of Modeling Power Distribution Networks for Mixed Signal Applications
Joong-Ho Kim*, Madhavan Swamhathan", and Youngsuk Sub**
*Dept. of Electrical and Computer Engineering, Georgia Institute of Technology Packaging Research Center 777 Atlanta Drive N.W., Atlanta, GA 30332-0250, USA
TEL:[phone], FAX: [phone] E-mail: [e-mail] and [e-mail] **Dept. of EECS, YeungNam University, KyongSan, 721-749, South Korea
Abstract: This paper presents a method for analyzing irregular shaped power distribution networks both in the fiequency and time domain for mixed signal applications. Using a two dimensional array of distributed RLCG circuits, the impedances of a power/ground plane pair are computed. For the efficient computation of the power distribution impedances at specific points in the network, a multiinput and multi-output transmission matrix method has been used. This method is 7X-13X faster than Spice and saves memory requirements. the results have been compared with Spice, in which T model was used for simulation. Using Inverse Fast Fourier Transform (FIT),a transient response in the time domain has also been generated from the frequency domain data. MODELING POWEWGROUND OF PLANE USING UNIT CELLS
A power/ground plane can be divided into unit cells with a lumped
INTRODUCTION
Mixed signal applications contain a combination of digital, RF and analog circuits. An important area in high-speed digital systems is the design of the power and ground planes arising in power distribution networks. A major challenge in the design of planes, which forms an integral part of the power delivery system (PDS) for gigahertz (GHz) packages and board, is the supply of clean power to the switching circuits. As clock speeds increase, and signal rise time and supply voltages decrease, the transient currents injected into the power distribution planes can cause voltage fluctuations and circuit delays [l]. This leads to unwanted effects on the PDS such as ground bounce, power supply compression and electromagnetic interference. An inherent property of the PDS is that it has a low impedance over the entire bandwidth of the signal so that the transient currents do not cause excess noise on the power distribution network. To predict and suppress this noise, efficient noise prediction methods are necessary. In a realistic packagehoard, which consists of numerous vias, decoupling capacitors, irregular geometries and multiple plane layers, the number of transmission line segments required may become very large, requiring large memory and CPU time for analysis. The transmission matrix method discussed in this paper offers a more efficient technique for solving these kinds of problems. It is based on a multi-input, multi-output transfer function which enables the matrix for the entire power distribution network to be computed as the product of the individual square matrices formed by 2N-port networks having N input ports and N output ports [2]. Once the unit cell parameters are computed, the transmission matrix method can be efficiently applied to any arbitrary shaped plane geometry. This paper is an extension of [3] and [4] when the unit cell has been modified and the method has been applied to arbitrary shaped power delivery systems arising in commercial mixed signal applications such as pagers, cellular phones etc. This paper discusses the use of the transmission matrix method with modified unit cells ( model) Il ' for computing impedances without decoupling capacitors or with decoupling capacitors between 2 or more ports for irregular geometries arising in power distribution networks. Where applicable,
element model for each cell, as described in [SI.Each cell consists of an equivalent circuit with R, L, C, and G components, as shown in Figure 1 for a rectangular structure.
Figure 1. (a) Plane structure (b) unit cell and equivalent circuit
The equivalent circuit parameters for a unit cell can be derived from quasi-static models provided the dielectric separation (d) is much less than the metal dimensions (a, b) [2], which is true for power/ground plane pairs. From the lateral dimension of a unit cell (w), separation between planes (d), dielectric constant ( E ), loss tangent of dielectric
( tan(6) ), metal thickness (t), and metal conductivity ( a ), the , equivalent circuit parameters of a unit cell can be computed as:
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0-7803-6569-0/01/$10.00 0 2001 IEEE
1117
In the above equation, E, is the permittivity of free space, p , is the , permeability of free space, and 6 , is the relative permittivity of the dielectric. The parameter Qc is the resistance of both the power and ground planes for a steady D.C. current where the planes are assumed to be of uniform cross section. The ac resistance RUC accounts for the skin effect on both conductors. The shunt conductance Gd represents the dielectric loss in the material between the planes.
