A Simplified Method of Calculating Busbar Inductance and Its Application for

Stray Resonance Analysis in an Inverter dc Link

KATSUTAKA TSUBOIMitsubishi Electric Corporation, Nagasaki, Japan

MINEO TSUJI and EIJI YAMADANagasaki University

SUMMARY

Sinusoidal PWM control is widely applied to power

electronic converters. The carrier frequency of recent con-

verter products is gradually being raised in order to decrease

current harmonics as much as possible. However, in the dc

filter circuits of large capacity converters, the capacitance

of a unit capacitor is relatively large, and the busbars are

relatively long. Therefore, the resonant frequency is lower

than those of small capacity converters. Consequently, there

is a concern that the carrier frequency will become close to

the resonant frequency, which will cause trouble. On the

other hand, it is often necessary to decrease the inductance

of the loop circuit consisting of a switching device and its

by-pass circuit, in order to suppress the switching surge

voltage.

In both cases described above, it is important to be

sure of the exact busbar inductance for reliable design. In

this paper, a simplified method of calculating busbar induc-

tance is proposed. This method is based on the partial

inductance theory. In particular, the characteristics of a pair

of busbars, and the use of the method are described. Finally,

an application to stray resonance analysis in the dc filter

circuit of a three level inverter is described in comparison

with experimental data. From these results, we confirmed

that the proposed method is usable to estimate busbar

inductance quickly and accurately. © 1999 Scripta Tech-

nica, Electr Eng Jpn, 126(3): 49�63, 1999

Key words: Busbar; inductance; PWM inverter;

stray resonance.

1. Introduction

Recently, high speed semiconductor switching de-

vices have been frequently applied in power electronics

apparatus, even for large capacity converters. Therefore it

is important to obtain the inductance of busbars, used as

connecting conductors in large capacity converters, in or-

der, for example, to minimize the loop inductance of com-

mutation circuits, snubber circuits, and so on so as to

decrease the switching surge voltage or to increase the

turn-off capability of GTOs, and to calculate stray resonant

frequency so as to avoid hazardous resonance with the

switching frequencies of semiconductor devices in power

electronics converters.

The latter stray resonance phenomena present an

especially important issue to be solved [1, 2] as PWM

applications to large capacity converters grow. In the dc

filter circuits of larger capacity converters, the capacitance

of a unit capacitor increases and the stray inductance also

increases because of relatively long busbars. Thus, the

resonant frequency decreases. On the other hand, the PWM

switching frequency tends to be increased in order to de-

crease harmonic components in the input or output current.

Even if the switching frequency of a single device is not

high, the equivalent frequency may be high as a result of

multi-level or multi-phase topologies. Furthermore, in large

capacity converters, since the voltage tends to be high, oil

capacitors are frequently used rather than electrolytic ca-

pacitors and these must be connected in multiple series for

high voltage applications. Therefore the dumping factor of

the circuit is reduced. For these reasons, the potential for

coincidence between the switching frequency and resonant

frequency increases and the system has a large resonance

factor. If the stray resonant frequency of the circuit matches

the switching frequency or any of its harmonics, unex-

pected phenomena such as overheating of main circuit

components due to irregular large oscillating currents is the

CCC0424-7760/99/030049-15

© 1999 Scripta Technica

Electrical Engineering in Japan, Vol. 126, No. 3, 1999Translated from Denki Gakkai Ronbunshi, Vol. 117-D, No. 11, November 1997, pp. 1364�1374

49

result. However, in most previous papers, the main goals

have been the explanation of resonance phenomena them-

selves and the proposal of control methods or circuit topolo-

gies to avoid them. We cannot find any reports that propose

a concrete method of estimating the stray resonant fre-

quency from the aspect of optimum design of busbars so as

to avoid resonant oscillation.

The inductance and resistance of busbars at high

frequency can be calculated by means of general purpose

electromagnetic field analyzing software tools using the

FEM and other techniques. However, it takes a very long

time for many busbar construction plans to be investigated

by a trial-and-error method. Therefore, a simplified method

of calculating inductance that can be used effectively is

required. If the fact that most busbars are not placed in

contact with or near magnetic materials is taken into ac-

count, simplification of the calculation is possible. Pre-

viously, some simplified calculation methods were

presented [3, 4]; however, one of them is a method that uses

existing dc inductance calculating tools, another seems not

to lead to the expected result. We cannot find reports that

present consistent procedures with reproducible results.

