GROUND PLANE IMPEDANCE ANALYSIS
OF
PRINTED CIRCUIT BOARD
G. M. Gonzsilez, R. Linares Y M, J. de la Rosa and W. H. Fonseca
ESIME-SEPI-IPN, National Polytechnic Institute of Mexico, Unidad Profesional Adolf0 Ldpez Mateos, Ed& 5, 3er.
Piso, Cal. Lindavista, C.P. 07738, Mtkico, D. F. Tel+Fax: (011525) 729-6000 ext. 54622. E-mail:
rlinares@nava.esimez.ipn.mx
Abstract- In this paper a theoretical and experimental analysis
of ground planes impedance for printed circuit boards is
presented. A frequency domain analysis of the impedance on
flat conductors is done. The realistic electromagneticeffects are
considered: the skin effect due to fmite conductivity of the
plane and the confinement effect of the distribution of current
lines at the contact points. The analytical results were proved by
different experimental methods. The experimental data and the
analytical results are in good agreement.The analytical model
may be used to design the ground plane in printed circuit board
with a good accuracy.
I. INTRODUCTION
Ground planes at electronic systems play an important role in
EMC problems. Safety reasons are the major purpose for
ground plane. But careless design can produce interference
problems. Different methods can be used to eliminate the
interference problems due to ground planes under different
situations, such as: single point reference, multiple point
reference and hybrid reference [l-3]. The basic problem is the
high impedanceof the ground plane, therefore the knowledge of
the behavior of this impedanceis required.
An important effect on internal impedanceof any conductor is
the skin effect, it has been examined in [4,5]. Recently [6] has
been shown that the internal resistance and the internal
inductive reactance are equal in circular cross section
conductors, but in rectangular cross section conductors this
parameterdependon the current distribution in accordancewith
the geometry of the conductors. That is not new, numerous
papers have analyzed the distribution of currents at different
frequencieson flat conductors [7-IO].
In modern high-sensitivity-density electronic equipment
working at high frequency a good ground plane is fundamental.
This topic is of great concern for the electromagnetic
compatibility (EMC), more particularly with coupling problems
between digital or analog circuits via a common ground plane.
The conventional notation of ground plane is valid at low
frequency or dc performance,since all conductors have a finite
impedance. But at high frequency, ground plane impedance is
not ideal and can produce unwanted voltages, which may
provoke malfunctions of the circuits. This paper presents a
theoretical and experimental analysis of the ground plane
0-7803-5057-X/99/$10.00 © 1999 IEEE
impedancefor realistic condition. Here, in accordancewith the
classic electromagnetic theory, a frequency domain analysis is
done. As results, the internal impedancefor a PCB ground plane
is obtained. The format of this paper is as follows: In section II,
the analytical formulation is introduced. In section III a
numerical evaluation is presented and in section IV the
experimental and the theoretical results are compared are
commented.
II. ANALYTICAL
FORMULATION
Maxwell began the analysis of high frequency currents in
conductors, a summary of the study of this phenomenon has
been developed in [l 11. In relation to the resistance of
rectangular conductors experimental data for different
dimension (width to thickness ration) are presentedin reference
[12]. The analysis of the internal impedanceof flat conductors
starts with Maxwell’s equation, in references [4-71 numerical
solution are given to this problems
The ground plane impedance can be derived analytically from
the model representedin figure 1, where the conductivity c
and the permeability p are considered homogeneous and
isotropic.
Figure 1. Groundplane
Using the elemental Ohm’s law equation
V=-
712
s
L
E.dL
where V is the electric potential [VI; E the electric field [V/m];
dL the differential path length, [ml; I the current [A]: and Z the
impedance[a 1.
In accordance with the boundary conditions, the equations
solutions can be referred as the Helmholtz’s expressions,which
in reference [13] have been solved by different numerical
methods.In this case
E
In this case, the electric field can be calculated from two
dimensions diffusion equations [4,5]
v2E,(x,y)-
jw,uoE,(x,Y)=O
V2E,(x,y)-j~~raE~(x,y)=0
G-4
=E
I
r(.,,~(L-y))+jc(.~(L-y))
I--
-m-T-
r( .$&G x) + j c(Jqu 0 x)
r<.,,G ~4+j c<.;G Y)
E 11=E2
r<,Gx>+jc<,Cx)
(2b)
(5)
The general solution of diffusion equations by different
methods is given in [13]. For the case of a ground plane the
solution applying the principle of superposition can be obtained
using the figure 2.
