NTIA Report 98-344 

Analysis and Calculations of the 

Ground Plane Inductance Associated 

with a Printed Circuit Board 

Christopher L. Holloway 

Edward F. Kuester 

U.S. DEPARTMENT OF COMMERCE 

William M. Daley, Secretary 

Larry Irving, Assistant Secretary 

for Communications and Information 

November 1997 

iii

CONTENTS 

Page 

FIGURES.........................................................................................................................................v 

ABSTRACT.....................................................................................................................................1 

1. INTRODUCTION .....................................................................................................................1 

2. CONCEPTS OF PARTIAL INDUCTANCE ............................................................................4 

 2.1 Partial Inductance of Wires..................................................................................................6 

3. NET INDUCTANCE OF THE GROUND PLANE AND TRACE ..........................................7 

 3.1 Self-Partial Inductance of the Ground Plane .....................................................................10 

 3.2 Self-Partial Inductance of the Trace ..................................................................................11 

 3.3 Mutual-Partial Inductance of the Ground Plane and Trace ...............................................12 

 3.4 Net Inductance ...................................................................................................................12 

4. NUMERICAL EVALUATION OF THE INTEGRALS.........................................................13 

5. CALCULATION OF THE NET AND PARTIAL INDUCTANCE.......................................15 

6. VALIDATION OF RESULTS ................................................................................................26 

7. EXCESS INDUCTANCE DUE TO CURRENT CONSTRICTION ......................................32 

8. CONCLUSION........................................................................................................................34 

9. ACKNOWLEDGMENTS .......................................................................................................35 

10. REFERENCES ........................................................................................................................36 

APPENDIX: GENERAL PURPOSE 10-POINT QUADRATURE WEIGHTS...........................39 

v

FIGURES 

Page 

Figure 1. a) Side and b) top view of a trace over a ground plane with attached cables. ...............3 

Figure 2. Trace over a ground plane: microstrip line....................................................................4 

Figure 3. Current loop. ..................................................................................................................5 

Figure 4. Definition of surfaces needed in the evaluation of the integral. ....................................9 

Figure 5. Self-partial inductance of the ground plane for a length of 2.54 cm. ..........................16 

Figure 6. Self-partial inductance of the ground plane for a length of 15.24 cm. ........................17 

Figure 7. Self-partial inductance of the ground plane for a length of 30.48 cm. ........................18 

Figure 8. Mutual-partial inductance of the ground plane for a length of 2.54 cm. .....................19 

Figure 9. Mutual-partial inductance of the ground plane for a length of 15.24 cm. ...................20 

Figure 10. Mutual-partial inductance of the ground plane for a length of 30.48 cm. ...................21 

Figure 11. Net inductance of the ground plane for a length of 2.54 cm........................................22 

Figure 12. Net inductance of the ground plane for a length of 15.24 cm......................................23 

Figure 13. Net inductance of the ground plane for a length of 30.48 cm......................................24 

Figure 14. Net inductance of the ground plane as a function of the trace height h, 

 the trace width w, the length l. .....................................................................................25 

Figure 15. Self-partial inductance of the trace. .............................................................................27

Figure 16. Net inductance of the trace for a length of 2.54 cm.....................................................28 

Figure 17. Net inductance of the trace for a length of 15.24 cm...................................................29 

Figure 18. Net inductance of the trace for a length of 30.48 cm...................................................30 

Figure 19. Inductance of a microstrip line. ...................................................................................33 

Figure 20. Illustration of current concentration at the end due to the via. ....................................34 

Figure 21. Illustration of a via of width w.....................................................................................35 

ANALYSIS AND CALCULATIONS OF THE GROUND PLANE 

INDUCTANCE ASSOCIATED WITH A PRINTED CIRCUIT BOARD 

Christopher L. Holloway1

Edward F. Kuester2

A knowledge of the net inductance of the ground plane can aid in the analysis and 

investigation of printed circuit board emissions. In this report, we present a method, 

based on the concept of partial inductance, to determine the net inductance of the 

ground plane associated with a microstrip line. This method is based on a previously 

derived expression for the current density on the ground plane. We show calculations 

for the net, self-partial, and mutual-partial inductance of the ground plane for various 

trace geometries of practical interest. We also illustrate how the classical transmission 

line inductance of a microstrip line can be obtained from the concept of partial 

inductance. Comparisons to different experimental results are also given. 

Key words: net inductance; partial inductance; ground plane inductance; common-mode 

emissions; microstrip line inductance; printed circuit board emission 

1. INTRODUCTION 

Printed circuit boards (PCBs) are the focus of many design efforts to comply with domestic and 

international electromagnetic compatibility (EMC) requirements for digital devices [1]. These 

boards are the root of most emission and immunity problems; hence, properly designed PCBs 

can offer a cost-effective approach to achieving EMC compliance [2] and [3]. 

PCBs produce two types of radiated emissions: differential-mode and common-mode [3]-[6]. 

Differential-mode or loop emissions are the result of currents that flow on a signal trace between 

integrated circuits (ICs), and return via a trace, grid, or plane. Fortunately, the signal-trace 

current produces fields that tend to cancel the fields from the ground-return current [4], [7], and 

[8]. Hence, typical signal currents must flow through relatively large PCB loops to produce 

emissions that exceed the regulatory limits. In these cases, the emissions can be reduced by 

simply placing the signal trace closer to the ground return and/or reducing the distance between 

the source and load ICs. 

 1

 The author is with the Institute for Telecommunications Sciences, National Telecommunications and 

Information Administration, U.S. Department of Commerce, Boulder Laboratories, Boulder, CO, 80303 2

 The author is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, 

CO, 80309-0424 
2

Common-mode emissions are predominantly caused by unbalanced interconnections between 

digital circuits. Even a slight difference in the geometry of the signal and ground-return 

conductors on a PCB or in an IC package can cause common-mode currents to flow. These 

currents are problematic because they flow in a common direction and produce fields that tend to 

add to one another [4], [7], and [8]. Hence, common-mode currents can be several orders of 

magnitude less than differential-mode currents, but produce the same level of radiated emissions. 

Attaching cables to the PCB simply increases the length of the antenna and reduces the resonant 

frequency at which the maximum radiated emissions occur. 

An unbalanced circuit on a PCB constructed with signal and ground-return trace pairs will 

radiate as an asymmetric folded-dipole [9]. Since the rectangular geometry of IC packages is 

inherently unbalanced, it is virtually impossible to construct balanced circuits on a PCB using 

ground-return traces. Even simple circuits will produce fields that exceed the regulatory limits 

[4], [8], and [9]. One solution to this difficult problem is to place a relatively large conducting 

plane beneath the PCB. The currents induced on this image plane will produce fields that tend to 

cancel the fields produced by the common-mode currents on the PCB, which causes a dramatic 

reduction in the total radiated emissions [9]-[11]. 

