lecture 7 modified nodal analysis and spice simulation ...zhuofeng/ee5780fall2013... · 0.0 0.3 0.6...
TRANSCRIPT
![Page 1: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/1.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.1
EE5780 Advanced VLSI CAD
Lecture 7 Modified Nodal Analysis and SPICE Simulation
Zhuo Feng
![Page 2: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/2.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.2
Introduction to SPICE■ Simulation Program with Integrated Circuit Emphasis
► Developed in 1970’s at Berkeley► Many commercial versions are available► HSPICE is a robust industry standard
▼Has many enhancements that we will use
■ Written in FORTRAN for punch-card machines► Circuits elements are called cards► Complete description is called a SPICE deck
![Page 3: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/3.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.3
Writing Spice Decks■ Writing a SPICE deck is like writing a good program
► Plan: sketch schematic on paper or in editor▼Modify existing decks whenever possible
► Code: strive for clarity▼Start with name, email, date, purpose▼Generously comment
► Test:▼Predict what results should be▼Compare with actual▼Garbage In, Garbage Out!
![Page 4: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/4.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.4
Example: RC Circuit* rc.sp* [email protected] 2/2/03* Find the response of RC circuit to rising input
*------------------------------------------------* Parameters and models*------------------------------------------------.option post
*------------------------------------------------* Simulation netlist*------------------------------------------------Vin in gnd pwl 0ps 0 100ps 0 150ps 1.8 800ps 1.8R1 in out 2kC1 out gnd 100f
*------------------------------------------------* Stimulus*------------------------------------------------.tran 20ps 800ps.plot v(in) v(out).end
R1 = 2K
C1 =100fF
Vin+
Vout-
![Page 5: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/5.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.5
Result (Textual)legend:
a: v(in)b: v(out)
time v(in)(ab ) 0. 500.0000m 1.0000 1.5000 2.0000
+ + + + + 0. 0. -2------+------+------+------+------+------+------+------+-
20.0000p 0. 2 + + + + + + + + 40.0000p 0. 2 + + + + + + + + 60.0000p 0. 2 + + + + + + + + 80.0000p 0. 2 + + + + + + + + 100.0000p 0. 2 + + + + + + + + 120.0000p 720.000m +b + + a+ + + + + + 140.0000p 1.440 + b + + + + + a + + + 160.0000p 1.800 + +b + + + + + +a + 180.0000p 1.800 + + b + + + + + +a + 200.0000p 1.800 -+------+------+b-----+------+------+------+------+a-----+-220.0000p 1.800 + + + b + + + + +a + 240.0000p 1.800 + + + +b + + + +a + 260.0000p 1.800 + + + + b + + + +a + 280.0000p 1.800 + + + + b+ + + +a + 300.0000p 1.800 + + + + +b + + +a + 320.0000p 1.800 + + + + + b + + +a + 340.0000p 1.800 + + + + + b + + +a + 360.0000p 1.800 + + + + + b + +a + 380.0000p 1.800 + + + + + +b + +a + 400.0000p 1.800 -+------+------+------+------+------+--b---+------+a-----+-420.0000p 1.800 + + + + + + b + +a + 440.0000p 1.800 + + + + + + b + +a + 460.0000p 1.800 + + + + + + b+ +a + 480.0000p 1.800 + + + + + + b +a + 500.0000p 1.800 + + + + + + +b +a + 520.0000p 1.800 + + + + + + +b +a + 540.0000p 1.800 + + + + + + + b +a + 560.0000p 1.800 + + + + + + + b +a + 580.0000p 1.800 + + + + + + + b +a + 600.0000p 1.800 -+------+------+------+------+------+------+---b--+a-----+-620.0000p 1.800 + + + + + + + b +a + 640.0000p 1.800 + + + + + + + b +a + 660.0000p 1.800 + + + + + + + b +a + 680.0000p 1.800 + + + + + + + b +a + 700.0000p 1.800 + + + + + + + b+a + 720.0000p 1.800 + + + + + + + b+a + 740.0000p 1.800 + + + + + + + b+a + 760.0000p 1.800 + + + + + + + b+a + 780.0000p 1.800 + + + + + + + ba + 800.0000p 1.800 -+------+------+------+------+------+------+------ba-----+-
