Electromagnetic Simulation for Power Integrity: From Analysis to

Electromagnetic Simulation for
Power Integrity:
From Analysis to Design
A. Ege Engin
[email protected]
Agenda
• Analysis
•
•
•
•
Equivalent-circuit modeling
of power planes
Key idea of efficient power
integrity simulation
Dielectric loss, surface
roughness, etc.
Frequency-domain
characterization
• Design
•
•
•
•
Goodness of a PDN
Design for target impedance
Sensitivity of a PDN
Incremental simulation for
“what-if”
Part 1: Analysis
SI/PI Challenges for IC Packaging
Microstrip line
Vias
Stripline
Chip
Package
Vdd
Vss
PCB
Vdd
Vss
Signal degradation / Jitter
Power supply noise
Electromagnetic interference
Non-Ideal Component Behavior
10
2
Measurement
RLGC model
Electrically short interconnect:
Lumped RLC model
mag(Z11) [Ohm]
10
1
L
10
0
R
10
-1
10
-2
C
10
7
G=ωCtanδ
10
8
10
Frequency [Hz]
9
Embedded Capacitor
Electrically long interconnect:
Transmission line
10
10
Sources of PI Issues
•
As transistors in an integrated circuit (IC) start switching,
current needs to be supplied by the power delivery network
(PDN). Because of the current flow through the impedance
of the PDN, power-supply voltage fluctuates from its DC
value. The power supply noise eventually degrades the
signal quality of high-speed signals, limiting the clock
frequency of the system.
Current
drawn by
the IC
low-power techniques,
such as clock gating to
reduce the current
Impedance
of the PDN
Decoupling capacitors,
Vdd/Gnd planes, lots of
Vdd/Gnd pads
Power-supply
voltage noise
Signal-integrity
problems
Control the return current
for I/Os
PDN Example
Package
PCB power /
power /
ground
Through hole
Wirebonds or
ground
planes & on- vias and BGA
flip-chip
planes & onpackage SMD solder balls
bumps
package SMD
capacitors
capacitors
Embedded capacitor
SMD capacitors
Vdd
Z
Gnd
Target Impedance
Vdd
Gnd
Embedded
capacitors
Noise Coupling in Multilayered
Structures
Signal to power coupling
SMD capacitor
Gap coupling
Gnd/Vdd
Horizontal
coupling
Gnd/Vdd
Aperture coupling
External
field
coupling
(EMI)
Vias and via
coupling
Gnd/Vdd
Embedded capacitor
Gnd/Vdd
Multi-Dimensional Electromagnetic Problem
Cavity Resonances in Distributed
Analysis
Voltage
Maxima
(a)
(b)
Voltage
Minima
(c)
(d)
Voltage on plane (a) 300MHz (b) 600MHz (c) 670MHz (d) 847MHz
How do the Cavity Resonances Affect
Impedance
•
•
•
a=b=250mm, εr=4, d=0.2mm
C=11.067 nF,
f10=f01=300MHz,f11=424MHz,
f20=f02=600MHz, f21=f12=671MHz,
f22=849MHz, f30=f03=900MHz
Some resonances may
be suppressed
depending on the
location of the
impedance port. Close
resonances may merge.
Equivalent Circuit Model of a Single
Plane Pair Based on FDM
 Based on the 5-point finite-difference approximation of
Helmholtz equation
R
L
L
R
R
L
R
R
R
L
L
L
C
L
R
R
G
R
L
R
R
G
L
L
C
L
R
R
L
L
R
R
G
L
DK, Loss tangent
L
C
R
G
Dielectric
thickness
R
R
L
R
Port 1
L
G
C
L
R
L
L
R
Port 2
L
L
L
C
R
R
C
L
G
R
Finite Difference Method
FDM results in a sparse block- diagonal matrix
ui,j+1
(T2  k 2 )u   jdJ z
 T2 u i , j
ui-1,j
1


1 
 2 1  4 1u i , j
h


1
YU  I
ui,j
ui+1,j
ui,j-1
h
h
A
B

B A  1/ Z B

B
 
Y 

 
B

B A 1/ Z


B








B
A
1/ Z
Y  2 / Z
 1/ Z Y  3 / Z
1/ Z


1/ Z Y  3 / Z
A





B  1 / Z







1/ Z

1/ Z Y  3 / Z
1/ Z 

 1 / Z Y  2 / Z 
Circuit Interpretations of FDM
• 5-point approximation results in the well-known bed-spring
model
• 9-point approximation includes diagonal inductances
•
More accurate representation of current paths
 T2 u i , j
1