Image currents can also lower emissions from PCBs constructed with a ground-return plane or an 

equivalent ground-return grid [2] and [12]-[15]. Indeed, common-mode currents cannot flow on 

a PCB with an infinite ground-return plane because the signal conductor and its ground-plane 

image are perfectly balanced. However, the finite ground-return plane on an actual PCB 

produces an imperfect image that causes the ground-return plane itself to radiate as a dipole 

antenna [16] and [17]. In this case, the ground-noise voltage produced when signal currents 

encounter the impedance of the ground-return plane (referred to simply as a ground plane 

throughout the rest of this report) is the source driving the dipole antenna [3]. Once again, 

attaching cables to the PCB simply increases the dipole length. Even PCBs with a solid groundreturn plane without attached cables can produce radiated emissions that exceed the Federal 

Communication Commission (FCC) Class-B limit [18]. 

The analysis of the actual problem of an imperfect image ground plane with unbalanced currents 

and attached cables (Figure 1) is a difficult one. However, by investigating the ground voltagedrop of a signal trace over a perfectly conducting infinitely large ground plane (microstrip line; 

Figure 2), one can determine how to reduce the ground voltage-drop (which is the source driving 

the emissions) of the actual problem. Using Faraday’s law, it can be shown that a trace above a 

ground plane produces a voltage-drop on the ground plane directly under the trace. Furthermore, 

Faraday’s law shows that this voltage-drop is related to the so-called net inductance [2] of the 

ground plane by the following [2] and [19]-[22]: 

,

t

I Vgr Lgr ∂

∂ = (1) 

where Lgr is the net inductance of the ground plane. From this equation, it is evident that an 

understanding of the net inductance is imperative to the effective reduction of the ground plane 

voltage-drop and the resulting common-mode emissions in the actual problem. 
3

Figure 1. a) Side and b) top view of a trace over a ground plane with attached cables. 
4

Figure 2. Trace over a ground plane: microstrip line. 

Attempts have been made to calculate the ground plane net inductance with little success. Most 

recently Leferink and van Doorn [23] introduced a closed form expression for the net inductance 

for the ground plane. The expression assumes that the current on the ground plane is uniformly 

distributed over the entire ground, which is not the case (see [24] and [25]). This expression 

indicates that if the width of the ground plane becomes very large, then the ground plane 

inductance approaches zero. 

In this report, we present a technique for calculating the net inductance of the ground plane, 

using the actual current distribution on the ground plane. This report is organized as follows: 

Section 2 illustrates the concepts of partial inductance and shows how they are related to the 

voltage drop across the ground plane. In Section 3, we introduce a procedure to calculate the net 

inductance of both the ground plane and the trace based on the knowledge of the current 

distribution on the ground plane. Section 4 discusses the numerical implementation of the results 

given in Section 3. In Section 5 we show results for the net, self-partial, and mutual-partial 

inductance of the ground plane and trace for various circuit geometries of practical interest. 

Finally, we discuss effects of current constriction due to vias on the net inductance of the ground 

plane. 

2. CONCEPTS OF PARTIAL INDUCTANCE 

Much has been written on the subject of partial inductance, for details see [2], [9], and [19]- 

[22]. The origin for this material dates back to the turn of the century with the work of 

Wien [26], Rosa [27], and Rosa and Grover [28]. This section provides a brief overview of 

the subject and its application to transmission lines. Assuming there is a rectangular loop of 

wire carrying current (Figure 3), the inductance of this loop is given by the classical
5

Figure 3. Current loop. 

expression: 

, I

B ds

L ∫s ⋅ = (2)

where B is the magnetic field flowing through the loop cross-section and I is the total current 

flowing through the loop. 

Recall that the magnetic field B can be represented in terms of a magnetic potential A by the 

following: 

B = ∇ × A. (3) 

With this expression and Stokes’ theorem, the inductance of the loop can be written as: 

, I

A dl

L ∫c ⋅ = (4)

where c is the contour of the loop. 

This integral can be broken into the following four segments: 

, 1 2 3 4

4 4 3 3 2 2 1 1

L L L L

I

A dl

I

A dl

I

A dl

I

A dl

L c c c c

= + + +

⋅ + ⋅ + ⋅ + ⋅ = ∫ ∫ ∫ ∫

 (5) 

where ci are the contours of each segment. Each Li represents the contribution of each segment to 

the total inductance, and is called the net inductance of each segment of line [2]: 

. I

A dl

L i c i

i

∫ ⋅

= (6) 

The net inductance of each segment can be expressed as the sum of self- and mutual-partial 

inductances for the segment, and can be written as: 

, 1

∑=

= ±

N

j

i pij L L (7) 
6

where N is the total number of segments. If i = j, then pij L corresponds to the self-partial 

inductance. If i ≠ j, then pij L corresponds to the mutual-partial inductance. The sign (±) is related 

to the relative orientation of the current in segment j to the current in segment i, and is 

determined by the sign of the dot product between directed current elements. The concept of 

mutual-partial inductance is equivalent to the mutual inductance associated with adjacent circuits 

that couple to one another [29] (i.e., the mutual-partial inductance is associated with coupling of 

the different segments). 

The partial inductance pij L is defined as the ratio of the magnetic flux penetrating the surface 

between segment i and infinity and current Ij that produces that flux. pij L can be written as: 

.

j

c ij i

p I

A dl

L i

ij

∫ ⋅

= (8)

The classical inductance of the loop can be written in terms of the net inductance. Using 

equations (5) and (7), 

∑ ∑∑ = ==

= = ±

N

i

N

i

N

j

i pij L L L

1 11

. (9)

The voltage drop across any segment of the loop can be expressed in terms of the net inductance 

of that segment. For example, the voltage across segment 1 can be expressed as: 

t

I V L

∂

∂ = 1

1 1 (10)

where 1 p11 p13 L = L − L and is the net inductance of segment 1. Equation (6) indicates that a 

mutual-partial inductance exists only between segments that are not perpendicular to each other. 

Thus, there is no p12 L or p14 L term in the expression for the net inductance of segment 1. 