In this paper, we offer a proposal for a new simplified

method of calculating busbar inductance, the calculated

results for some simple models, and applications to the

analysis of stray resonance phenomena. As a preliminary,

the derivation of a formula for calculating the inductance

of a fine wire element, which is the basic component of a

busbar, and an estimate of the approximation error caused

by simplifying the formula, are presented in section 2. The

calculation methods and calculated results for the example

of single busbar and two or three busbars mounted in

parallel are described in section 3. Interactions between two

busbars are selectively studied as basic characteristics that

will be applied in order to synthesize the inductance in the

actual apparatus. Finally, the calculation method and calcu-

lated results for practical busbar systems with complicated

construction are described in section 4. A stray resonance

analysis of an NPC (neutral point clamped) converter is

carried out by the proposed method, and the result is com-

pared with experimental data. It is confirmed that the reso-

nant frequency of the apparatus can be calculated easily and

accurately by using only construction data on the busbars.

2. Fundamentals of Inductance Calculation

2.1 Definition of issues and some assumptions

Normally, the directions of busbars in large capacity

practical apparatuses are limited to three alternatives: (1)

vertical, (2) horizontal and parallel to the face of the cubicle,

and (3) horizontal and parallel to the side of the cubicle.

Since these directions are orthogonal, the inductance be-

tween the two busbars in different directions is very small

even if the finite cross sections of the busbars are taken into

account. It can therefore be neglected and interactions

between busbars need be considered only within the same

directional busbar group. For this reason, the fundamental

inductance calculations may be carried out among busbars

that are parallel to each other.

Before deriving the formula, we assume the follow-

ing.

(i) All the current in a busbar flows parallel with the

center axis in the direction of its length.

(ii) In any cross section perpendicular to the center

axis, all positions are at equal potential.

In case of the connecting point of two busbars with

different widths or in case of two parallel busbars with

extremely different length, assumption (i) cannot be satis-

fied strictly. However, in most practical cases, this assump-

tion will be acceptable. If it is not satisfied under the original

conditions, it will be approximately satisfied by partition-

ing a busbar conceptually into appropriately short lengths.

If assumption (i) is satisfied, no current component in

parallel with the cross section of the busbar can exist.

Consequently assumption (ii) can be derived secondarily.

Based on assumption (i), each busbar can be consid-

ered a set of fine wire elements, each having a square cross

section with small sides as shown in Fig. 1. The self-

inductance and resistance of a fine wire element will be

calculated by assuming that the skin effect can be neglected

and hence that the density of the current can be considered

uniform. Mutual inductance between wire elements will be

calculated by assuming that all currents are concentrated on

the center axis. The inductance and resistance of busbars

under high frequency conditions will be synthesized by

aggregating the characteristics of the fine wire elements.

Fig. 1. Division of busbar into fine wire elements.

50

2.2 Self-inductance and resistance of a fine

wire element

The self-inductance L is defined by

where i is the current density, I the total current in a

conductor, v the total cubic volume of a conductor, A the

vector potential caused by i, m the permeability, and rl is the

distance between the point where A is defined and an

infinitesimal volume element dv.

Since the current density is assumed to be uniform,

if a busbar is placed in a field where the permeability ratio

is 1.0, the self inductance of a fine wire element can be

expressed by means of Eqs. (1) and (2) as

where m0 is the permeability in vacuum, S the area of cross

section of a fine wire element, l the length of a fine wire

element, l1 and l2 are the position of an infinitesimal volume

element dv1 or dv2 in the direction of the length, and q is the

distance between two infinitesimal volume elements dv1

and dv2 in the direction perpendicular to the length. (Refer

to Fig. 2).

Performing integration with respect to l1 and l2, Eq.

(3) can be rewritten as

If l / q >> 1, Eq. (4) can be approximated by

where R is the geometrical mean distance (GMD), which

depends on the shape of the cross section of a fine wire

element, and can be expressed as

For a rectangle with sides a and b, they can be calculated

as

where x1 and x2 are variables that indicate the position of

infinitesimal area elements dS1 and dS2 in the direction of

side a, and y1 and y2 are the corresponding variables for b.

Since the cross section is assumed to be a rectangle

in this case, by substituting b = a into Eq. (7) and further

substituting Eq. (7) into Eq. (5), the self inductance of a fine

wire element can be finally expressed as

where a is the length of a side of the cross section of a fine

wire element.

We will next investigate the error caused by approxi-

mating Eq. (4) by Eq. (5). By assuming that the integrands

in Eqs. (4) and (5) are F1(l / q) and F2(l / q), and calculating

the approximation error of the integrand (F2 - F1) /F1, it is

recognized that the smaller l / q, the larger the error. To

determine the value of l / q, using l / qmax where qmax is the

maximum outer diameter of a fine wire element, equal to

Ö̀`̀2a , we calculated the maximum error (F2 - F1) /F1

within a fine wire element with l / a = 20, 100, 1000, and

obtained �2.9%, �0.36%, �0.023%, respectively. From this

we infer that, even in the worst case, the error of the

inductance of a fine wire element can be kept below this

value. The actual inductance error is smaller because in

most of the integration area q < qmax, the integrand is larger

and the error ratio smaller than under the condition

(1)

(2)

(3)

Fig. 2. Details of fine wire element.