1 
 2 1  4 1u i , j
h


1
Z/2
Z/2
Y
5-point Approximation
 T2 u i , j
2 / 3 1/ 6
1 / 6
1 
 2 2 / 3  10 / 3 2 / 3u i , j
h
1 / 6
2 / 3 1 / 6 
3Z/4
3Z/4
Y
9-point Approximation
Finite-Difference Time Domain
(FDTD)
Vi,j+1
Iyi,j
Ixi-1,j
Ixi,j
Vi,j
Vi-1,j
Vi+1,j
Iyi,j-1
Iz
Vi,j-1
t
t
2
t
Iyi , j  Iyi , j
t
Vi , j
t
2
t
 Vi , j
t
2
t
2
y
x
t t
t
 (Vi , j  Vi , j 1 )
L
t
 ( Ixit1, j  Ixit, j )  ( Iyit, j 1  Iyit, j )  Izit, j 
C
FDTD Simulation Example
-3
1
x 10
Iz [A]
0.8
V
0.6
0.4
10cm
Iz
εr=4, d=100um
10cm
0.2
0
0
0.2
0.4
0.6
Time [s]
0.8
1
-8
x 10
FDTD Simulation Result
-4
0
x 10
SPICE
FDTD
Voltage [V]
-1
-2
-3
-4
-5
-6
0
0.2
0.4
0.6
Time [s]
0.8
1
-8
x 10
Finite Element Method
FEM simulation done with the PDE toolbox in Matlab. Current source is in the
middle.
Color: abs(u) Height: abs(u)
1.5
1.5
1
1
0.5
0.5
0
0.1
0.05
0.1
0.05
0
0
-0.05
-0.05
-0.1
-0.1
FEM
0
0.1
0.05
0.1
0.05
0
0
-0.05
-0.05
-0.1
-0.1
FDM
Modeling of Multiple Plane Pairs
-10
mag(S12) [dB]
Plane pairs get coupled
through the apertures
-20
-30
-40
Multilayer FDM
Sonnet
Measurements
-50
-60
-70
0
Port 1
200
phase(S12) [degrees]
Port 2
0.8
0.6
Voltage [V]
0.4
0.2
0
1
2
3
4
Frequency [Hz]
5
6
9
x 10
Multilayer FDM
Sonnet
Measurements
100
0
-100
-0.2
-0.4
-0.6
0
0.2
0.4
0.6
Time [s]
0.8
1
-8
x 10
Switching noise voltage at port 1
-200
0
2
4
Frequency [Hz]
6
9
x 10
Losses
The Debye model requires real (and positive) coefficients:
Standard curve fitting does not work!
RC curve fitting approach can be used to obtain the coefficients assuming that
the permittivity is available at some discrete frequency points.
-3
3.785
8.5
Measured
RC Vector Fitting
3.78
Measured
RC Vector Fitting
8
3.775
7.5
3.77
7