2.1 Partial Inductance of Wires 

The net and partial inductance of parallel wires has been obtained in the past and can be found in 

[2], [19]-[20] and [27]. The self- and mutual-partial inductances are given by the following: 

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟ + ⎠

⎞ ⎜

⎝

⎛ + − ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟ + ⎠

⎞ ⎜

⎝

⎛ = ln + 1 1

2

2 2

l

r

l

r

r

l

r

l L l pii π

μ (11) 

and 

ln 1 1 , 2

2 2

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟ + ⎠

⎞ ⎜

⎝

⎛ + − ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟ + ⎠

⎞ ⎜

⎝

⎛ = +

l

d

l

d

d

l

d

l L l pij π

μ (12) 
7

where d is the separation of the two wires, r is the radius of the wires, and l is the length of the 

wires. When the length l of the wires is large compared to the radius r and the separation d, 

equations (12) and (11) can be approximated by: 

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟ −

⎠

⎞ ⎜

⎝

⎛ ≈ 1 2 ln

2 r

l L l pii π

μ for << 1

l

r (13) 

and 

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟ − + ⎠

⎞ ⎜

⎝

⎛ ≈ l

d

d

l L l pij 2

1 2 ln

2π

μ for << 1

l

r and << 1

l

d . (14) 

For parallel conductors of rectangular cross-section, the mutual-partial inductance is given by 

equation (14) and the self-partial inductance is given by: 

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟ −

⎠

⎞ ⎜

⎝

⎛ ≈ 1 8 ln

2 w

l L l pii π

μ for << 1

l

w and << 1

l

w , (15) 

where w and t are width and thickness of the trace, respectively. See [19] and [22] for 

expressions of the net and partial inductance for wires of different orientations. 

3. NET INDUCTANCE OF THE GROUND PLANE AND TRACE 

The problem of interest in this report is that of a trace suspended over a ground plane (see Figure 

2). The equation given in Section 2 for the partial inductance (see equation (8)) is for conductors 

with a current filament. However, the current density in the ground plane is not a constant, nor is 

it concentrated under the trace. The current spreads out from under the trace, and in previous 

work ([24] and [25]) the following expression was derived for the current density on the ground 

plane: 

, 2

2

tan

2

2 ( ) tan 1 1

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ + ⎟ +

⎠

⎞ ⎜

⎝

⎛ − = − −

h

w x

h

w x

w

I J x gr π

 (16) 

where W is the width of the trace and h is the height of the trace above the ground. Thus, the 

procedure given in Section 2 for calculating the net, self-partial, and mutual-partial inductance 

must be modified to account for this type of current distribution. 

The basic concept of inductance must be used to obtain an expression of the net inductance of 

the ground plane and the trace. The inductance of this microstrip geometry of length 1 is given 

by: 

2

2

I

U L = , (17) 

where U is the stored energy within the volume bound by the ground plane and the trace, and is 

given by: 

∫ = ⋅ v U B H dv

2

1 . (18) 
8

The inductance can then be given by: 

∫ = ⋅ v

B H dv

I

L 2

1 . (19) 

When B is replaced with ∇ x A , L can be written as: 

∫ = ∇ × ⋅ v A H dv

I

L 2

1 . (20) 

The integrand can be rewritten with the aid of a vector identity as: 

H ⋅(∇ × A) = ∇ ⋅(A × H )+ A ⋅(∇ × H ). (21) 

The H field in this expression is related to the volume current density by Maxwell’s equations: 

v ∇ × H = J . (22) 

For this problem, the volume current density Jv is zero. Thus, equation (20) reduces to the 

following: 

( ) ∫ = ∇ ⋅ × v A H dv

I

L 2

1 . (23) 

With the divergence theorem, this integral becomes: 

( ) ∫ = × ⋅ s A H an ds

I

L 2

1 , (24) 

where s consists of three parts (Figure 4). The first is the surface of the ground plane sgr, the 

second is the surface of the top and bottom portions of the trace top

tr s and bottomtr s , and the third is 

the surface surrounding the ground plane and trace s1. The fields decay to zero as the surface s1 is 

moved to infinity. Therefore, the surface integral in equation (24) reduces to an integration only 

over sgr, top

tr s , and bottomtr s . 

The integrand of this integral can be rewritten as: 

( ) ( ) an A H A H × an ⋅ × = ⋅ . (25) 

When equation (25) is substituted into equation (24), the following is obtained: 

 (26) 

The boundary conditions on Maxwell's equations allow the H to be related to the current 

density on the ground plane Jgr and on the trace Jtr:

 (27) 
9

Figure 4. Definition of surfaces needed in the evaluation of the integral for calculating the 

 inductance. 

The minus sign accounts for the current on the ground plane flowing in the opposite direction 

(the minus z direction) from that of the current on the trace. 

We showed in [30] that for most practical microstrip geometries, the current densities on the top 

and bottom surface of the trace are equivalent and are approximated by: 

2

bottom tr

tr

top

tr

J J ≈ J ≈ , (28) 

where Jtr is the total current density on the strip. Thus the inductance of the microstrip line 

reduces to the following: 

 (29) 

where Jgr is the current density on the ground plane as given in equation (16), Jtr is the total 

current density on the trace (see section 3.2), and Az is the magnetic potential produced by both 

the ground plane and trace current densities. 

Equation (29) can be broken into two parts: 

L = Lgr + Ltr , (30) 

where Lgr is the net inductance of the ground plane, and 

() ( ) ∫ ∫ ∞

= = −∞ = − = x gr z

l

z gr J x A x z y dx dz

I

L , , 0 1

0 2 , (31) 

where the minus sign indicates that the current on the ground plane flows in the minus z 

direction. Ltr is the net inductance of the trace: 

() ( ) ∫ ∫ = = − = = / 2

/ 0 2 2 , , 1 w

x w tr z

l

z tr J x A x z y h dx dz

I

L (32) 
10

where y = 0 corresponds to the location of the ground plane on the y axis and y = h corresponds 

to the location of the trace on the y axis. 

To obtain an expression for these net inductances, an expression for the magnetic vector 

potential Az is needed. Az has contributions from the currents on both the ground plane and the 

trace, and is determined from a standard quasi-static Green's function approach: 

A ( ) x z y A (x z y) A (x z y) tr

z

gr

z z , , = , , + , , , (33) 

where A ( ) x z y gr

z , , is the portion of the total potential produced by the currents flowing on the 

ground: 

( ) () ( ) ∫ ∫ ∞

′=−∞ ′= = − ′ ′ ′ ′ = ′ ′ x gr

l

z

gr

z A x,z, y J x G x, x ,z,z , y, y 0 dx dz

0 μ 0 (34) 

and A ( ) x z y tr

z , , is the portion of the total potential produced by the current flowing on the trace: 

( ) () ( ) ∫ ∫ ′=− ′= = ′ ′ ′ ′ = ′ ′ / 2

/ 0 2 0 , , , , , , , wx w tr

l

z

tr

z A x z y μ J x G x x z z y y h dx dz , (35) 

where G is the free space Green’s function given by: 