+ + + + +
![Page 6: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/6.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.6
Result (Graphical)
t(s)0.0 100p 200p 300p 400p 500p 600p 700p 800p 900p
0.0
0.5
1.0
1.5
2.0 v(in)
v(out)
![Page 7: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/7.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.7
Sources■ DC Source
Vdd vdd gnd 2.5
■ Piecewise Linear SourceVin in gnd pwl 0ps 0 100ps 0 150ps 1.8 800ps 1.8
■ Pulsed SourceVck clk gnd PULSE 0 1.8 0ps 100ps 100ps 300ps 800ps
PULSE v1 v2 td tr tf pw per
v1
v2
td tr tfpw
per
![Page 8: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/8.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.8
SPICE Elements
Letter ElementR ResistorC CapacitorL InductorK Mutual InductorV Independent voltage sourceI Independent current sourceM MOSFETD DiodeQ Bipolar transistorW Lossy transmission lineX SubcircuitE Voltage-controlled voltage sourceG Voltage-controlled current sourceH Current-controlled voltage sourceF Current-controlled current source
![Page 9: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/9.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.9
Units
Letter Unit Magnitudea atto 10-18
f fempto 10-15
p pico 10-12
n nano 10-9
u micro 10-6
m mili 10-3
k kilo 103
x mega 106
g giga 109
Ex: 100 femptofarad capacitor = 100fF, 100f, 100e-15
![Page 10: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/10.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.10
DC Analysis* mosiv.sp
*------------------------------------------------* Parameters and models*------------------------------------------------.include '../models/tsmc180/models.sp'.temp 70.option post
*------------------------------------------------* Simulation netlist*------------------------------------------------*nmosVgs g gnd 0Vds d gnd 0M1 d g gnd gnd NMOS W=0.36u L=0.18u
*------------------------------------------------* Stimulus*------------------------------------------------.dc Vds 0 1.8 0.05 SWEEP Vgs 0 1.8 0.3.end
Vgs Vds
Ids
4/2
![Page 11: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/11.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.11
I-V Characteristics
Vds
0.0 0.3 0.6 0.9 1.2 1.5 1.8
Ids(A)
0
50
100
150
200
250
Vgs = 1.8
Vgs = 1.5
Vgs = 1.2
Vgs = 0.9
Vgs = 0.6
■ NMOS I-V►Vgs dependence►Saturation
![Page 12: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/12.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.12
MOSFET ElementsM element for MOSFET
Mname drain gate source body type+ W=<width> L=<length>+ AS=<area source> AD = <area drain> + PS=<perimeter source> PD=<perimeter drain>
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Z. Feng MTU EE5780 Advanced VLSI CAD7.13
Transient Analysis* inv.sp
* Parameters and models*------------------------------------------------.param SUPPLY=1.8.option scale=90n.include '../models/tsmc180/models.sp'.temp 70.option post
* Simulation netlist*------------------------------------------------Vdd vdd gnd 'SUPPLY'Vin a gnd PULSE 0 'SUPPLY' 50ps 0ps 0ps 100ps 200psM1 y a gnd gnd NMOS W=4 L=2 + AS=20 PS=18 AD=20 PD=18M2 y a vdd vdd PMOS W=8 L=2+ AS=40 PS=26 AD=40 PD=26
* Stimulus*------------------------------------------------.tran 1ps 200ps.end
a y
4/2
8/2
![Page 14: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/14.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.14
Transient Results
(V)
0.0
1.0
t(s)0.0 50p 100p 150p 200p
v(a)
v(y)
tpdr = 15pstpdf = 12ps
tf = 10ps
tr = 16ps
0.36
1.44
1.8
■ Unloaded inverter► Overshoot► Very fast edges