tan 
3.765
3.76
6.5
6
3.755
5.5
3.75
5
3.745
3.74
0
x 10
2
4
6
8
Frequency (Hz)
10
12
14
9
x 10
4.5
0
2
4
6
8
Frequency (Hz)
10
12
14
9
x 10
Skin Effect and Surface Roughness
Assume the Hammerstad model for the surface roughness:
𝛼 = 𝑘𝛼𝑐 + 𝛼𝑑
Where: 𝑘 = 1 +
2
atan
𝜋
𝑟𝑚𝑠 2
𝛿
1.4
For a transmission line, 𝑅~𝑘/𝛿
-9
3
30
2.5
2
20
Inductance (H)
Resistance (Ohm)
RC Vector Fitting
Simulated
RC Vector Fitting
25
15
1.5
10
1
5
0.5
0
x 10
0
1
2
3
4
5
6
Frequency (Hz)
7
8
9
10
9
x 10
0
0
1
2
3
4
5
6
Frequency (Hz)
7
8
9
10
9
x 10
Effect of Surface Roughness
RT5880
Rolled (less rough)
Electrodeposited
Materials courtesy of Rogers Corp.
Effect of Surface Roughness
RO4350
LoPro (less rough)
TWS foil
Examples of SI/PI Coupling via Return
Currents
Helen K. Pan et al, Intel, PIERS 2007
Sung-Ho Joo, IEEE MWCL Oct07
Scott McMorrow,
PCDesign, 2002
Larry Smith, Sun Microsystems,
EPEP’99
Modal Analysis – Key Idea of Efficient
Signal and Power Integrity Simulation
A stripline can be routed between a power and a ground plane.
Where is the return current in such a case?
Is=Ip+Ig
h2
Ip=?
Ig=?
h1
Power Ip
Is
r
Ground Ig
L
R
R
L
L
L
R
R
R
L
C
R
L
R
R
G
L
L
R
G
R
R
G
L
R
L
C
L
L
C
G
R
R
R
L
R
R
L
R
G
L
L
C
L
C
L
L
L
C
R
R
G
C
L
L
L
R
L
R
R
L
G
R
R
R
R
L
L
L
C
L
R
G
R
L
C
G
R
Modeling of Transmission Lines
Considering Non-Ideal Supply Planes
•
•
The signal line and the power/ground planes
can be regarded as a multiconductor
transmission line (MTL)
Power/Ground planes can be modeled at
various levels of complexity
h2
r
h1
stripline
µ-strip
Parallel-Plate and Stripline Modes
• The transformation matrices are defined such that
the two modes are the
• parallel plate mode (i.e., no current flows on the signal
line)
• stripline mode (i.e., both planes at the same potential)
r
Vdd
h2
h1
Parallel-Plate
V1
k*I1
h1
k 
h1  h2
Signal
Stripline
I1
k*V1
Vss
V2
Vdd
k*I2
I2
Vss
k*V2
Signal
Summary of Modal Decomposition
Methods
Transmission line type
Coupling
factor (k)
Inhomogeneous medium (ε1
different than ε2)
Microstrip referenced to ground
k=0
Modal decomposition works
k=-1
Modal decomposition works
k=h1/(h1+h2)
Capacitive coupling terms need
to be considered
k can be
obtained
from a 2D
field solver
Capacitive coupling terms need
to be considered
Signal
Ground
Power
ε1
ε2
Microstrip referenced to power
Signal
Power
Ground
ε1
ε2
Stripline
Power
Signal
Ground
ε1
h2
ε2
h1
Conductor-backed CPWG
Signal
Power
Ground
ε1
ε2
Power
• Model
• Measurement
• S12: near-end coupling
between signal line and
power plane
S12, dB
Experimental Validation
-15
-20
-25
-30
-35
-40
-45
0
2
4
6
8
10
freq, GHz
Vss
Vdd
S12, degrees
100
50
0
-50
-100
0
2
4
6
freq, GHz
8
10
Modeling of Microstrip-Stripline Vias
Solid Lines: Model
Circles: Measurement
Via inductance and
capacitance neglected
mag(S12), dB
0
-10
-20
-30
-40
-50
0
1
2
3
4
5
6
4
5
6
freq, GHz
phase(S12), degrees
•
•
•
Signal
µstrip
 model
100
0
-100
-200
k*V2 I2
I1 k*V1
Signal
stripline
Vss
Vdd
200
Vss
parallel
plate
V1+-
k*I1
parallel
plate
k*I2
+V2
Vdd
0
1
2
3
freq, GHz
Measurement of a PDN with a VNA
•
•
•
•
•
Since typical PDN impedance is very small, the probe inductance
has to be removed (or deembedded) from measurements
Assume we want to measure the input impedance of a PDN
Using a VNA we get the S-parameters of the setup in the figure
below including the probe inductance
We always use two probes. For the input impedance, they are
placed as close as possible to the input impedance port
It turns out that the transfer impedance is actually not affected by
the probe inductance (based on the T-model below)!
Measured impedance
PDN input impedance
Measurement of Input vs. Transfer
Impedance
Probe
Error box
tip 1
𝒁𝒑
DUT
𝒁𝟏𝟏 − 𝒁𝟏𝟐
𝒁𝟐𝟐 − 𝒁𝟏𝟐
𝒁𝟏𝟐
Probe
Error box
tip 2
𝒁𝒑
Probe
Error box
tip 1
𝒁𝒑
𝑫𝑼𝑻
Input impedance: 𝒁𝒎
=
𝒁
𝟏𝟐
𝟏𝟏
DUT
𝒁𝟏𝟏 − 𝒁𝟏𝟐
Error box
𝒁𝟐𝟐 − 𝒁𝟏𝟐
𝒁𝟏𝟐
𝑫𝑼𝑻
Transfer impedance: 𝒁𝒎
=
𝒁
𝟏𝟐
𝟏𝟐
𝒁𝒑
Probe
tip 2
Part 2: Design
Concept of Target Impedance
•
•
Frequency-domain target impedance is commonly used
to design power distribution networks
It is based on the consideration that the power supply
will have less voltage noise than allowed ripple for the
given maximum current and power supply voltage.
i
+
VDD
-
Z
+ vL +
VIC
-
Not always true
but useful!
IC
The concept of target impedance:
If mag(Z) < mag(ZT)
Then (VL) < (Vdd x ripple)
How Robust is the Target Impedance
Approach?
An example where the voltage noise is
larger than that predicted by the target
impedance
X. Hu, W. Zhao, P. Du, Y. Zhang, A. Shayan, C. Pan, A. E. Engin,