( )( )( ) 2 2 2

1

4

1

x x z z y y

G

− ′ + − ′ + − ′

= π

. (36) 

With equations (31), (34), and (35), the net inductance of the ground plane reduces to: 

 (37) 

and with equations (32), (34), and (35), the net inductance of the trace reduces to 

 (38) 

3.1 Self-Partial Inductance of the Ground Plane 

The self-partial inductance of the ground plane is defined as the inductance of the ground caused 

by the magnetic potential produced solely by the current flowing on the ground plane. 
11

Thus, from equation (37) the self-partial inductance of the ground plane can be expressed as: 

J () ( ) ( ) x J x K x x dx dx

I

LS gr x gr x gr ⎥⎦

⎤ ⎢⎣⎡ = ′ ′ ′ ∫ ∫

∞

′=−∞

∞

=−∞ , 4 2

0

π

μ , (39) 

where the kernel K (x, x′) contains the integrals over z and z′, and is given by: 

( ) ( )( )

dz

x x z z

dz K x x

l

z

l

z ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

− ′ + − ′

′ ′ = ∫ =0 ∫′=0 2 2 , . (40) 

This double integral is fairly simple and can be evaluated explicitly. Leaving out the details, it 

can be shown that once the z and z′ integrals are evaluated, the kernel K (x, x′) reduces to the 

following: 

K( ) x, x′ = K (x, x′) + K (x, x′) 1 2 , (41) 

where K1 (x, x′) is the nonsingular part, and is expressed as: 

( ) ( ) ( ) ⎟

⎠

⎞ ⎜

⎝

⎛ ′ = − ′ − − ′ + + + − ′ + 2 2 2 2

1 K x, x 2 x x 2 x x l 2l ln l x x l (42) 

and K2 (x, x′) is the singular part, and is expressed as: 

K (x, x′) = −2l ln ( x − x′) 2 . (43) 

Therefore, the self-partial inductance of the ground is given by: 

 (44) 

Unfortunately, the two integrals in this expression cannot be evaluated in closed form and must 

be determined by numerical means. In Section 4, the numerical integration of these two integrals 

and the integrals evaluated in the next two sections are discussed. 

3.2 Self-Partial Inductance of the Trace 

The self-partial inductance of the trace is defined as the inductance of the trace caused by the 

magnetic potential produced solely by the current flowing on the trace. To calculate this quantity, 

two different expressions for the current density on the trace are used. For a w/h ≥ 0.5, the 

current density on the trace is approximated by a constant distribution: 

w

I Jtr = (45) 

and for w/h < 0.5, the current density is approximated by the Maxwell distribution: 

2

2

2

x

w

I Jtr

⎟ − ⎠

⎞ ⎜

⎝

⎛

=

π

 . (46) 
12

For the constant distribution, the self-partial inductance of the trace can be expressed as: 

K( ) x x dx dx

w

LS w

x w

w

x w tr = ′ ′ ∫′=− ∫ =− , 4

/ 2

/ 2

/ 2

/ 2 2

0

π

μ (47) 

and for the Maxwell distribution, the self-partial inductance becomes 

( ) ( )

K( ) x x dx dx

x

w

x

w

LS w

x w

w

x w tr ′ ′

⎟ − ⎠

⎞ ⎜

⎝

⎛ ⎟ − ′ ⎠⎞ ⎜

⎝

⎛

= ∫′=− ∫ =− ,

2

1

2

1

4

2

2

/ 2

/ 2

2

2

/ 2

/ 2 3

0

π

μ , (48) 

where the kernel K(x,x') is given in equation (41). Note that equation (48) has an added 

singularity caused by the Maxwell distribution. Once again, this integral cannot be evaluated in 

closed form, and must be determined numerically. 

3.3 Mutual-Partial Inductance of the Ground Plane and Trace 

The mutual-partial inductance of the ground plane is defined as the inductance of the ground 

caused by the magnetic potential produced solely by the current flowing on the trace. Using 

equation (37), the mutual-partial inductance of the ground can be obtained. Once again the z and 

z′ integral can be evaluated explicitly. If the x and x′ integrals are interchanged, the following is 

obtained: 

J () ( ) x J x K x x dx dx

I

LM gr m x tr

w

x w gr = ′ ′ ′ ∫ ∫

∞

′=− =−∞ ( ) , 4

/ 2

/ 2 2

0

π

μ , (49) 

where Jtr is given by either equation (45) or (46). Note that the mutual-partial inductance is 

defined as a positive quantity. In this equation, Jgr is given by equation (16) and the kernel 

Km(x, x′) is given by: 

 (50) 

Once again, this integral cannot be evaluated in closed form, and must be determined numerically. 

The mutual-partial inductance of the trace is obtained by using equation (38). Since the free 

space Green’s function in this expression is symmetric in x and x′, it can be shown that the 

mutual-partial inductance of the trace is identical to the mutual-partial inductance of the ground 

plane LMtr = LMgr.

3.4 Net Inductance 

With the self- and mutual-partial inductance obtained, the net inductance of the ground plane can 

be expressed as: 

Lgr = LSgr − LM gr , (51) 
13

and the net inductance of the trace is given by 

Ltr = LStr − LMtr . (52) 

4. NUMERICAL EVALUATION OF THE INTEGRALS 

The integrals needed for the self-partial inductance of the ground are evaluated first. From 

Section 3, the self-partial inductance can be expressed as: 

LSgr = L1 + L2 , (53) 

where L1 corresponds to the part of the inductance that has a nonsingular kernel in its integral 

and L2 corresponds to the part of the inductance that has a singular kernel in its integral. These 

two terms are given by: 

J x J () ( ) x K x x dx dx

I

L gr x gr x ⎥⎦

⎤ ⎢⎣

⎡ = ′ ′ ′ ∫ ∫

∞

′=−∞

∞

=−∞ ( ) , 4 2 1

0

1

π

μ

 (54) 

J x J () ( ) x K x x dx dx

I

L gr x gr x ⎥⎦

⎤ ⎢⎣

⎡ = ′ ′ ′ ∫ ∫

∞

′=−∞

∞

=−∞ ( ) , 4 2 2

0

2

π

μ

where K1(x, x′) and K2(x, x′) are the nonsingular and singular kernels, respectively, and are given 

in equations (42) and (43). 

The L1 integral is fairly well behaved, and the only difficulty is that the limits of this double 

integral extend to infinity. The integral over x' (the inner integral) is evaluated with a 10-point 

quadrature rule (see [31]). The integration to ±∞ can be difficult to handle, but taking advantage 

of how the current on the ground is distributed can aid in evaluating these integrals. We showed 

in [24] that the amount of current that spreads out from under the trace is a function of the trace 

height h. For large h, the current spreads out far from the edge of the trace, and for small h the 

current is concentrated under the trace. To account for the rapid change in the ground plane 

current density in the vicinity of x = w/2, the integral is subdivided into fine cells around x = w/2. 