(4)

(5)

(6)

(7)

(8)

51

q = qmax. By numerical integration, it was confirmed that

the actual inductance errors of fine wire elements with l / a

of 20, 100, and 1000 are �0.74%, �0.10%, and �0.007%,

respectively. The cautions which should be observed when

calculating the inductance are described below. The synthe-

sized loop inductance of two busbars with self inductance

L and mutual inductance M, and which are placed parallel

and close to each other is expressed by 2(L - M). Since the

value of M is nearly equal to L, the value of the synthesized

inductance is severely affected by the calculation error of L

and M. Therefore, it is important to decide how finely the

busbar should be subdivided in order to obtain proper

accuracy within a reasonable calculation time, depending

on the purpose of analysis. In ordinary systems such as are

described in this paper, a value of 100 for l / a will be

sufficient. However, under higher frequency conditions, in

addition to assuring the accuracy of the inductance of a fine

wire element, finer subdivision of a busbar is also necessary

in order to satisfy the previous assumption that no skin

effect appears in a fine wire element. In this case, the

recommended thickness a of the fine wire element is equal

to or less than the depth of the skin effect, i.e., Ö̀```````2r / (wm0)(where r is the volume resistivity of busbar material). This

consideration is very important when the high frequency

resistance must be calculated exactly.

The basic resistance of a fine wire element from

which the high frequency resistance of a busbar can be

synthesized is defined by

2.3 Mutual inductance between fine wire

elements

The mutual inductance M between two conductors is

given by

where i1 and i2 are the current densities of conductors 1 and

2, I1 and I2 are the total currents of conductors 1 and 2, v1

and v2 are the total cubic volumes of conductor 1 and 2,

A2 is the vector potential within conductor 1 caused by i2distribution, and rl is the distance between two infinitesimal

volume elements dv1 and dv2.

By substituting Eq. (11) into Eq. (10), we obtain

Assuming that two fine wire elements are placed in parallel

in a space where the relative permeability is 1.0 and that

their thicknesses are negligibly fine, Eq. (12) can be rewrit-

ten as

where l1 and l2 are the lengths of fine wire elements 1 and

2, and rl is the distance between two infinitesimal length

line elements dl1 and dl2 on fine wire elements 1 and 2,

respectively.

Defining the relative position between two fine wire

elements using c, d(³ 0) as shown in Fig. 3, Eq. (13) can be

expressed as

where

Under the condition d = 0, c ¹ 0, l1 = l2 = 1, which fre-

quently appears within the same busbar, Eq. (14) can be

simplified as

Since this formula uses only the interval c between the

center axes of fine wire elements as information on relative

position between them, the calculation can be carried out

easily. In case of large c, it will be acceptable to assume that

the currents are concentrated on the center axes, neglecting

the cross sectional shapes of the fine wire elements, and to

use the above equation. However, in case of small c, some

(11)

(9)

(10)

(12)

(13)

(14)

(15)

Fig. 3. Relative position of two wire elements.

52

estimation is necessary when neglecting the shape of these

cross sections. To obtain M, there is a more precise approach

as shown below. In this method, assuming l1 = l2 and d = 0,

Eq. (12) can be modified similarly to Eq. (3). Consequently,

similarly to Eq. (6), the GMD R¢ between the cross sections

of two fine wire elements can be obtained as

where S1 and S2 are the cross sectional areas of fine wire

elements 1 and 2, and q is the interval between infinitesimal

area elements dS1 and dS2 contained in areas S1 and S2,

respectively.

By using R¢, similar to Eq. (5), the mutual inductance

between fine wire elements can be expressed as

On the other hand, under the condition l / c >> 1, Eq. (15)

can be written as Eq. (17) whose component R¢ is replaced

by c. Therefore it can be understood that c is used instead

of R¢ in Eq. (17) in an approximate method and R¢ is used

in a strict calculation. Although the latter case is more true

to the principle, time-consuming calculation of R¢ is needed

for each arrangement of two fine wire elements, and so the

procedure becomes very complicated. The error of distance

between two wire elements caused by the approximation

(c - R¢) /R¢ was calculated by the method of numerical

integration as shown in Fig. 4. In this figure, one of a pair

elements is fixed at the origin, and the other is placed at

regular positions around the origin element. Although the

error decreases quickly as the interval between the elements

increases, the maximum error of about �0.65% appears on

a pair of adjacent elements. The effect of the error of R¢ on

the error of the mutual inductance (DM /M) / (DR¢ /R¢) can

be expressed as -1 /{log(2l / c) - 1} which can be derived

by using dM/dR¢ obtained by differentiating Eq. (17). In

case of a pair of adjacent elements at l / a = 100, the value

of l / c is 100 and the value of -1 /{log(l / c) - 1} becomes

�0.23. Consequently, even in case of a pair of adjacent

elements, the error of the mutual inductance is �0.23 times

smaller than �0.65% and is sufficiently small in value, at

+0.15%. Therefore, simplified Eqs. (14) and (15) can be

used effectively in most practical applications.