Using the unit cell, a distributed network of RLCG elements can be generated for rectangular planes, as shown in Figure 1. In the figure, a total of N x M unit cells have been used to represent the rectangular plane. Since this is a circuit model, it can be simulated in Spice by generating the Modified Nodal Analysis (MNA) equations. To obtain good accuracy, a unit cell size that is 10 times less than the wavelength at the highest frequency of interest was used. For the rectangular structure, the T model was used in Spice for simulation, as described in [3].
(3) where [TA],[TB],[Tc], and [TD]are N x N matrices. In Eq. (3), the [ Imatrix is of the form: T
00 i
2zro
...
0 0-
...
00
...
...................................
00 00
01
i 00
i i
10
...
00
TRANSMISSION MATRIX METHOD
XYaYbO YbYcYb
As shown in Figure 1, using a distributed network of RLCG elements, the rectangular plane can be divided into N x M unit cells. Consider a column of unit cells ( Nx 1 unit cells), which is shown as a dashed line in Figure 1. The N x 1 unit cells can be represented as a 2N x 2N matrix formed by N input ports and N output ports. This is shown in Figure 2 for the n equivalent circuits for the unit cells.
... ... 0 0 01 00 . .. . . . . . ... ... .. . .
.,
'
00
00
Yc Yb : 0 0
YbYu
'
,,,
10
01
00
As can be seen in Eq. (3A), the transmission matrix for a column of
unit cells is sparse, which enables reduction in memory and CPU time when applied to realistic structures. Using the 2 x 2 block matrix representation in Eq. (3), the transmission matrix can be used
Figure 2. Equivalent circuit for a column of unit cells
to
In Figure 2, the input and output ports are indexed as 1 to N and N+l 2N, respectively. The transmission matrix for the 2N-port network can be derived in terms of the node voltages and port currents as:
5
VN
1 '
-
r
l
Since the network is reciprocal, det[TJ=I. The 2 N x 2 N transmission matrix for the overall power distribution network which consists of a cascade of two or more networks can now be obtained by multiplying the individual matrices [2]. For the rectangular plane in Figure 1, since all the matrices for the column of unit cells are the same, the response of the entire geometry can be obtained as a single 2 N x 2 N matrix. The cascade connection of 2N-port networks is T 4. shown in Figure 3 using the [ Imatrix representation in Eq. ( )
3.1
"'
T1.N
:
TN.I
"'
...... = ..........................................
TN+~,I
*S.
TN,N
TN+SN
1
TN+I.N+I
'0'
TN+I,UI
_IN,
'
LTZN.1
..-
TW,N
Tlh',h'+I
."
?N.W]
where M=l+m+n Figure 3. Cascade connection
The above transmission matrix can be rewritten in the simpler form:
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For a cascade connection of 'M' [TI matrices where [q] ITn] and represent the input and output matrices of the entire structure, the total voltage and cumnt equationscan be derived as:
ZN+I.N+l z2N,N+1
'.'
'"
ZN+1,2N Z2N,2N
---
I, I;,, 1 2 -= - -I, = IOuf+ 14
where
-1 2 = fi Vjn
where I , = YnVmf
--
As mentioned earlier, since the network is reciprocal, Z B = Zc and ZLj = Z j , i , Using the relation between the transmission matrix and the impedance matrix, the impedance of the power plane becomes:
where
-
YI = T;/Tc1
-
Y,, = Tc,,Tz
[ z B ] = [ z C l = ~ ~ r l [ z A l = ~ ~ ~ z C [l z D 1 = [ z C i r b ]
(8)
A block representation of Eq. ( 5 ) showing the input and output voltage and current variables is shown in Figure 4.
During the design of the power delivery system, the impedance at specific points on the network is often desired. This can either be the self impedance at a port or the trans-impedance between ports. Using Eq. (8), the self impedance and trans-impedance can be computed as: Tc1.1 TC2,l
det
TAI,I zAi,j = LT;N,l
' jth
TLv3 ;".