We also showed in [24] that at a location of about 100 w from the edge of the trace, the current is 

very small. The contribution of this small amount of current to the total integral in equations (44) 

and (49) is negligible. For assurance, these integrals are extended to ±(100000 w). Therefore, the 

domain of the x' integral is divided into the following subdomains: 

(55) 
14

Similar expressions exist for the negative portion of the integral. Over each of these sub domains, the integral is approximated by the following 10-point Guassian quadrature formula: 

∫ ∑=

≈

N

i

i i

b

a

f x dx A f x

1

( ) ( ), (56) 

where the weights Ai and abscissae xi for a general purpose 10-point quadrature rule are given by 

Harris and Evans [31], and are listed in the Appendix. The details of the implementation of this 

technique can be found in [32]. 

The domain of the x integral (the outer integral) is divided into the same subdomains as in 

equation (55). Over each of these subregions, the integral is also approximated by a 10-point 

quadrature formula. 

The L2 integral poses an added difficulty, in that its kernel is singular and must be treated 

carefully. Harris and Evans [31] introduced an efficient general purpose 10-point quadrature 

formula for integrals with end point singular behavior. The weights Ai and abscissae xi for this 

approach are given in [31], and are listed in the Appendix. As before, the inner and outer 

integrals (the x and x' integrals) are subdivided into the sub domains given in equation (55). Once 

again, the outer and inner integrals are evaluated with the general purpose 10-point quadrature 

rule. In order to take advantage of the efficient general purpose quadrature formula for endpoint 

singularities given in [31], the singular behavior of the kernel must be at the end point of the 

integrals given in equation (55). Therefore, a change of variable is needed for the inner integral 

to ensure that the singular behavior of the kernel in each subregion occurs at an end point. By 

using a change of variables where 

t = x − x′ and dx′ = −dt , (57) 

the integrals in equation (55) can be rewritten as: 

J ( ) x x x dx J ( ) x t (t )dt gr

t x b

t x a gr

b

x a

′ ln − ′ ′ = − − ln ∫ ∫

= −

′= = − . (58) 

The numerical integration of the mutual-partial inductance [equation (49)] for the constant 

distribution is very similar to the evaluation of L1. The main difference is that the outer limit of 

the integral is not extended to ±∞, but is evaluated to ±w/2. The domain of the outer integral of 

equation (49) (the x' integral) is subdivided into the following subdomains: 

 (59) 

and the 10-point quadrature formula is used over each subdomain. The inner integral of equation 

(49) (the x integral) is subdivided as indicated in equation (55), and is evaluated with the general 

purpose 10-point quadrature procedure. 
15

When the Maxwell current distribution is used in equations (48) and (49), the integrals have the 

following form: 

f x dx

x

w

b

a ( )

2

1

2

2

⎟ − ⎠

⎞ ⎜

⎝

⎛ ∫ . (60) 

This integral needs to be modified to handle the singularity in Jtr. By using a change of variables 

where: 

sin( ) 2θ w

x = and θ dθ w dx cos( ) 2 = , (61) 

the integral in equation (60) can be rewritten as: 

θ dθ w f

b w

a w sin( )) 2

(

sin (2 / )

sin (2 / )

1

∫ 1

−

− . (62) 

The evaluation of the self-partial inductance of the trace is similar to the determination of L1 and 

L2. The only difference is that for the trace, the limits of integration are ±w/2. 

5. CALCULATION OF THE NET AND PARTIAL INDUCTANCE 

In this section, we present results of the net inductances Lgr,tr (as well as the self-partial LSgr,tr

and mutual-partial inductance LMgr,tr) of both the ground plane and trace for various trace 

geometries of practical interest. Figures 5-7 show results for the self-partial inductance of the 

ground LSgr as a function of the trace width w and the trace height h for trace lengths l of 2.54 cm 

(1 in), 15.24 cm (6 in), and 30.48 cm (12 in), respectively. Figures 8-10 show results of the 

mutual-partial inductance of the ground LMgr as a function of the trace width w and the trace 

height h for trace lengths l of 2.54 cm, 15.24 cm, and 30.48 cm, respectively. 

These figures indicate how the self- and mutual-partial inductances of the ground vary as a 

function of different geometries. The more important quantity needed to assess printed circuit 

board emissions is the net inductance (LSgr – LMgr) of the ground plane. This quantity can be 

obtained from equation (51). Figures 11-13 show results of the net inductance of the ground 

plane as a function of the trace width w and the trace height h for lengths l of 2.54 cm, 15.24 cm, 

and 30.48 cm, respectively. Figure 14 show results for the net inductance of the ground plane as 

a function 6f the trace height h, the trace width w, and the trace length l.

From these figures, it is apparent that the net inductance of the ground is less sensitive to the 

trace width w than it is to the trace height h. Two orders of magnitude change in the trace width 

results in only about a factor of 1.2-2.5 change in the net inductance of the ground. Whereas if 

the height of the trace is changed from 0.254 mm (10 mils) to 2.54 mm (100 mils), the net 

inductance changes by about a factor of 50. The smaller the height of the trace over the ground, 

the lower the net inductance, and more importantly, the lower the voltage-drop on the ground 

plane. 
16

Figure 5. Self-partial inductance of the ground plane as a function of the trace width w and the 

 height over the ground h for a trace length l of 2.54 cm (1 in). 
17

Figure 6. Self-partial inductance of the ground plane as a function of the trace width w and 

 the height over the ground h for trace length l of 15.24 cm (6 in). 
18

Figure 7. Self-partial inductance of the ground plane as a function of the trace width w and the 

 height over the ground h for trace length l of 30.48 cm (12 in). 
19

Figure 8. Mutual-partial inductance of the ground plane as a function of the trace width w and 

 the height over the ground h for trace length l of 2.54 cm (1 in). 
20

Figure 9. Mutual-partial inductance of the ground plane as a function of the trace width w and 

 the height over the ground h for trace length l of 15.24 cm (6 in). 
21

Figure 10. Mutual-partial inductance of the ground plane as a function of the trace width w and 

 the height over the ground h for trace length l of 30.48 cm (12 in). 
22

Figure 11. Net inductance of the ground plane as a function of the trace width w and the height 

 over the ground h for trace length l of 2.54 cm (1 in). 
23

Figure 12. Net inductance of the ground plane as a function of the trace width w and the height 

 over the ground h for trace length l of 15.24 cm (6 in). 
24

Figure 13. Net inductance of the ground plane as a function of the trace width w and the height 

 over the ground h for trace length l of 30.48 cm (12 in). 
25

Figure 14. Net inductance of the ground plane as a function of the trace height h, the trace width 

w, and the trace length l. 
26

While this reduction in the voltage-drop (net inductance) we calculated is for an idealized 

perfectly conducting infinitely large ground plane with balanced currents flowing on the ground 

and trace, it illustrates how the voltage-drop of the actual problem (finite ground plane with 

unbalanced currents on the ground and trace) varies as a function of the height h and width w of 

the trace. Therefore, the reduction in the voltage-drop on the ground plane (due to the reduction 

of the trace height h), decreases the source driving the common-mode emissions, and more 

importantly results in a dramatic reduction of the common-mode emissions of the board. 