and C. K. Cheng, “On the Bound of Time-Domain Power
Supply Noise Based of Frequency-Domain Target Impedance,”
ACM/IEEE System Level Interconnect Prediction, pp. 69-76, 2009.
Possible Metric for the Goodness of a
PDN
𝒁
PDN1
PDN2
𝒁𝑻
𝒇
Which PDN is better?
Problem Definition
•
Assume the impedance is simulated at m distinct
frequencies f1, f2, …, fm.We want to minimize
𝐹(ℎ)) = maxj 𝑍𝑗 ℎ
𝐹(ℎ
•
− 𝑍𝑗𝑇
Where the impedance at frequency fj is Zj and 𝑍𝑗𝑇 is
the target impedance at that frequency. The variables
h that can be controlled are the locations and values
of the decoupling capacitors.
Example for Minimax Optimization
The values of four decoupling capacitors are
optimized using the minimax algorithm to reduce
the input impedance of a 20 cm x 20 cm board
•
0.7
Before optimization
After optimization
0.6
mag(Z11) [Ohm]
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
3.5
4
4.5
5
8
x 10
Ref: Engin, A. E.; , "Efficient Sensitivity Calculations for Optimization of Power Delivery Network Impedance,“ IEEE
Transactions on Electromagnetic Compatibility, May 2010
Optimization Algorithms
•
•
Heuristic: Simulated annealing, genetic algorithms, etc.
Gradient-based: Needs efficient sensitivity calculations
for the impedance.
All the terms are already available from
the simulation of input impedance 𝑍𝑖
Sensitivity
•
Differential sensitivity
•
𝜕𝑍
𝜕𝐶
•
𝜕𝑍
𝜕𝐿
•
•
𝜕𝑍
or 𝜕𝑅 : Is the PDN impedance really that sensitive to the ESL or
ESR of the decaps?
Exact calculation based on adjoint method.
Large-change sensitivity
•
•
•
: Needed to find the search direction to minimize the PDN
impedance, or just have quick idea of which capacitors are not
effective at certain frequencies.
Fast design-space exploration.
What if some decaps are removed, added, or replaced? What if
the geometry is slightly modified? Can we do an incremental
simulation using previous data?
Tracking sensitivity
•
•
Simulated cavity resonances don’t match measured data due to
incorrect value of dielectric constant or loss tangent?
Frequency-dependent behavior of complex permittivity.
Fast Design-Space Exploration
•
•
•
In all optimization methods, the same PDN needs to be
simulated with only small changes (such as the location or
value of a decoupling capacitor) to explore design space
The simulation is typically restarted from scratch, without
using any information from previous simulations
This procedure can be greatly improved by applying an
incremental simulation for fast design-space exploration
Can be solved incrementally if the
nominal solution is available for
Simulation of PDNs with Variable
Dielectric Constant and Loss Tangent
•
•
•
•
The dielectric constant (DK) and loss tangent (DF) of the substrate
between the power and ground planes has a big impact on the PDN
resonances
The DK and DF are functions of frequency, with an unknown
frequency variation for most cases
This impacts the accuracy of the simulation
When simulations are being correlated to measurements, many
iterative simulations are necessary, resulting in a 100X-1000X
increase in simulation time
Tracking Sensitivity
•
•
The tracking sensitivity
algorithm is an efficient
method to calculate the
changes in the network
matrix of a circuit,
when a global
parameter, such as
temperature, is
continuously varied.
In our case, the global
variable (K) we consider
is the complex
permittivity of the
dielectric.
Can be solved by extracting K as a
variable, based on several
simulations of the nominal matrix
Ref: Engin, A. E.; , "An Arnoldi Algorithm for Power Delivery Networks with Variable Dielectric Constant and Loss
Tangent,“ accepted for publication at IEEE Transactions on Electromagnetic Compatibility
Tracking Sensitivity Based on Complex
Arnoldi Iteration
To approximate this system:
Rewrite as:
Using complex Arnoldi iteration,
obtain the factorization:
The approximate solution is then
given by:
which can be
expressed as a
rational function
of K:
𝑵
𝑽=
𝒋=𝟏
𝒇𝒋 𝒈𝒋
𝟏 − 𝑲𝝀𝒋
Block Arnoldi Algorithm for
Complex Matrices:
Tracking Sensitivity Example
• Even though the nominal simulation was
done for zero loss, the tracking sensitivity
algorithm was able to “track” the correct
complex permittivity
Summary
• PDN analysis:
•
•
Efficient electromagnetic simulation techniques
well understood and implemented
Refinements in improving the accuracy for vias,
gaps, and other 3D discontinuities
• PDN design:
•
•
•
•
Target impedance approach provides a design
methodology
Goodness of a PDN
Sensitivity of the PDN impedance to decaps,
geometry
Efficient what-if simulations