![Page 15: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/15.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.15
Subcircuits■ Declare common elements as subcircuits
■ Ex: Fanout-of-4 Inverter Delay► Reuse inv► Shaping► Loading
.subckt inv a y N=4 P=8M1 y a gnd gnd NMOS W='N' L=2 + AS='N*5' PS='2*N+10' AD='N*5' PD='2*N+10'M2 y a vdd vdd PMOS W='P' L=2+ AS='P*5' PS='2*P+10' AD='P*5' PD='2*P+10'.ends
a b c d eX1 X2 X3 X41
2
4
8
16
32
64
128 fX5256
512
Shape input
DeviceUnderTest Load
Load onLoad
![Page 16: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/16.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.16
FO4 Inverter Delay* fo4.sp
* Parameters and models*----------------------------------------------------------------------.param SUPPLY=1.8.param H=4.option scale=90n.include '../models/tsmc180/models.sp'.temp 70.option post
* Subcircuits*----------------------------------------------------------------------.global vdd gnd.include '../lib/inv.sp'
* Simulation netlist*----------------------------------------------------------------------Vdd vdd gnd 'SUPPLY'Vin a gnd PULSE 0 'SUPPLY' 0ps 100ps 100ps 500ps 1000psX1 a b inv * shape input waveformX2 b c inv M='H' * reshape input waveform
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Z. Feng MTU EE5780 Advanced VLSI CAD7.17
FO4 Inverter Delay Cont.X3 c d inv M='H**2' * device under testX4 d e inv M='H**3' * loadx5 e f inv M='H**4' * load on load
* Stimulus*----------------------------------------------------------------------.tran 1ps 1000ps.measure tpdr * rising prop delay+ TRIG v(c) VAL='SUPPLY/2' FALL=1 + TARG v(d) VAL='SUPPLY/2' RISE=1.measure tpdf * falling prop delay+ TRIG v(c) VAL='SUPPLY/2' RISE=1+ TARG v(d) VAL='SUPPLY/2' FALL=1 .measure tpd param='(tpdr+tpdf)/2' * average prop delay.measure trise * rise time+ TRIG v(d) VAL='0.2*SUPPLY' RISE=1+ TARG v(d) VAL='0.8*SUPPLY' RISE=1.measure tfall * fall time+ TRIG v(d) VAL='0.8*SUPPLY' FALL=1+ TARG v(d) VAL='0.2*SUPPLY' FALL=1.end
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Z. Feng MTU EE5780 Advanced VLSI CAD7.18
FO4 Results
(V)
0.0
0.5
1.0
1.5
2.0
t(s)0.0 200p 400p 600p 800p 1n
a
b
c
d
e
ftpdf = 66ps tpdr = 83ps
![Page 19: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/19.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.19
Power Measurement■ HSPICE can measure power
► Instantaneous P(t)►Or average P over some interval
.print P(vdd)
.measure pwr AVG P(vdd) FROM=0ns TO=10ns
■ Power in single gate►Connect to separate VDD supply►Be careful about input power
![Page 20: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/20.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.20
■ How is SPICE simulator created?►N equations in terms of N unknown Node voltages►More generally using modified nodal analysis
G(v)i 1v2v
3v
4vC
R
v
![Page 21: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/21.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.21
Time Domain Equations at node 1:
If we do this for all N nodes:
N dimensional vector of unknownnode voltages
vector of independent sources
nonlinear operator
0)()()(12
4131 vvGR
vvdt
vvdC
0))(),(),(( tutxtxF Xx
)0(
)(tx
)(tu
F
![Page 22: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/22.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.22
■ Closed form solution is not possible for arbitrary order of differential equations
■ We must approximate the solution of:
■ This is facilitated in SPICE via numerical solutions
0))(),(),(( tutxtxF Xx
)0(
![Page 23: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/23.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.23
■ Basic circuit analyses►(Nonlinear) DC analysis
