Aperture Coupling Method for EMI Analysis
of Microwave Circuits within Multilayer Cavities
T. Yang1 and J. L. Volakis1,2
1
Radiation Lab., EECS Dept., The University of Michigan, Ann Arbor, MI 48109
2
ElectroScience Lab., The Ohio State University, Columbus, OH 43212
taesiky@umich.edu, volakis.1@osu.edu
Introduction
With increasing density of RF/microwave integrated circuits and systems, printed circuit board
(PCB) becomes more multilayered and metallized and demands more vertical structure. This trend
makes electromagnetic interference (EMI) and signal integrity analysis of such systems a
challenging task. Hence, efficient modeling and accurate prediction of EMI effects on the high
speed RF/microwave integrated circuits is becoming an important design requirement.
This paper presents a simple and efficient approach for the EMI analysis of PCB enclosed within
cavities having multilayered substrates, possibly separated by metallic plate with apertures. The
cavity Green’s function is used to model the enclosure structure and internal circuit components
and the method of moment (MoM) is applied (referred to as the MoM-GF technique). Specifically,
the method employs magnetic currents on the non-metallic region at each layer interface instead of
electric currents on the metallized portions. By employing the surface equivalence principle, each
layer is separated and modeled as a single cavity (see Fig.1). That is, the original multilayer cavity
is replaced by cascaded cavities. Because only single cavity is considered in the proposed method,
there is no need for a multilayer cavity Green’s function (GF) whose analytical derivation becomes
quite inefficient and time consuming with a large number of layers. In addition, for such cases in
which there exist vertical interconnects and vias connected to planar passive circuits enclosed in a
multilayer cavity, the aperture coupling method does not require (electric) junction current
modeling. This is because the proposed method considers only magnetic currents in the layer
interface and, hence, there is no planar electric current to be otherwise coupled to vertical metals.
Thus, for each single cavity, horizontal magnetic currents at the apertures (or non-metallic areas)
and vertical electric currents at the vias/loads are unknowns to be determined.
Compared to a method using electric currents on the planar passive microwave circuits placed in
the layer interface, the present method is more efficient when the cavity of interest is composed of
multilayered substrates, large planar metallic portions in each layer interface, and numerous
vertical interconnects embedded in each layer.
Formulation
To illustrate the approach, consider the general geometry consisting of cascaded interior cavities or
a cavity with layered substrates separated by metallic interfaces containing an arbitrary distribution
of apertures as shown in Fig.1 (a). For electromagnetic wave penetration analysis, we employ the
surface equivalence principle at the interface apertures and introduce equivalent magnetic current
densities over both sides of the PEC planes of the same but opposite magnitudes.
Referring to the configuration in Fig.1, the computation of the magnetic currents at layer interfaces
are performed by enforcing magnetic field continuity as follows.
Top surface of layer 1:
H ti  H ta ( M 1 )  H tb1 ( M 1 )  H tb1 ( J1 )  H tb1 ( M 2 )
(1)
Bottom surface of layer N:
H tb ( N 1) ( M N )  H tbN ( M N )  H tbN ( J N )
(2)
Bottom surface of layer n (or top surface of layer n+1):
H tbn ( M n )  H tbn ( J n )  H tbn ( M n 1 )  H tb ( n 1) ( M n 1 )  H tb ( n 1) ( J n 1 )  H tb ( n 1) ( M n  2 ) (3)
where Jn refers to the electric currents on the vertical posts or via within the nth layer. Thus,
tangential electric field must be zero on the vias of the nth layer.
Etbn ( M n )  Etbn ( J n )  Etbn ( M n 1 )  0
(4)
The above equations are discretized using roof-top basis functions for representing magnetic
currents and pulse basis for the vertical interconnects/posts along with Galerkin’s testing. Doing so,
we obtained the general matrix system
i
  M 1     I  
 YMa Mb1  YMb1J  YMb1M   0