3. Basic Characteristics of Busbar Inductance

3.1 Process of analysis

1) One busbar system

Partitioning a busbar into n pieces of fine wire ele-

ments, the circuit equation within the busbar is given by

where Vi is the voltage across the both ends of each fine

wire element, Ii is the current through each fine wire ele-

ment, w is the angular frequency of Vi and Ii, L is the self

inductance of a fine wire element, Mij is the mutual induc-

tance between i-th and j-th fine wire elements, and r is the

resistance of a fine wire element.

As a simplified expression, Eq. (18) can be expressed

as

The value of M is constant even if l1 and l2 are interchanged

with each other in Eq. (14), and therefore Mij = Mji, and Z

is a symmetric matrix. Setting Y = Z�1, and assuming that

V1 through Vn are each equal to V according to Assumption

(ii) described in section 2.1, the current vector I can be

obtained as

Accordingly, the total current IT of the busbar is

(16)

(17)

Fig. 4. Error of distance between two wire elements.

(18)

(19)

(20)

(21)

53

Consequently the constants of a busbar, namely, self-induc-

tance L and resistance R, can be expressed as follows:

where Im and Re denote the imaginary and real parts of the

variable following in parenthesis.

By using the above equation, calculating V that cor-

responds to a given total current IT, and substituting it into

Eq. (20), the currents of all the fine wire elements can be

obtained. These currents show the current distribution

within a busbar caused by the skin effect.

2) Two-busbar system

The circuit equation of a two-busbar system can be

expressed as

where it is assumed that busbars 1 and 2 are subdivided into

m and n of fine wire elements, respectively. V1 and I1, or

V2 and I2 are column vectors with components m and n that

represent the voltages and currents of all the fine wire

elements within busbar 1 or 2. Z11 and Z22 are m ´ m and

n ´ n symmetric matrices corresponding to Z in Eq. (19) for

busbars 1 and 2, respectively. Z12 and Z21 are m ´ n and

n ´ m matrices, respectively, that represent the mutual im-

pedances between busbars 1 and 2, that is, between the

voltages of the fine wire elements within each busbar and

the currents within the other busbar as shown below:

Depending on the characteristic of each matrix above, the

combined impedance matrix in Eq. (23) and its inverse

matrix also become symmetric. Defining Y as its inverse

matrix, and applying Assumption (ii) described in section

2.1 to busbars 1 and 2, the current of each fine wire element

can be obtained as

where 1 is an m or n dimensional column vector all of whose

components are equal to one, and V1, V2 are voltages across

both ends of busbars 1 and 2.

The total currents of busbars IT1 and IT2 can be ex-

pressed as shown below in the same manner as for the

one-busbar system.

From this equation, V1 and V2 can be solved as follows

where z11, z12, z21, and z22 are components of the inverse

matrix obtained from the matrix in Eq. (27).

z12 is equal to z21, because of the identical two non-

diagonal components in the matrix in Eq. (27), which are

the sum of all the components in each of the two transpose

matrices. Since each z is obtained from a matrix whose

elements are comprised of information related only to

construction and material, its value is not affected by the

busbar current. On the other hand, Eqs. (1), (2), (10), and

(11) mean that the inductance of the conductor that has a

finite volume is affected by the current distribution in the

conductor. However, depending on the results described

above, it can be understood that the circuit constants are not

affected by the �total� currents of the busbars and that the

busbar system maintains linearity regardless of the skin

effect at high frequency if the frequency is constant.

From the given total currents IT1 and IT2, calculating

V1 and V2 by using Eq. (28) and substituting these values

into Eq. (26), the current distribution in busbars 1 and 2 can

be obtained. Furthermore, by using the z in Eq. (28), the

circuit constants of the two-busbar system can be calcu-

lated. An interesting feature in this stage is that z12 and z21

are not pure imaginary numbers, but include a real part. Due

to this real part, it is observed that the effective resistance

of a busbar may differ depending on the method of connect-

ing two parallel busbars, i.e., connected in parallel (currents

of two busbars are in the same direction) or connected in

serial (currents of two busbars are in opposite directions).

In this paper, this real part is called the �mutual resistance.�

L1, L2, R1, R2, M12, M21, R12, and R21, which represent the

self inductance and resistance of busbar 1 and 2 and the

mutual inductance and mutual resistance between busbars

1 and 2, can be expressed as follows:

(25)

(22)

(23)

(24)

(26)

(27)

(28)

54

According to the direction of the current, the effective

resistance of a busbar in the two-busbar system is expressed

by R1 + R12 (same current direction) or R1 - R12 (opposite

current directions). However, even in the latter case, the

effective resistance does not fall below the dc resistance.