TLN,N
'T I6 n
Figure 4. Block diagram of entire system
In Eq. (9,
and
are N x N input admittance matrices seen
...
.
looking into the I f h column from the input ports and (I + rn + column from the output ports, respectively. Using Eq. (3,the overall
2N x 2N transmission matrix
r
b']for the entire structure is:
1
where
[Mc,,~1 is the ( N - 1) x (N - 1)
matrix obtained fiom
k 1 by A
L
1
deleting the j f h row and the ifh column.
APPLICATION IRREGULAR GEOMETRIES TO
It is important to note that if index '1' or 'n' are zeros, Tyl or TYn are 2N x 2N identity matrices, which represent open circuits. Using the transmission matrix of the network, the 2N x 2N impedance matrix [Z] of the network can be derived, which can be simplified and represented as:
The transmission matrix modeling approach discussed in the previous section can be extended to arbitrary shaped geometries. As an example, to explain the approach of the transmission matrix, the method is applied to an L-shaped plane, as described in Figure 5. ---. ---..
' \~wger~ohunn
ZN,I
...
ZN,N
Figure 5. Top view of L shaped plane
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The modeling method for the L-shaped plane is similar to that for the rectangular plane with the difference that two different matrix sizes are required for the L-shaped structure. This is because the size of the [ Imatrix in Figure 3 is not constant as indicated by dashed lines in T Figure 5. Since the open circuit can be represented as an identity matrix (as explained earlier), the matrix size for the smaller column can be expanded to match the larger column in Figure 5. To fiuther illustrate the procedure, assume that the smaller column ) can be represented as a 2k x 2k square matrix ( [TShkxZk formed by 2k-port networks having k input ports and k output ports. let Similarly, the larger column be represented as a 2 N x 2 N matrix with N > k . The matrix representing the smaller column can be expanded and written as:
RESULTS
To check the accuracy of the transmission matrix method with modified unit cells (n model) shown in Figure 2, the results have been compared with Spice (T model) for a rectangular plane structure shown in Figure 1 (a). This structure is currently being used in Sun workstations. The test structure consists of a 10.6 by 8.2 rectangular pair of planes with 1 mil thick FR4 dielectric with relative permittivity E , = 4. The conductor planes are made of copper
( 6,= 5.8 x IO S/m) with a thickness of 1.2 mils and dielectric loss tangent tan(6) = 0.02 at 1 GHz. Using a unit cell size of 0.25 by 0.25, the PDS was divided into 3 3 x 4 2 unit cells. An excitation point (Port 1) was located at (x = 3, y = 1) and an observation point (Port 2) at (x = 6, y = 9). The frequency response of transimpedance for the plane is shown in Figure 6 and has been compared with Spice. The results from the two models show good correlation over a f?equency range 10 MHz to 1 GHz.
212 bnculamcn wlnl)
T s ~ 0 i TSD
where
[ ~ s j = [?A~ ~ ~ ?B] 2 ~ TSC TSD
j
lo
J
WtasuancYW)
As a result, using a single matrix size for the columns, the matrix of the smaller column is expanded to match the matrix size of the larger column, by changing the elements at the interface. After the matrix expansion, the impedance computation for the L-shaped plane is similar to the rectangular plane, as described in the previous section.
INCORPORATION OF DECOWLINC CAPACITORS
104 1 0
lop
In the transmission matrix method, decoupling capacitors can readily be included into the matrices. The impedance of a decoupling capacitor is represented using the following equation.
Figure 6. Rectangular plane without decoupling capacitors solid line :transmission matrix (TI model) dashed line : Spice (T model) Resonant frequencies caused by the reflection of incident waves at the plane edges result in build-up of energy between the planes, which can induce excessive simultaneous switching noise (SSN). To nullify the resonances and suppress simultaneous switching noise, decoupling capacitors with a low impedance response are often attached to the PDS. Since a capacitor is nonideal, the effective series resistance (ESR) the effective series inductance (ESL) values of a and capacitor affect the frequency response of the PDS. In this section, three different capacitors with measured ESR, ESL, and C values, which are shown in Table 1, were attached to the rectangular plane.