▼Finds the DC operating point of the circuit▼Solves a set of nonlinear algebraic eqns
►AC analysis▼Performs frequency-domain small-signal analysis▼Require a preceding DC analysis▼Solves a set of complex linear eqns
►(Nonlinear) transient analysis▼Computes the time-domain circuit transient response▼Solves a set of nonlinear different eqns▼Converts to a set nonlinear algebraic of eqns using
numerical integration
![Page 24: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/24.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.24
■ SPICE offers practical techniques to solve circuit problems in time & freq. domains
► Interface to device models▼Transistors, diodes, nonlinear caps etc
► Sparse linear solver
► Nonlinear solver – Newton-Raphson method
► Numerical integration
► Convergence & time-step control
![Page 25: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/25.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.25
■ Circuit equations are usually formulated using
►Nodal analysis ▼N equations in N nodal voltages
►Modified analysis▼Circuit unknowns are nodal voltages & some branch
currents▼Branch current variables are added to handle
– Voltages sources– Inductors– Current controlled voltage source etc
■ Formulations can be done in both time and frequency
![Page 26: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/26.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.26
How do we set up a matrix problem given a list oflinear(ized) circuit elements?
Similar to reading a netlist for a linear circuit:
* Element Name From To Value
3
2
1
1
RRRI
2110
0201
1005101mA
![Page 27: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/27.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.27
The nodal analysis matrix equations areeasily constructed via KCL at each node:
1I 1R
2R
3RmA1
5
10010
1 2
0
JvY
![Page 28: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/28.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.28
■ Naïve approach► a) Write down the KCL eqn for each node► b) Combine all of them to a get N eqns in N node voltages
■ Intuitive for hand analysis
■ Computer programs use a more convenient “element” centric approach► Element stamps
![Page 29: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/29.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.29
Instead of converting the netlist into a graph and writing KCL eqns, stamp in elements one at a time:
Stamps: add to existing matrix entries
RR
RR11
11 Fromrow
Torow
Fromcol.
Tocol.
j
i
i j
i
jRY
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Z. Feng MTU EE5780 Advanced VLSI CAD7.30
RHS of equations are stamped in a similar way:
II
i
jI
From row
Torow
ij
J
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Z. Feng MTU EE5780 Advanced VLSI CAD7.31
Stamping our simple example one element at a time:
10002521
1001110
3
2
1
1
RRR
mAI
01
2
1
322
221 IVV
GGGGGG
![Page 32: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/32.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.32
We know that nonlinear elements are firstconverted to linear components, then stamped
EQI EQG
![Page 33: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/33.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.33
For 3 & 4 terminal elements we know that the linearized models have linear controlled sources
Gdsv
D DG
S S
gsv
gsmvg dsr
We can stamp in MOSFETs in terms of a completestamp, or in terms of simpler element stamps
![Page 34: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/34.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.34
Voltage controlled current source
kv 0I
k
kmvgi
p
q
Voltmeter
mm
mm
gggg
k
p
qrow
row
colcol
Large value that does not fall on diagonal of Y!