Mutual resistance R12 affects only the increase R1 due to the

skin effect. Although the mutual resistance is usually posi-

tive, it also becomes negative in some particular arrange-

ments of busbars.

3) More-than-three busbar system

Even in the case of a more-than-three busbar system,

it is similarly possible to express the circuit equations of the

system faithfully, and to obtain the circuit constants of the

busbars by solving these equations. However, the size of the

matrices becomes larger and the calculation time longer. In

order to solve this problem, the system can be treated as a

combination of pairs of busbars, and the constants can be

obtained from each pair of busbars by using the linearity of

the system. It is not a problem to assume that the total

current of the busbars other than the pair under investigation

is zero; however, in principle, it is not permissible to neglect

the existence of those busbars whose total currents are taken

to be zero. Even if its total current is zero, an eddy current

can flow within the busbar, which also affects the current

distribution within the other busbars and results in changes

of the constants of these busbars. However, the degree of

its influence on the constants varies from case to case, and

it can be neglected in most practical cases. Therefore it is

recommended that the procedure be applied for the two-

busbar system in practical investigations, repeating it many

times. Estimates in some representative similar cases show

that the result given by this procedure is nearly equal to that

given by the more rigorous procedure.

3.2 Results of analysis

For a matrix calculating tool, MATLAB (The Math-

Works, Inc.) 4.2c was used. Usually in practical apparatus,

the cross sections of busbars in a group running in an

identical direction are arranged systematically; namely, all

the sides of the cross sections are placed parallel or perpen-

dicular to each other. By using this attribute, programming

the calculation can be accomplished simply. The size of the

program is approximately 250�300 lines. It takes a PC with

a 133 MHz CPU and 32 MB of RAM about 40 seconds per

case to calculate the constants for a system that consists of

400 fine wire elements. If the cross section has symmetry

with respect to a horizontal line and/or a vertical line, the

program can be refined and the size of the matrix can be

reduced. Therefore, the scale of the applicable busbar sys-

tem can be enlarged or the time for the calculation can be

shortened.

Examples of two busbar systems are shown in Fig. 5.

The calculated constants and current distributions of these

busbar systems are shown in Figs. 6 and 7 under various

conditions of the total current in each busbar. The vertical

axis shows the ratio of the actual RMS current to the RMS

current allocated uniformly from total current for each fine

wire element. The mean square of the above values over the

cross section of a busbar is equal to the ratio of the ac

resistance of a busbar with skin effect to the dc resistance

of the same busbar. In Fig. 6(b), in which two identical total

currents flow in the same direction, the skin effect appears

markedly compared with Fig. 6(a), in which busbar 1 is

open-circuited and the total current flows only through

busbar 2. On the other hand, in Fig. 6(c), in which two

identical total currents flow in opposite directions, the skin

effect is reduced compared with Fig. 6(a). From this, it is

obvious that the ratio of the resistance with skin effect to

the dc resistance, i.e., the effective resistance of the busbar,

is larger in the former case (same current direction), so that

the effect of �mutual resistance� previously described can

be visually understood.

By investigating the circuit constants of various kinds

of busbar pairs, it is recognized in most cases that the ac

inductance is not changed much even at higher frequencies

compared with the resistance. The reason is that the busbar

inductance is changed almost solely by the inner induc-

tance, which is rather small compared with the outer induc-

tance if the current is concentrated symmetrically at the

ends of a busbar�s cross section by the skin effect. However,

sometimes the change of the inductance due to the fre-

quency becomes large in certain some particular arrange-

ments of busbars. Concerning arrangements (a) and (b) in

Fig. 5, the changes of the inductances with frequency are

shown in Fig. 8. In this figure, the change of the inductance

in arrangement (b) is larger than that in (a). The reason also

can be understood by observing the current distribution at

(29)

Fig. 5. Models of two busbar systems (cross section).

55

higher frequencies. In Fig. 7, which shows the current

distribution in arrangement (b), even when the total current

of another busbar is zero, the current distribution of a busbar

is not completely symmetric. From this it can be expected

that the influence of the frequency on the self inductance is

relatively large. When the total currents flow through each

busbar, the directions of these currents greatly affect the

position of current concentration spots. From this it can also

be expected that the influence of the frequency on the

mutual inductance is relatively large.

In case of a busbar construction with less influence

of the frequency on the inductance, and when only rough

values are required, instead of the ac inductance, the dc

inductance which has been given as a complete analytical

equation [5] may be used. However, taking into account the

fact that the change in the inductance due to frequency

cannot be easily forecast and that the equation for dc

inductance is more complicated than that in this paper, our

proposed method can be used conveniently not only in the

case when accuracy is needed but also in many other

practical cases.