The substantial reduction of the net inductance of the ground plane Lgr for small values of h can 

be explained by referring to equations (44) and (49). These two equations represent the self- and 

mutual-partial inductance of the ground, and as h gets small these two expressions approach one 

another for two reasons. First, one of the differences in these integrals is that the outer integral 

for the mutual-partial inductance is evaluated to ±w/2 with J = I/w. From equation (16) and [24] 

it can be shown that as the height of the trace becomes small, the current of the ground plane is 

concentrated under the trace and very little current spreads out past the trace edges. For very 

small h values, the current distribution on the ground plane can be approximated as a constant 

(e.g., Jgr ≈ I/w). Therefore, for very small h values, the limit to infinity on the integral in 

equations (44) and (49) can be evaluated to ±w/2. If this approximation is made for small h 

values, then the only difference in the self- and mutual-partial inductance is the kernel (K1, K2, 

and Km). Based on equations (42), (43), and (50), it is seen that for very small h values, 

lim[ ( 1 2 )] 0

0

− + ⇒ →K m K K

h

. (63) 

Therefore, for small h values, the net inductance of the ground plane Lgr approaches zero: 

lim 0

0 = − ⇒ → gr gr gr h

L LS LM . (64) 

That is to say that for small h values, the mutual-partial inductance LMgr approaches the selfpartial inductance LSgr. 

The results for the self-partial inductance of the trace LStr are shown in Figure 15. Note that the 

self-partial inductance of the trace is not a function of h (the trace height). This is because we 

have assumed that the current density on the trace is not a function of h. The net inductance of 

the trace Ltr can be obtained from equation (52). Figures 16-18 show results of the net inductance 

of the trace Ltr for various geometries. Note, that for the same geometries, the net inductance of 

the trace Ltr is significantly larger than that of the net inductance of the ground Lgr. 

6. VALIDATION OF RESULTS 

For validation, we compared the ground plane inductance obtained with the techniques 

presented here to measured data. There is limited measured data of the ground plane net
27

Figure 15. Self-partial inductance of the trace as a function of the trace width w for trace length 

l of 2.54 cm (1 in), 15.24 cm (6 in), and 30.48 cm (12 in), respectively. 
28

Figure 16. Net inductance of the trace as a function of the trace width w and the height over the 

 ground h for trace length l of 2.54 cm (1 in). 
29

Figure 17. Net inductance of the trace as a function of the trace width w and the height over the 

 ground h for trace length l of 15.24 cm (6 in). 
30

Figure 18. Net inductance of the trace as a function of the trace width w and the height over the 

 ground h for trace length l of 30.48 cm (12 in). 
31

inductance. Basically, the only measured data are that of Ott [33] and Leferink [34]. In [33], Ott 

measured the ground plane inductance for a trace of w = 2.54 mm (100 mils), h = 1.575 mm (62 

mils), and l = 15.24 cm (6 in). In these measurements, Ott determined the ground plane 

inductance for a 2.54-cm (1 in) segment at the center of the 15.24-cm (6 in) line. By performing 

the measurements at the center of the line, the end effects can be minimized. The current at the 

ends of the line is forced to constrict (due to vias), which causes the inductance at the ends to 

increase. (The increase of inductance due to current constriction at the ends is discussed in the 

next section.) For the above geometry, Ott measured a net inductance of the ground plane Lgr to 

be approximately 0.5 nH. Using our model, we calculated the ground plane net inductance Lgr for 

this same geometry to be 0.42 nH. Ott indicates that his measurement error is about ±20%. Based 

on this margin of error, our results correlate well with Ott’s measurements. 

Leferink [34] measured the ground plane inductance for a length of l = 0.3 m (11.81 in), h = 1.6 

(63 mils), and w = 2.2 mm (87 mils). Leferink’s so-called immunity method determined the 

ground plane inductance to be 0.96 nH. We calculated the ground plane net inductance Lgr for 

this geometry to be 0.91 nH. Once again, good correlation between the theory and measurement 

was demonstrated. 

Notice that the geometry that Leferink investigated has essentially the same height and width of 

that investigated by Ott. The main difference is that Leferink measured the net inductance for a 

length of l = 0.3 m, which is about a factor of 10 larger than in Ott’s example. The good 

correlation to both Ott’s and Leferink’s measurements illustrates that our results properly 

account for the behavior of the net inductance as a function of length. 

Comparison of a third experiment by Dockey and German [17] allowed us to investigate how 

well our model can predict differential ground plane inductance. Dockey and German [17] 

measured the difference between the emissions of two traces with different heights. In the first 

trace, h = 1.575 mm (62 mils) and w = 2.286 mm (90 mils); and in the second trace, h = 0.1778 

mm (7 mils) and w = 0.381 mm (15 mils). The width of the traces was changed to ensure that the 

impedance of both the traces was the same. Dockey and German measured on average a 12- to 

13-dB reduction in emissions for the smaller h value. The reduction in the height corresponded to 

a small cross-sectional area of the current loop (the trace and ground plane loop). The new crosssectional area was only 11% of the original area. Since the emissions due to the differentialmode currents are proportional to the cross-sectional area of the loop, these measured reductions 

in emissions cannot be accounted for by the differential-mode current (the loop antenna), and are 

associated with common-mode emissions (or a change in the ground plane net inductance Lgr).

For a trace height of 1.575 mm (62 mils) and a width of 2.286 mm (90 mils) our model 

predicted a ground plane inductance of 0.42 nH; and for a trace height of h = 0.1778 mm 

(7 mils) and a width of 0.381 mm (15 mils), our model predicted a ground plane inductance 

of 0.09 nH. If it is assumed that the voltage drop across the ground plane is the
32

dominant emission source, then the ratio of these two calculated ground inductances would 

indicate about a 13-dB reduction in emissions. Our model correlates well with the measured 

differential ground plane inductance of Dockey and German. 