0 0
0


 1 1   1 1   1 2 
   J     0 
  Z Jb1M   Z Jb1J   Z Jb1M   0
 1  
0
0

0





1 1 
1 1  
1 2 

   M 2     0 


 Y b1  Y b1  Y b1 b 2  Y b 2  Y b 2   0

0    J    0 
  M 2 M1   M 2 J1   M 2 M 2   M 2 J 2   M 2 M 3 

 2  
b2
b2
b2
  0








0
Z
Z
Z
0



0
   J2 M 2   J2 J2   J2 M3   
     M 3    0 




 
b2
b2
b 2  b3
b3








0
0
Y
Y
Y
Y



0
J

0
   M 3 M 2   M 3 J 2   M 3 M 3   M 3 J3 
    3      
  


 

 0
0 0  Z Jb33M 3   Z Jb33J3    
0      
  0

     







 

 

    


b ( N 1)  bN
bN
 YM N J N 




   YM N M N
  0
 M    0 

 N  

bN
bN




0















Z
Z


M
M
M
J

 N N   N N    J N     0 

 
 

where all matrix elements take the form of double modal series and the details are available in [1].
It is interesting to note that by invoking reciprocity, coupling admittance matrices YMbij Ji  and


YMbi M  can be shown to be equal to the negative transpose of  Z JbiM  and YMbi M  ,
 i j
 j i
 i j
respectively, for j = i or i+1.
Results
To validate the technique, coupled voltage is computed at the 50Ω load of interdigital filter
enclosed within a rectangular cavity having a slot illuminated by Ey polarized plane wave
incidence. The details of all dimensions are given in Figs.2 and 3 where the cavity and enclosed
filter geometry is identical except that for Fig.3, to increase structure complexity, a second plate
containing three rectangular apertures is added inside the cavity. The present method (Magnetic
GF in legend of Figs.) is compared with HFSS (Ansoft) and another MoM-GF method [1] (Electric
GF in legend of Figs.) using electric currents in the metallic parts of the layer interface. In all cases,
close agreement is exhibited for broad frequency band. We observe two large coupling peaks at
around 2GHz and 6 to 8 GHz, corresponding to the resonances of the filter and the cavity plus
aperture. The overall coupled voltages in Fig. 2 are higher than those in Fig. 3 where the second
set of apertures is non-resonant at the frequency of interest.
Conclusion
A simple and efficient technique is presented to model PCB enclosed within a cavity having
multilayer substrates. The method employs magnetic currents in non-metallic region of layer
interface instead of electric currents in planar metallic parts and is only associated with single layer
cavity GF. Examples were given to demonstrate the validity of the method.
(Ei, Hi)
M1
-M1
J1
(ε1, μ1)
Layer 1
M2
-M2
J2
(ε2, μ2)

M3
Layer 2








-MN
(εN, μN)
JN
(a)
Layer N
(b)
Fig. 1. (a) General geometry for the multilayered cavities with apertures and passive
microwave circuits under EMI illumination. (b) Connectivity of the cascaded cavities via
the surface equivalence principle.
-3
Enclosed Interdigital Filter
x 10
7
Magnetic GF
Electric GF
HFSS
6
Induced Voltage (V)
5
Ki
Ey
x
y
4
z
0
ε1
50 Ω
d1
3
ε2
d2
2
1
0
1
2
3
4
5
Frequency (GHz)
6
7
8
Fig. 2. Voltage at the load of an interdigital filter enclosed a cavity. The dimensions are given as
follows: cavity (29 x 40 x 4), aperture (19 x 2), strip (1 x 20), d1=1, d2=3, ε1=1, ε2=3.48 in
[mm].
-3
7
x 10
6
Ey
Voltage at the output (V)
z
5
y
Ki
x
A1
ε1
4
3
2
Magnetic GF
Electric GF
HFSS
A2
A3
d1
A4
ε2
ε3
d2
d3
1
0
2
3
4
5
Frequency (GHz)
6
7
8
Fig. 3 Coupling onto the microwave filter placed below the second set of apertures. The
dimensions of the apertures and the cavity are given as follows: cavity (29 x 40 x 4), aperture #1
(19 x 2), aperture #2 (3 x 16), aperture #3 & 4 (15 x 2), d1=2, d2=d3=1, ε1=ε2=1, ε3=3.48 in
[mm]
References
[1] Taesik Yang, “Coupling onto RF Components Enclosed in Canonical Structures”, Ph.D.
dissertation, EECS Dept., University of Michigan, Feb. 2006