary to consider lugs positioning as a key parameter to follow during measurement and keep them straight to avoid them adding inductance to the measurement setup. To illustrate this comment, in addition to the measure of the paragraph III A), 5 measures of the same circuit were done, changing only the position of the sensor voltage clips (Figure 23).

<figure><img src="image_24.png" /><figcaption>Fig. 23 - Bus bar inductance measurement circuit: (a), (b), (c), (d), (e)</figcaption></figure>

The Table 2 summarizes the inductance values for each position.

<table><caption>Table 2 - Inductance values for each case:</caption><thead><tr><th>Frequency (kHz)</th><th>Case</th><th>Measured inductance (nH)</th></tr></thead><tbody><tr><td rowspan="6">171</td><td>(0)</td><td>97.3</td></tr><tr><td>(a)</td><td>145</td></tr><tr><td>(b)</td><td>144</td></tr><tr><td>(c)</td><td>112</td></tr><tr><td>(d)</td><td>58.4</td></tr><tr><td>(e)</td><td>108.5</td></tr><tr><td colspan="2">Average:</td><td>110.9</td></tr></tbody></table>

The Table 3 summarizes the obtained values in measurement and simulation.
<table><caption>Table. 3 - Simulated and measured inductance values of the bus bar composed of 1 conductor</caption><thead><tr><td>Frequency (kHz)</td><td>Simulated inductance (nH)</td><td>Measured inductance (nH)</td></tr></thead><tbody><tr><td>171</td><td>83.5</td><td>110.9 +/- 32.3</td></tr></tbody></table>

According to these results, simulated and measured values are of the same order of magnitude, with a difference of 25%. But measures values are very variable. Explanation comes down to the setup used to measure the inductance, since the simulation environment is considered as ideal compared to the measurement setup. This setup has its own inductance, such as the cables to power the bus bar. Also, the positioning of the bus bar and the measuring terminals affect the inductance value. More inductance of the circuit is high, more setup influence is important.

## B - 2-conductors bus bar

The Table 4 summarizes the obtained values in measurement and simulation for the 2-conductors bus bar.

<table><caption>Table. 4 - Simulated and measured inductance values of the laminated bus bar composed of 2 conductors</caption><thead><tr><td>f (kHz)</td><td>Simulated inductance (nH)</td><td>Measured inductance. ΔV Method (nH)</td><td>Measured inductance. RLC Bridge (nH)</td></tr></thead><tbody><tr><td>164</td><td>9.25</td><td>9.82</td><td>9.65</td></tr></tbody></table>

According to these results, values for 2-conductors laminated bus bar are close, with a difference of only 6% between the measured and simulated values. The difference is not significant and comes from the measurement setup.

## IV. Conclusion

The determination of inductance values was approached both theoretically and experimentally. Theoretical calculations were conducted using simulation software such as Ansys and COMSOL Multiphysics, while experimental measurements were performed using dedicated measurement equipments and different methods.

When inductance is high, the correlation is not fully effective, mainly due to very sensitive measurement setup. But obtained results are of the same order of magnitude.

Since laminated bus bars are a very efficient way to reduce inductance, it was important to correlate simulated and measured inductance. Results are very close to simulated values, with a deviation lower than 6%.

For both single conductor and laminated bus bars, deviation between simulation and measurement could be reduce by mastering of the setup.

To improve the measures, future works will focus the mastering of the measurement setup, considering impact of sensors positioning.

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