An example of a system with three parallel busbars

at even intervals is shown in Fig. 9. Concerning this system,

two inductances, in each case obtained from faithful calcu-

lation among these three busbars and from approximate

calculation based on the pair of busbars, are shown in Fig.

10 as a function of the interval. In the system with thicker

busbars at narrower intervals, the results of approximate

calculation are a little different from the real values ob-

tained from faithful calculation. However, the difference is

negligibly small in such practical systems as are described

below because of their normal thickness and proper inter-

vals.

Fig. 6. Current distribution of two busbars arranged as Fig. 5(a).

Fig. 7. Current distribution of two busbars arranged as Fig. 5(b).

56

4. Stray Resonance in dc Link

An experimental filter circuit was made as a model

of a dc link for a converter. Some parts of this filter consist

of many busbars with various widths and directions, even

though each part is drawn simply as a straight line in the

circuit diagram. Accordingly, at first, the general procedure

to obtain a single synthesized constant from mixed busbars

connected in series will be described. Second, using the

above procedure, the process of stray resonance analysis

and the calculated result will be shown.

4.1 Inductance of multiply bent busbars

A busbar model for calculating synthesized induc-

tance is shown in Fig. 11. Busbars in this model can be

classified into two groups based on direction, vertical or

horizontal. Since there is no electromagnetic interference

between these groups, as previously described, the total

inductance can be obtained as the sum of the inductances

calculated for each group. Accordingly the procedure will

be described only for the horizontal group.

Fig. 8. Frequency characteristics of inductance.

Fig. 9. Model of three-busbar system.

Fig. 10. Inductance change due to the number of

busbars.

Fig. 11. Model of multiply bent busbar.

57

All the busbars in the horizontal group should be

numbered sequentially along a hypothetical loop current in

an arbitrary direction of rotation as shown in Fig. 11.

Assuming the current in each busbar in the same direction

as the loop current is positive, and the voltage against the

loop current is positive, the circuit equation of this group

can be expressed as

where n = 11 in the case of Fig. 11.

In Eq. (30), I1 to In are currents through each busbar

and V1 to Vn are the voltages across them. Each component

Zij in the impedance matrix is a self impedance or a mutual

impedance between busbars i and j. It corresponds to z11,

z12, or the like in Eq. (28) and can be obtained by using the

method proposed previously. Therefore, similarly, Zij = Zjiis established even in this system. The diagonal matrix with

components C11 to Cnn is implemented for the matching of

directions of voltage and current. The value of each com-

ponent is defined as shown below based on the current

direction through the respective busbar:

Cii = 1 when the current flows from left to right

through the i-th busbar.

Cii = �1 when the current flows from right to left

through the i-th busbar.

Equation (30) can be expressed simply as

Considering the magnitude of the current through each

busbar to be equal to a common value I, the magnitude of

the loop current, then V can be obtained as

where each component in Z¢ equals Sg × Zij.Here, Sg equals 1 when the currents through the i-th

and j-th busbars are in the same direction, and it equals �1

when these are in the opposite directions. Finally the syn-

thesized impedance of this group can be obtained as

The inductances of this system obtained from each pair of

busbars at 1 kHz are shown in Table 1.

Even for the same busbar, the self inductance varies

rarely and is little dependent on the associated busbars

selected to make a pair (about 0.6% in this example). Since

the self inductance becomes minimum when the associated

busbar is the most influential one, namely very close and

long, the minimum values are used as the representative

values in this case. The synthesized inductance of this

system obtained from Table 1 by the above procedure is 656

nH. On the other hand, 653 nH was obtained as the meas-

ured value with a HP4284A LCR meter in L-R mode at 1

kHz. For such a small inductance, the measured value is

easily affected by the circumstances. However, practical

reproducibility has been accomplished by means of proper

arrangement of the measuring system including the meas-

ured object, proper length of the leads, the compensation of

the leads, and so forth.

Although all the digits of this measured value are not

necessarily significant, the calculated value shows good

agreement with the measured value.

4.2 Analysis of stray resonance

The experimental model prepared for studying stray

resonance in the dc link is shown in Fig. 12. This model

represents a dc link circuit for three level inverters as shown

in Fig. 13, and so consists of three buses, denoted P, C, and

N, and four banks of capacitors between P and C, and

between N and C. There are many arms of the busbars,

whose inductances cause complicated stray resonance char-

acteristics cooperating with capacitor banks. Although this

model has the same busbar construction as Fig. 9, the

calculation procedure based on a pair of busbars can be

applied, because the intervals between busbars are suffi-

ciently large to neglect approximation errors, as described

in section 3. Since the difference between inductances at 1

kHz and 10 kHz is below 1% in this model, the values at 1

(30)

(31)

(32)

(33)

Table 1. Inductances in multiply bent busbar

58

kHz are used over the entire frequency range for all circuit

constants. The volume resistivity r is assumed to be 17.5 ´10�9 Wm. The inductance and resistance, calculated and

synthesized for every busbar arm in the model based on

above assumptions, are shown in Table 2. The negative

values in Table 2 are caused by the specific definition of the

positive current direction in this model. Mutual resistances

are neglected because these are less than 1/16 of the self

inductances, even in the critical arms. Measured values

were used for the constants of the leads in the capacitor

banks.