![Page 35: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/35.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.35
All other types of controlled sources includevoltage sources
Voltage sources are inherently incompatiblewith nodal analysis
Grounded voltages sources are easily accommodated
1
0
v2
21
2
1
vv
![Page 36: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/36.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.36
But a voltage source in between nodes is more difficult
Node voltages and are not independentk
k
![Page 37: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/37.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.37
We no longer have N independent node voltagevariables
So we can potentially eliminate one equation and one variable (section 2.3 of reference [1])
But the more popular solution is modified nodalanalysis (MNA)
i
Vk
Create one extra variable and one extra equation
![Page 38: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/38.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.38
Extra variable: voltage source current
Allows us to write KCL at nodes and
Extra equation
Advantage: now have an easy way of printingcurrent results - - ammeter
Vvvk
k
![Page 39: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/39.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.39
Voltage source stamp:
Vi0111
1
row
row
row
col colcol
k
k
1N
1N
![Page 40: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/40.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.40
Current-controlled current source (e.g. BJT) has to stamp in an ammeter and a controlled current source
12 ii 1i
![Page 41: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/41.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.41
In general, we would not blindly build the matrixfrom an input netlist and then attempt to solve it
Various illegal ckts are possible:
Cutsets of current sources
![Page 42: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/42.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.42
Loops of voltage sources
Dangling nodes
![Page 43: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/43.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.43
■ Once we efficiently formulate MNA equations, an efficient solution to is even more important
■ For large ckts the matrix is really sparse► Number of entries in Y is a function of number of elements
connected to the corresponding node
■ Inverting a sparse matrix is never a good idea since the inverse is not sparse!
■ Instead direct solution methods employ Gaussian Elimination or LU factorization
JvY
![Page 44: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/44.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.44
ckt file
dc nonlinear soln. open C’s ; short L’s
insert TR models for L’s and C’s
“dc” nonlinear soln. for equivalent CKT
finaltt
0t
ttt
?
Done
■ TR analysis flow (Chap. 4 of ref. 2)
![Page 45: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/45.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.45
■ One-step integration approximation
dtdvCi C
v i
tt
t
diC
tvttv )(1)()(
)()(2
)()(
)(
ttititttit
titdi
tt
i
Forward Euler (FE)
Backward Euler (BE)
Trapezoidal (TR)
![Page 46: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/46.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.46
)(ti
1t 2t t12 ttt
)()( 12
1titdttit
t Forward Euler
■ FE is explicit, and no nonlinear iterations are required► Extremely difficult to use in practice
![Page 47: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/47.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.47
)(ti
1t 2t t
)()( 22
1titdttit
t Backward Euler
■ BE is implicit► Much more robust than FE► Can also make unstable responses appear stable
![Page 48: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/48.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.48
)(ti
1t 2t t
)()(2
)( 212
1tititdttit
t
Trapezoidal
■ TR is implicit too► Works similarly to BE► Incurs less error
![Page 49: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/49.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.49
■ FE Capacitor Companion Model
dtdvCi C
v i
)()()( tiCttvttv
)( tti
)()( titditt
t
![Page 50: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/50.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.50
■ BE Capacitor Companion Model► Thevenin
)()( ttitditt
t
dtdvCi C
v i
CtReq
)( tti )(tvVeq
![Page 51: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/51.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.51
■ BE Capacitor Companion Model► Norton
tCGeq
)(tvt
CIeq
)( ttv
)( tti
![Page 52: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/52.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.52
■ TR Capacitor Companion Model► Thevenin
)()(2
)( ttititditt
t
dtdvCi C
v i
CtReq 2
)( tti )(
2)( ti
CttvVeq
![Page 53: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/53.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.53
■ TR Capacitor Companion Model► Norton
tCGeq
2)()(2 titv
tCIeq
)( ttv
)( tti
![Page 54: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/54.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.54
■ TR Inductor Companion Model► Norton
)()(2
)( ttvtvtdvtt
t
vi
L dt
Ldiv dvL
titti )(1)()(
LtGeq 2
)()(
2titv
LtIeq
)( ttv
)( tti
![Page 55: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/55.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.55
■ TR Inductor Companion Model► Thevenin
tLReq
2
)( tti )(2)( ti
tLtvVeq
![Page 56: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/56.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.56
■ Nonlinear DC analysis (Chapter 10.6 – 10.8 of Ref. 2)
►Store device equations and their partial derivatives w.r.t. branch voltages for efficient N-R procedure:
▼Insert linearized models into MNA formulation (first order Taylor series at operating point)
▼Solve the linear ckt to complete one N-R iteration
▼Use the solution as operating pt. for next linearization step
![Page 57: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/57.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.57
Diode equations and companion model for N-R
21 vvvd 1
2
)( 1 TVdv
eIi sd
Td
Vv
T
s
d
d eVI
vi
)( 11 kd
kd
kd
dkd
kd vv
viii
)( kd
kd
dkd v
vii
kd
d
vi
StoredModelEqns.