For a final validation of the results presented here, we calculated the inductance of the microstrip 

line with net and partial inductance. An expression for the inductance of a microstrip line with an 

accuracy of 2% was derived by Wheeler [35] and is given by: 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + + +

2

2

2

2

2

0

8

1 1 32 ln 1

4 h

w

w

h L l TEM

π

π

μ . (65) 

The inductance of the microstrip line can also be approximated with net inductance by summing 

the net inductance of the ground Lgr and trace Ltr: 

L ≈ Lgr +Ltr . (66) 

Figure 19 shows a comparison of the inductance of a microstrip line obtained from these two 

expressions for l = 30.48 cm (12 in) and various values of h and w. This comparison shows that 

the deviation of the inductance of the microstrip line obtained by equation (66) from that of 

equation (65) ranges from 2-5%. Secondly, notice that the net inductance of the trace Ltr

dominates the net inductance of the ground plane Lgr and is the dominant term in the microstrip 

inductance given in equation (66). 

7. EXCESS INDUCTANCE DUE TO CURRENT CONSTRICTION 

A discussion is in order concerning the effects of current constriction on the ground plane at the 

ends of a trace. In this study, we used the ground plane current distribution given in [24] (see 

equation (16)) which was derived assuming that the length of the microstrip was infinite. That is, 

effects of the ends were not taken into account. 

On the ground plane, the current behaves as follows: in the region away from the ends (Region B 

in Figure 20), the current spreads out from under the trace and is governed by equation (16). In 

the regions close to the ends (Regions A and C in Figure 20) the current on the ground plane is 

forced back under the trace. This occurs because the current on the ground is forced into the via 

that connects the trace to the ground. Figure 20 shows how the current flows in these three 

regions. 

The constriction of the ground plane current at the ends of the trace causes the inductance of the 

line to increase. The calculations in the previous section do not account for the added inductance 

due to current constriction. An exact calculation of this added inductance requires solving a 

three-dimensional electromagnetic problem. Kok and De Zutter [36] and Swanson [37] analyzed 

a similar problem found in MIMIC circuits and indicate that this type of calculation can be very 

involved. A detailed calculation of this added inductance due to the current constriction at the 

end of the trace will be a topic of future investigation. 
33

Figure 19. Inductance of a microstrip line as a function of the trace width w and the height over 

 the ground h for trace length l of 30.48 cm (12 in). 
34

Figure 20. Illustration of how the current of the ground plane is forced to concentrate at the end 

 due to the via. 

In this section, a qualitative discussion of the added inductance is given for the case when h << 

w. For convenience, it is assumed that the via from the trace to the ground plane is simply an 

extension of the trace itself (Figure 21). The current on the ground will then be forced inward at 

the ends. At the end of the trace, the current on the ground plane will be distributed within the 

±w/2 region. We showed [24] that the amount the current on the ground plane spreads out from 

the edge of the trace is decreased by making h small. If h is small, the current is constricted 

under the trace and very little current spreads out from the ±w/2 region. For small h values, the 

current distribution close to the end (or via) is the same as the current distribution far from the 

end. The distribution of the current on the ground plane is uniform over the whole length of line. 

Therefore, for small h values, the added inductance associated with current constriction at the 

ends is negligible. 

Finally, it should be noted that besides the added inductance caused by the current constriction, 

there is an additional inductance at the end of the trace that is associated with the connection of 

components and devices to the trace. This inductance may also need to be considered. 

8. CONCLUSION 

In this report, a model for calculating the net inductance of a ground plane and the trace 

was presented. The calculation of these inductances is based on a previously derived 

expression for the current distribution on the ground plane. Due to the complicated integrals 

in the formulas given here, it is not possible to develop closed form expressions for these 

net inductances. Given, however, are curves for the calculated net inductances (as well as the 

self- and mutual-partial inductance) of the ground plane and trace for various trace geometries of 

practical interest. These curves can be used in designing printed circuit boards. Results
35

Figure 21. Illustration of a via of width w. 

from this model were compared to three different sets of experimental data and excellent 

correlation was demonstrated. 

It was shown that the net inductance of the ground plane is affected more by changes in the 

height h of the trace than by changes in the width w of the trace. Moderate changes in the height 

h of the trace can drastically decrease the net inductance of the ground plane, and more 

importantly can dramatically reduce the voltage-drop across the ground plane, and in turn lower 

the printed circuit board emissions. In summary, emissions from the printed circuit board trace 

can be reduced by making h smaller because of the following two reasons. First, the added end 

inductance due to current constriction is negligible, and second, the smaller the value of h, the 

smaller the net inductance of the ground plane. 

9. ACKNOWLEDGMENTS 

The authors would like to thank R.F. German of German Training and Consulting, LLC for his 

helpful discussions, encouragement, and technical assistance. 
36

10. REFERENCES 

[1] C.R. Paul, Introduction to Electromagnetic Compatibility, New York: John Wiley & 

Sons, Inc., 1992, chapter 2. 

[2] Ibid., chapter 13. 

[3] D.M. Hockanson, J.L. Drewniak, T.H. Hubing, T.P. Van Doren, F. Sha, and M.J. Wilhelm, “Investigation of fundamental EMI source mechanisms driving common-mode 

radiation from printed circuit boards with attached cables,” IEEE Trans. Electromag. 

Compat., vol. 38, no. 4, pp. 557-566, 1996.

[4] op.cit. Paul, chapter 8. 

[5] H.W. Ott, Noise Reduction Techniques in Electronic Systems (second edition), New 

York: John Wiley Interscience, 1988, chapter 11. 

[6] J.R. Bergervoet, “EMC measurements and models connecting the system level with the 

module level,” Philips J. Res., vol. 48, no. 1-2, pp. 63-81, 1994. 

[7] C.R. Paul, “A comparison of the contributions of common-mode currents and 

differential-mode currents in radiated emissions,” IEEE Trans. Electromag. Compat., vol. 

31, no. 2, pp. 189-193, 1989. 

[8] C.R. Paul and D.R. Bush, “Radiated emissions from common-mode currents,” in Proc. 

1987 IEEE Internat. Symp. Electromag. Compat., Atlanta, Ga., 1987, pp. 197-203.

[9] R.F. German, H.W. Ott, and C.R. Paul, “Effect of an image plane on printed circuit board 

radiation,” in Proc. 1990 IEEE Internat. Symp. Electromag. Compat., Washingtion, 1990, 

pp. 284-291. 

[10] J.T. Fessler, K.W. Whites, and C.R. Paul, “Effect of image plane dimensions on radiated 

emissions,” in Proc. 1992 IEEE Internat. Symp. Electromag. Compat., Anaheim, CA, 

1992, pp. 106-111. 