If all the mutual inductances in this model are taken

into account, the circuit becomes very complicated and hard

to handle, particularly when the system is simulated with

the inverter by a simulator based on the circuit diagram.

From Table 2, it is recognized that the mutual inductance

between two arms, each in a different neighboring span, is

smaller than that between two arms within the same span,

but is not small enough to be neglected. However, since the

sum of the currents through three arms P, C, and N within

the same span is equal to zero and the distances between

the centers of spans are relatively longer than the busbar

intervals, the effects from the three arms which are con-

tained in another span almost completely cancel each other.

Consequently, if only the inductances within the same span,

which are surrounded by the thick frame in Table 2, are

considered, the resonant frequency can be obtained with

practical accuracy, as described below.

To simplify the analysis further, even the mutual

inductances within the same span should be incorporated

into the self inductances. This conversion process is as

Fig. 12. Experimental filter circuit of dc link.

Fig. 13. Three level inverter.

Table 2. Circuit constants in dc link

Fig. 14. Equivalent circuit of a span of P, C, and N

busbars.

59

follows. Fig. 14(a) shows the equivalent circuit of a span.

Its circuit equation can be expressed as

In practical analysis, let us use two line-to-line voltages and

two loop currents as shown in Fig. 14(b). The new variables

in Fig. 14(b) and the old ones in Fig. 14(a) are related as

follows:

By substituting Eqs. (34) and (36) into Eq. (35), we obtain

On the other hand, the circuit equation of the equivalent

circuit without mutual impedance as shown in Fig. 14(c)

can be expressed as

By comparing Eqs. (37) and (38), and considering Zij = Zjiwithin the same span, we obtain

Figure 15 is the equivalent circuit for the experimental

model shown in Fig. 12. Since the frequency characteristics

of the experimental model were measured from the termi-

nals indicated by an arrow in Fig. 12, the input of the

equivalent circuit is arranged accordingly. The arm imped-

ances Zn in Fig. 15 are given by

where Rn is the resistance of each arm, Ln is the inductance

of each arm converted from the values inside the thick frame

in Table 2 by using Eq. (39), and Cn is the capacitance of

the capacitor bank. The third term of Eq. (40) should be zero

in the arm without capacitor. The circuit equation of Fig. 15

can be expressed as

By simplifying Eq. (41) as V = Z × I, the loop currents in

the equivalent circuit can be expressed as

(36)

(34)

(35)

(37)

(38)

Fig. 15. Equivalent circuit of dc link.

(39)

(40)

(41)

60

Since all the components except for the seventh in V are

zero, Eq. (42) can be rewritten as

Consequently the synthesized circuit impedance Z0 with

respect to the input is equal to V/ I7, that is, 1/Y77. The

frequency characteristics obtained from the calculated bus -

bar inductances are shown by solid lines in Fig. 16. On the

other hand, the characteristics in Fig. 17 were measured

with an LCR meter scanned from 1 kHz to 10 kHz in

impedance mode. Similarly, care was taken to ensure accu -

racy. The calculated resonant frequencies have good agree -

ment with the measured values. For the calculation of

frequency characteristics, the arm resistances are compen -

sated by the following formula based on the values at 1 kHz:

Equation (44) is an approximate equation obtained from the

calculated frequency characteristics of various similar bus-

bars. By the effect of this compensation, the resonant fac-

tors in both figures also agree with each other.

Up to this point, the mutual inductances between the

different spans have been neglected. However, it is easy to

take them into account when making the analysis for only

the dc link itself by means of matrix calculation as described

below. After converting each 3 ´ 3 matrix which is shown

outside the thick frame in Table 2 into a 2 ´ 2 matrix by

using such converting matrices as in Eq. (37) and adding

them to the corresponding components in the impedance

matrix in Eq. (41), we can obtain the complete equation for

this model. The resultant characteristics from this equation

are shown by dotted lines in Fig. 16. These characteristics

are almost the same as those obtained by neglecting mutual

inductances between different spans.