ZCw = R + j m L + .d i
where R is the Equivalent Series Resistance (ESR), L is the Equivalent Series Inductance (ESL), and C is the capacitance. The transmission matrix for decoupiing capacitors can be represented as follows:
1
1
1
where Y q j =zwp,i
and if there is no decoupling capacitor in the
C(F) O.le-6 1Oe-6 1500e-6
ESL (H)
0.5e-9 1e-9 7e-9
ESR (a)
0.05 0.02 0.01
ith
row, then Ycap,i = O
. As
an example, let the decoupling
capacitors be connected between the kfh column and the (k +I)* column of unit cells shown in Figure 1. Then the transmission matrix of the entire structurecan be updated as: (13) t ~ l m d .[7,1k x[Tlcup [TIR+I ... =
As mentioned earlier, the rectangular plane is a realistic structure which contains 675 decoupling capacitors of three different types, as shown in Table 1. Each capacitor was included into the transmission matrix, as described in Eq. (12) and (13). The trans-impedance were computed at the ports defined earlier, which are shown in Figure 7. The results were found to agree with Spice over a frequency range 1 kHz to 1 GHz.
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x 1oJ
I
-10
10
io
IO*
io
10
$8
1
-(O
.
0.2
.
0.4
.
0.6
.
0.8
1
.
1.2
*
1.4
.
1.6
.
1.8
I
2
Isl(iaPmCY (nt)
Time (-lid)
x 10
Figure 7. Rectangular plane with decoupling capacitors II model) solid line :transmission matrix ( dashed line :Spice (T model) Using Inverse Fast Fourier Transform (IFFT), a transient response in the time domain can be generated from the frequency domain data The equivalent circuit diagram using a two port impedance matrix is o shown in Figure 8. T compute the transient response of the plane, the time signal source is fmt changed to a fiequency-domain representation using Fast Fourier Transform (FFT). The frequency response at the output is next computed by evaluating the product of the transfer impedance obtained f o the impedance matrix and the im current source. The output transfer voltage is then converted to a time-domain representation using IFFT.
(b) Figure 9. Transient response of output voltage (a) without (b) with decoupling capacitors
As is typical of a resonant cavity, initially the rectangular plane builds up energy, then reaches the steady state and fmally decays to zero after the power is turned off, as shown in Figure 9 (a). The presence of the decoupling capacitors on the plane reduces the coupling between ports 1 and 2, as shown in Figure 9 (b).
Arbitram Geometrv
In consumer mixed signal applications, the power distribution structure is typically of irregular shape. As an example of an irregular geometry, a Motorola Bravo Plus pager was selected. Figure 10 shows the top view and two port locations P1 and P2.
Figure 8. Equivalent circuit for transient response
A time signal output voltage was generated f o the frequency rm impedance data from 5 MHz to 1 GHz. A one ampere current source (peak to peak) with open source resistance, consisting of 25 clock pulse waveforms having a width of 1.25 ns, rise time of 0.25 ns, fall time of 0.25 ns, and a period of 3.5 ns, w s used and sampled up to a 200 ns using a sampling interval of 0.25 ns. The source was placed at Port 1 defined earlier. After 93.25 ns, the source was turned off to understand the effect of the source on the cavity. The period of the source signal was set to the inverse of the first maximum resonant frequency (285.7 MHz). In Figure 9, the response of the rectangular Figure 10. Motorola Bravo Plus pager plane plane cavity has been captured in the time domain. The board was assumed to have a 2 0 0 p thick FR-4 dielectric, a
20 p n thick copper planes ( uc = 58x10 S/m), dielectric of relative permittivity 6, = 4, and dielectric loss tangent tan...