[11] T.A. Jerse, C.R. Paul, and K.W. Whites, “The effect of finite image plane width on the 

radiation from an electric line source,” in Proc. 10th Internat. Zurich Symp. and

Technical Exhibition Electromag. Compat., Zurich, 1993, pp. 201-206. 

[12] T.S. Smith and C.R. Paul, “Effect of grid spacing on the inductance of ground grids,” in 

Proc. 1991 IEEE Internat. Symp. Electromag. Compat., Cherry Hill, New Jerry, 1991, 

pp. 72-77. 

[13] op.cit. Ott, chapter 10. 
37

[14] R.F. German, “Use of a ground grid to reduce printed circuit board radiation,” in Proc. 

6th Symp. and Technical Exhibition on Electromag. Compat., Zurich, 1985, pp. 133-138. 

[15] H.W. Ott, “Digital circuit grounding and interconnection,” in Proc. 1981 IEEE Internat. 

Symp. Electromag. Compat., Boulder, CO, 1981, pp. 292-297. 

[16] R.W. Dockey, “Asymmetrical mode radiation from multi-layer printed circuit boards,” in 

Proc. 1992 EMC/ESD Internat., Denver, CO, 1992, pp. 247-251. 

[17] R.W. Dockey and R.F. German, “New techniques for reducing printed circuit board 

common-mode radiation,” in Proc. 1993 IEEE Internat. Symp. Electromag. Compat., 

Dallas, TX., 1993, pp. 334-338. 

[18] Title 47, Code of Federal Regulations, Part 15, Subpart B, section 15.109, October 1, 

1995. 

[19] A.E. Ruehli, “Inductance calculations in a complex integrated circuit environment,” IBM 

J. Res. Develop, vol. 16, no. 5, pp. 470-481, 1979. 

[20] C.R.Paul, “Modeling electromagnetic interference properties of printed circuit boards,” 

IBM J. Res. Develop., vol. 33, no. 1, pp. 33-50, 1989. 

[21] C. Hoer and C. Love, “Exact inductance equation for rectangular conductors with 

applications to more complicated geometries,” J. Res. Nat. Bureau of Standards-C. Eng. 

Instrum., vol. 69C, no. 2, pp. 127-137, 1965. 

[22] F.W. Grover, Inductance Calculations: Working Formulas and Tables, New York: Dover 

Publications, Inc. (by the Instrument Society of America), 1973. 

[23] F.B.J. Leferink and M.J.C.M. van Doorn, “Inductance of printed circuit board ground 

planes,” in Proc. 1993 IEEE Internat. Symp. Electromag. Compat., Dallas, TX., 1993, pp. 

327-329. 

[24] C.L. Holloway and E.F. Kuester, “Closed-form expressions for the current density on the 

ground plane of a microstrip line, with application to ground plane loss,” IEEE Trans. 

Microwave Theory and Tech., vol. 43, no. 5, pp. 1204-1207, 1995. 

[25] C.L. Holloway, “Edge and surface shape effects on conductor loss associated with planar 

circuits,” MIMICAD Technical Report No. 12, Electromagnetic Laboratory, Department 

of Electrical and Computer Engineering, University of Colorado at Boulder, Appendix G, 

1992. 

[26] M. Wien, “Ueber die Berechnung und messung kleiner selbstpotentiale,” Annalen der 

Physik und Chemie, band 53, heft 5, pp. 928-947, 1894. 
38

[27] E.B. Rosa, “The self and mutual inductance of linear conductors,” Bulletin of the Bureau 

of Standards, vol. 4, no. 2, pp. 301-344, 1907.

[28] E.B. Rosa and F.W. Grover, “Formulas and tables for the calculation of mutual and selfinductance,” Bulletin of the Bureau of Standards, vol. 8, no. 1, pp. 1-230, 1912. 

[29] C.T.A. Johnk, Engineering Electromagnetic Fields and Waves, New York: John Wiley & 

Sons, 1975, chapter 5. 

[30] C.L. Holloway and E.F. Kuester, “Edge shape effects and quasi-closed form expressions 

for the conductor loss of microstrip lines,” Radio Science, vol. 29, no. 3, pp. 539-559, 

1994. 

[31] C.G. Harris and W.A.B. Evans, “Extension of numerical quadrature formulae to singular 

behaviors over finite intervals,” Intern. J. Computer Math., Section B, vol. 6, pp. 219-

227, 1977. 

[32] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes: 

The art of Scientific Computing, New York: Cambridge University Press, 1986, Chapter 

4, pp. 120-126. 

[33] H.W. Ott, “Empirical determination of ground-plane inductance,” Regional Symp. on 

EMC, IEEE EMC Society, Rocky Mountain Chapter, Boulder, CO, 1994. 

[34] F.B.J. Leferink, “Inductance calculations; experimental investigations,” in Proc. 1996 

IEEE Internat. Symp. Electromag. Compat., Santa Clara, CA., 1996, pp. 235-240. 

[35] H.A. Wheeler, “Transmission-line properties of a strip on a dielectric sheet on a plane,” 

IEEE Trans. Microwave Theory and Tech., vol. 25, no. 8, pp. 631-647, 1977. 

[36] P.A. Kok and D. De Zutter, “Scalar magnetostatic potential approach to the prediction of 

the excess inductance of grounded via’s and via’s through a hole in a ground plane,” 

IEEE Trans. Microwave Theory and Tech., vol. 42, no. 7, pp. 1229-1237, 1994. 

[37] D.G. Swanson, “Grounding microstrip lines with via holes,” IEEE Trans. Microwave 

Theory and Tech., vol. 40, no. 8, pp. 1719-1721, 1992. 
39

APPENDIX: GENERAL PURPOSE 10-POINT QUADRATURE WEIGHTS 

The weights Ai and abscissae xi used in this report are from Harris and Evans [1] and are listed in 

the Table below. These are the values used for the integrals with both the singular and 

nonsingular kernels. 

General Purpose 10-point Quadrature Weights Ai and Abscissae xi Values 

xi Ai

0.22950371738283398583 

0.63647584009176348145 

0.90150720533183638718 

0.992838312235203529446 

0.9999843442623408409287 

0.45011008253896641997 

0.34830268517741692339 

0.17446797661827909007 

0.026962997721603785257 

0.0001562579437337813003 

REFERENCE 

[1] C.G. Harris and W.A.B. Evans, “Extension of numerical quadrature formulae to singular 

behaviors over finite intervals,” Intern. J. Computer Math., Section B, vol. 6, pp. 219-

227, 1977. 
40