If only the resonant frequencies are needed, fre-

quency swept calculations are not necessary. The solution

can be obtained rapidly applying the following method to

Z¢, a matrix constructed from Z in Eq. (41), by removing

the 7th row and 7th column. Assuming that A is a matrix Z¢whose components Zn are replaced by 1 / Cn (but Zn should

be 0 in the arms without a capacitor), that B is a matrix with

the corresponding replacement by Ln, and that all resis-

tances can be neglected, the circuit equation of Fig. 15 with

open-circuited input can be expressed as

Multiplying both members of Eq. (45) by jw, setting

w2 = l, and rearranging, we obtain the following equation

as a necessary and sufficient condition for I ¹ 0.

Consequently, by using the eigenvalues l of matrix B-1A,

the resonant frequencies can be expressed as

(42)

(43)

(44)

Fig. 16. Calculated resonant characteristic of dc link.

Fig. 17. Measured resonant characteristic of dc link.

(45)

(46)

(47)

61

These frequencies are 2.102, 2.874, 3.917, 4.609, 5.201,

and 6.114 kHz when only the mutual inductances within

the same spans are considered, and 2.107, 2.880, 3.930,

4.597, 5.188, and 6.121 kHz when all the inductances are

considered.

From these analyses, the validity of this simplified

method of calculating busbar inductance and the feasibility

of accurate forecasting of resonance phenomena in dc links

are confirmed. The proposed method will also be useful for

anticipation of various problems caused by resonance and

for efforts to avoid them.

5. Conclusions

In this paper, we showed the simplified calculating

method for busbar inductance and current distribution in

busbars at higher frequencies, which has become more

necessary in recent power electronics apparatus. From our

calculated results, we studied typical characteristics of bus-

bar inductances at high frequencies. Furthermore, we

showed that the electrical constants obtained from pairs of

busbars have sufficient accuracy for practical applications.

Applying these procedures to stray resonant analysis in dc

links, which are widely used in recent large capacity in-

verters, we showed that the resonant frequencies can be

calculated with good accuracy, by a comparison with ex-

perimental results.

Acknowledgments

We express our sincere appreciation to Dr. M. Kawa-

mura, Mr. M. Adachi of Mitsubishi Electric Corporation,

and Mr. K. Takahira of Mitsubishi Electric Engineering for

their kind cooperation.

REFERENCES

1. Nakamichi Y, Nozawa H, Okui A, Ikeda H. Suppres-

sion of higher harmonic resonance in power feeding

circuit containing PWM inverter. 1991 National Con-

vention of IEE Japan.

2. Okui A, Kaga S, Ikeda H, Kinoshita K. Harmonic

resonance at dc-link of inverter with large capacity

and measure for its suppression. Papers of Tech Mtg

on Semiconductor Power Converter IEE Japan

1996;SPC96-55:19�28.

3. Schanen JL, Clavel E, Roudet J. Modeling of low

inductive busbar connections. IEEE Industry Appli-

cations Mag 1996;9:39�43.

4. Ruehli AE. Inductance calculations in a complex

integrated circuit environment. IBM-J Res Develop

1972;9:470�481.

5. Hoer C, Love C. Exact inductance equations for

rectangular conductors with application to more

complicated geometries. J Res Natl Bureau Stand-

ards-C Engineering and Instrumentation 1965;69C:

127�137.

AUTHORS (from left to right)

Katsutaka Tsuboi (member) graduated from Waseda University and joined Mitsubishi Electric Corporation in 1967. He

has been engaged in designing, developing, and testing many kinds of power electronics products. At present, he is a member

of the Power Electronics Department of the Nagasaki Works, Mitsubishi Electric Corporation; a part-time lecturer at Kumamoto

University; a doctoral student at Nagasaki University; and a member of the Institute of Electrical Engineers of Japan.

Mineo Tsuji (member) obtained a Ph.D. in March 1981 from Kyushu University. He became a lecturer at Nagasaki

University in April 1981 and an associate professor in October 1983. His current interest is power electronics and applications

of control theory. He is a member of IEEE; the Institute of Electrical Engineers of Japan; the Society of Instrumentation and

Control Engineers of Japan; the Society of Systems Control and Information Processing of Japan; and the Japan Society of

Applied Electromagnetics. Dr. Tsuji received a Paper Award from the Institute of Electrical Engineers of Japan.

62

AUTHORS (from left to right)

Eiji Yamada (member) obtained his D.Eng. degree in 1964 from Kyushu University, where he was appointed a research

associate. He became an associate professor at Nagasaki University in 1968 and a professor in 1974. He has been involved in

research on power electronics and control of electric apparatus. He is a senior member of IEEE and a member of the Institute

of Electrical Engineers of Japan; the Institute of Electronics, Information and Communication Engineers of Japan; the European

Power Electronics and Drives Association; the Japan Society of Applied Electromagnetics; and the Society of Instrumentation

and Control Engineers of Japan. Dr. Yamada received a Paper Award from the Institute of Electrical Engineers of Japan.

63