E
r(JGy)+jc(~Y)
III = E3
r(Gx)+jc(JwD
E
x)
r(,hGu)+jc<,fiY)
IvzE4 r(J&G(a-x))+
jc( JijZ(a-x))
where
r[,,fi
[ w o variable)/212
(variable)]= 2 cos(nrr’2) ” r.,,,’
a, sen(nn /2)[ ,,/w,Dcs(variable)y / 2p”
~(nlj2
C[\~CD,D
c (variable)]= c
n=O
(6)
As ground planes are referencedfor single point or multi point,
we consider now that the ground plane is feed by a current
Z(jw) as the figure 3 shows. The current distribution can be
deducedwith the following boundary conditions
Figure 2. Superpositionprinciple
The solution of the problem can be the solution of Laplace’s
equation with the following boundary condition:
@x,0) = E,
E(x, L) = E,
-W,
Y> = E4
Eta, Y> =
-4
Then, in accordancewith superpositionprinciple,
E,,(qO)=&(3)
Ixl<b
for El
E,,(x,O)=O
b<jxl<a
E,(+a,y)=O
E,(x,O) = 0
O<y<L
forE2and E4
/xl < b for El
E,(x,L)=O
lxl<a
713
for EI
for E3
coth( ,/SF
c
1
1
a) *
O” [cos(nn /2) + jsen(nn /2)]( J&GO2)2”
2n+l
p,”
n=O
L2n+1- (-L)2n+*
rt,bGa+jct,bGa)
Confinement of
cure
m [cos(nn /2) f jsen(nn /2)]( J&G02)2n
c-
+A
n=o
w2
~_~~
(2L)2mn+’
2n+l
r(,Ga)+jc(&Ea)
1
(10)
Figure 3. Groundplane geometry.
The voltage on the ground plane can be calculated by the
equation (9) and their impedance by equation (IO), the skin
effect and the confinement effect are included.
Applying these boundaries conditions to equation (2) the
electric field component in y can be obtained by the method of
separation of variables, then:
cW~~x)
*
I
~_
EJx,y)=I
~-
senh(fi,u
46
-
x) + j c(-&lj&
r<&G
+-I
20 bt
!
b=1x10m3 m
,
thickness
1~10~~ m,
a,
the
,I.I~ = 4~ x10m7 H/m and 0 = 5.8~10~ S/m. Applying
equation (10) for different dimensions of L and a the results
obtained are shown in figure 4.
(Y>>
x) I
(Ll,>>,.M&aL-y))
r(..&jG
EVALUATION
We consider for numerical evaluation, a copper rectangular
ground plane with the following characteristics: length L, width
0 a)
(u>>+j CC&G
r( ,/&E
III. NUMERICAL
x) + j c( .&jz
x)
I
50
45
(9)
5
;:
Substituting (9) into (1 a) and integrating from 0 to 2L the
voltage is obtained.
[cos(m /2) + j sen(nn /2)]( ,!G
/ 2)2n
Figure 4. Evaluation of groundplane impedance
IV. EXPERIMENTAL
r(*a)+jc(&$Ea)
Finally the ground plane impedance can be written as:
RESULTS
The behavior of the ground plane impedance was measured on
typical glass- epoxy printed circuit board with a copper side of
0.3 mm thickness. The dimensions are 1Ocm x 27cm.
Measurements were made in accordance with the diagram
shown in figure 5.
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It can be seen that figures 6 and 7 show differences. That is
because the source impedance and the measurement set
impedance are not totally frequency compensated. But, the
behaviors are similar.
CONCLUSION
Figure 5. Schemefor measurements of groundplane
impedance
The impedance ZG~ corresponds to the ground plane
impedance. The source impedance Z, and measurement set
Z, are compensatedin frequency until 500 MHz. The voltage
V, is measuredthrough Z, and ZG~ is obtained by equation
(11).
zGP
= k
! vm ) - lhn
- z,
(11)
The behavior of the ground plane impedance as of the
frequency function is shown in figure 6. Also the impedanceof
the ground plane was measuredwith a network analyzer , the
result is shown in the figure 7.
Frequency (MHz)
Figure 6. Ground Plane impedance k4easured
Figure 7. Ground Plane Measured with network analyzer.
This paper introduces an analytical method to estimate the
impedance behavior of the ground planes in the frequency
domain. The method was validated experimentally, where
analytical and experimental results are in good agreement.The
impedance magnitude is a function of the ground plane
dimensions. The results show that a 1Ocmx 30cm ground plane
can be used until a maximal frequency of 100 MHz, as figures
4, 6 y 7 show. Finally, these results may be used to obtain the
dimensions of the ground plane in printed circuit boards with a
good accuracy
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