![Page 58: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/58.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.58
Diodes are modeled by Norton equivalentcompanion models
Some sort of voltage limiting scheme isrequired to make N-R iterations robust
Solve to obtainand so on…
1kv
)( kd
kd
dKdeq v
viiI
kd
deq v
iG
![Page 59: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/59.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.59
■ Recap: linear transient analysis►Replace each of C’s and L’s by a companion
model – numerical integration▼Forward Euler, Backward Euler, Trapezoidal ▼Norton or Thevenin models
►Solve the equivalent linear circuit at the current time step
►Update all the companion models and move to the next time step
![Page 60: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/60.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.60
■ Example: BE capacitor companion models►Thevenin
)()( ttitditt
t
dtdvCi C
v i
CtReq
)( tti )(tvVeq
![Page 61: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/61.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.61
►Norton
tCGeq
)(tvt
CIeq
)( ttv
)( tti
![Page 62: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/62.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.62
■ We combine the above two in nonlinear transient analysis
tnt 1nt
)(tv
Numerical integration
N-R iterations
![Page 63: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/63.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.63
■ Start from some initial condition at time t0
■ Move to the next time step by replacing all the C’s & L’s by a companion model
► Solve the equivalent nonlinear DC problem at the new time point▼Use the solution at the previous time step as the initial guess
for N-R
▼ Iterate till convergence
■ Repeat till reaching the ending time
■ Nonlinear dynamic elements need to be handled more carefully
► Charge conservation – more on this later
![Page 64: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/64.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.64
What about nonlinear elements with more than2 terminals?
Nonlinear equations:
Once again, model by first 2 terms of Taylor series
1i 2i
2v1v3i
),( 2111 vvgi ),( 2122 vvgi
213 iii Port Equations
![Page 65: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/65.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.65
22
11
1
121111
11 ),( v
vgv
vgvvgiii
kk
kkkkk
22
21
1
221222
12 ),( v
vgv
vgvvgiii
kk
kkkkk
22
2
2
11
1
2
1
1
212211331
3 ),(),(
vvg
vgv
vg
vg
vvgvvgiii
Kk
kkkkkkk
![Page 66: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/66.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.66
3-Terminal MOSFET Stamp:2v1v
Consider the current contribution at each nodeNote that we must translate port voltages to node voltages
a b
c
22
11
1
121111
11 ),( v
vgv
vgvvgiii
kk
kkkkk
11ki
cb
ca
vvv
vvv
2
1
ckk
bk
ak
kkkkk vvg
vgv
vgv
vgvvgiii
2
1
1
1
2
1
1
121111
11 ),(
![Page 67: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/67.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.67
3-Terminal MOSFET Stamp:2v1v
Most solvers are setup to solve for instead ofv 1kv
2
2
1
2
2
1
1
1
2
2
2
1
1
2
1
1
2
2
1
2
2
2
1
2
2
1
1
1
2
1
1
1
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
vg
kkkkkkkk
kkkk
kkkk
a
b
c
a b cv
),(
),(
),(
),(
212
211
212
211
kk
kk
kk
kk
vvg
vvg
vvg
vvg
a b
c
c
b
a
v
v
v
![Page 68: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/68.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.68
MOSFETs (3 terminal)
Triode Region
dsds
dsTHgsds vvvvvi
1
2)(
2
THgsds vvv
0 THgs vv
LWC ox
di
si
gidsv
gsv
![Page 69: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/69.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.69
Saturation Region
Cutoff Region
dsTHgsds vvvi 1)(
22
THgsds vvv
THgs vv 0dsi
![Page 70: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/70.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.70
dc port equations:
),( dsgsdsd vvii
0gi
),( dsgsdsds vviii
)(
)(
),(
1
1
1
kds
kds
kds
ds
kgs
kgs
kgs
ds
kds
kgsds
kd
kd
kd
vvvi
vvvi
vviiii
![Page 71: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/71.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.71
dsi
kdsi
gsv
kdsv dsv
THgsds Vvv
m
kgs
ds gvi
dskds
ds Gvi
kds
kds
kgs
km
kds
kds vGvgii 1
![Page 72: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/72.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.72
kds
kds
kgsm
kds
kds vGvgii 1
Equivalent ckt model for N-R
kgs
kgs
kgs vvv 1
kds
kds
kds vvv 1
kgsv
G
S
1kdsi
kgs
km vg k
dsi kdsG k
dsv
![Page 73: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/73.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.73
Stamping in terms for all of these 2-terminalelements is equivalent to applying MOSFET stamp
Note the large off-diagonal terms that are createdby sgm '
Could also build models to solve forand directly1k
dsv1k
gsv
![Page 74: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/74.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.74
Simple Example
INV
1R
2R
3R
1MDDV
1 2
0
DDV3R
12kv
kdsG
kEQI
11kk
mvg11kv2RINV
1R
kds
kds
kgs
km
kds
kEQ vGvgiI
kgs
dskm v
ig
kds
dskds v
iG
![Page 75: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/75.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.75
Now formulate the nodal equations forthis linearized equivalent ckt
DDV3R
12kv
kdsG
kEQI
11kk
mvg11kv2RINV
1R
![Page 76: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/76.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.76
■ and values change at each N-R iteration
■ Damping methods control the N-R convergence► More of a problem for BJTS
■ In general, we would include the 4-th terminal of the MOSFET (body effect)
dsm Gg , EQI
![Page 77: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/77.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.77
■ We use dc nonlinear algorithms find solution at t=0 AND for all timepoints
■ Energy storage elements are properly considered by numerical integration
■ How about nonlinear capacitors?
Ct
2
![Page 78: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/78.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.78
Conventional NL device models► ids = f(vds, vgs, vsb) -- 30+ parameters► Accurate evaluation can be very expensive
NR requires full evaluation of models► Derivatives required for Jacobian
▼Expensive to evaluate -- can consume 50-80% of computation time
SC-based approach ► No derivatives to re-organize – approximate Jacobian► Will work with table models, measured data► Reduced model evaluation time, but is based on:
▼Slower convergence (depending on the Japprox)▼Matrix update required for varying timesteps▼Selection of representative linear model in Japprox
![Page 79: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/79.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.79
Newton-Raphson
Good convergence properties
Reliable when implemented with a step-limiting mechanism (damped)
Requires explicit differentiation
n-D case: Jacobian Matrix stamping (update) and re-factorization x0
x2
x1
![Page 80: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/80.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.80
Successive Chord J(x) represents a fixed gradient
(Jacobian)
No explicit differentiation
Reliable when implemented with a step-limiting mechanism (damped)
n-D case: Single Jacobian factorization improvement by optional updates
x0
x2
x1
![Page 81: Lecture 7 Modified Nodal Analysis and SPICE Simulation ...zhuofeng/EE5780Fall2013... · 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I ds ( A) 0 50 100 150 200 250 V gs = 1.8 V gs ... SPICE offers](https://reader030.vdocuments.mx/reader030/viewer/2022040806/5e46dd4de24e754ad75436e5/html5/thumbnails/81.jpg)
Z. Feng MTU EE5780 Advanced VLSI CAD7.81
NR has varying R, I SC has fixed R
Linearized network changes for each NR step, i.e. J(xn)
Single linearized network for SC steps, Japprox
Newton-Raphson Successive Chord