Filename: “AET_ch8”
Multi-Domain Modeling
1.1 MEMS Accelerometers Modeling1
Sensing is typically performed in two steps:
Primary
Transducer
tethered
seismic
mass
Secondary
Transducer
capacitive
electrical
signal
Transform acceleration to mechanical displacement
Transform mechanical displacement to electrical signal
mechanical
displacement
environmental
acceleration


Microflexural Structure
Daspot: (D)
 Model frictional resistance as damping force proportional to rate of movement
(velocity)
Seismic Mass (M):
 Model inertia as force proportional to rate of velocity (acceleration)
Spring (K):
 Model structural elasticity as spring force proportional to movement
F applied Fmass  Fdamping  Fspring
F (t)  M
d 2x
dx
D
 Kx
2
dt
dt
Inertia, dissipative, elasticity characteristics determine transient (time domain) and
bandwidth (frequency domain) response.
F (t )
d 2x
dx
 G (t )  2  20
 0 x
M
dt
dt
K
0 
M
D

2 KM
  0 1   2
x
1
 2
a 0
Where:
ω0
natural resonant frequency - oscillation with no damping/forcing
ξ
damping factor – actual damping/critical damping
ω
damped resonant frequency
x/a
primary transducer transfer function – displacement/unit acceleration
Capacitive Sensing
Seismic mass forms one plate of a parallel plate capacitance. Movement of mass changes
area/gap between parallel plates and , consequently the capacitance.
Differential Capacitance Sensing
Easier to detect relative (differential) change rather than absolute change
Comb finger capacitances connected in parallel to add to form C1 and C2.
Sensing requires a bridge circuit with complementary AC source VS.
VS
+
VC1
-
+
C1
+
I
VS
C2
VX
I  2C12sVS  2
C1C2
sVS
C1  C2
VX  VS  VC1  VS 
C1 = C2
C1 ≠ C2
I
1  C1C2
  C1 - C2 
 VS 
2
sVS   
VS

sC1
sC1  C1  C2
  C1  C2 
=> no acceleration
=> acceleration
Signal Detection is based on
Sin x Sin y = 0.5[cos(x-y) – cos(x+y)]
vsource1
env_force1
ENV_FORCE
mag_ac := 0.16e-9*5*9.8
mag_dc := 0.0
freq := 100.0
VSOURCE
comb_drive1
comb_drive
mag_ac := 300.0e-3
mag_dc := 0.0
freq := 1.0 Meg
vsource2
m := 0.16e-9
k := 2.6455
d := 4.0e-6
a := 2.0e-6*110.0e-6
do := 1.5e-6
r1
VSOURCE
mag_ac := -300.0e-3
mag_dc := 0.0
freq := 1.0Meg
LIBRARY IEEE;
USE IEEE.ENERGY_SYSTEMS.ALL;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
USE IEEE.MECHANICAL_SYSTEMS.ALL;
ENTITY comb_drive IS
GENERIC(m : MASS := 0.16*nano;
d : DAMPING := 4.0e-6;
k : STIFFNESS := 2.6455;
a : REAL := 2.0e-6 * 110.0e-6;
do : REAL := 1.5e-6);
PORT(TERMINAL proof_mass, ref : TRANSLATIONAL;
TERMINAL top_el, mid_el, bot_el : ELECTRICAL);
END ENTITY comb_drive;
r := 3.0Meg
ARCHITECTURE bcr OF comb_drive IS
--Free quantities
QUANTITY vel : VELOCITY;
QUANTITY qtm, qbm : CHARGE;
QUANTITY dtm, dbm : DISPLACEMENT;
QUANTITY ctm, cbm : CAPACITANCE;
--branch quantities
QUANTITY pos ACROSS force THROUGH proof_mass TO ref;
QUANTITY vtm ACROSS itm THROUGH top_el TO mid_el;
QUANTITY vbm ACROSS ibm THROUGH bot_el TO mid_el;
BEGIN
--compute displacement of comb drive
--mechanical dynamics
--compute change in capacitance
--electrical dynamics
--compute generated current
END ARCHITECTURE bcr;
LIBRARY IEEE;
USE IEEE.MATH_REAL.ALL;
USE IEEE.MECHANICAL_SYSTEMS.ALL;
ENTITY ENV_FORCE IS
GENERIC(MAG_AC, MAG_DC : FORCE :=0.0;
FREQ : REAL := 0.0);
QUANTITY not allowed in GENERIC
PORT(TERMINAL PT1, PT2 : TRANSLATIONAL);
END ENTITY ENV_FORCE;
ARCHITECTURE SINE OF env_force IS
QUANTITY FORCE THROUGH PT1 TO PT2;
BEGIN
FORCE == MAG_AC*SIN(MATH_2_PI*FREQ*NOW);
END ARCHITECTURE SINE;
LIBRARY IEEE;
USE IEEE.MATH_REAL.ALL;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
ENTITY vsource IS
GENERIC(MAG_AC, MAG_DC : VOLTAGE := 0.0;
FREQ : REAL := 0.0);
PORT (TERMINAL p, m : ELECTRICAL);
END ENTITY vsource;
ARCHITECTURE sine OF vsource IS
--CONSTANT,
QUANTITY v across i THROUGH p TO m;
BEGIN
v == mag_ac*SIN(MATH_2_PI*freq*NOW)+mag_dc;
END ARCHITECTURE sine;
Signal Detection
The modulated signal in r1.v can be demodulated by mixing the signal with a local
oscillator with the same frequency as the carrier.
[A(t)Sin x] Sin y = 0.5A(t)[cos(x-y) – cos(x+y)]
If x=y
[A(t)Sin x] Sin x =0.5A(t)[1-cos2x]=0.5A(t)-0.5A(t)cos2x
A(t) can be recovered by low pass filtering.
electrical2sfg2
vsource3
Electrical2Sfg
VSOURCE
vsource1
env_force1
ENV_FORCE
mag_ac := 0.16e-9*5*9.8
mag_dc := 0.0
freq := 100.0
VSOURCE
comb_drive1
comb_drive
mag_ac := 300.0e-3
mag_dc := 0.0
freq := 1.0 Meg
Electrical2Sfg
vsource2
m := 0.16e-9
k := 2.6455
d := 4.0e-6
a := 2.0e-6*110.0e-6
do := 1.5e-6
electrical2sfg1
VSOURCE
r1
r := 3.0Meg
mag_ac := -300.0e-3
mag_dc := 0.0
freq := 1.0Meg
mul1
gs1
G(s)
gs1=6.28e3/(s+6.28e3) 1kHz cutoff frequency
1.2 Electrical and Thermal Modeling
Thermal Components3
 Temperature Source (T) :Across Quantity like V, unit K(Kelvin)
 Heat Flow Source (H)
:Through Quantity like I, unit J/s(Joules/sec)
 Thermal Resistance (RTH) :Resistive component like R, unit K*s/J
 Thermal Capacitance (CTH): Capacitive component like C, unit J/K
Thermal Resistance (RTH):
T(t)= RTH H(t)
Thermal Resistance VHDL-AMS Model
library ieee;
use ieee.thermal_systems.all;
entity rth is
port(
quantity k: in real:= 1.0;
terminal th1, th2: thermal);
end entity rth;
-- linear coefficient (k*s/J)
architecture beha of rth is
quantity t across h through th1 to th2;
begin
t == h*k;
end beha;
Thermal Capacitance (CTH):
T 
1
CTH
t
 Hdt  T
0
0
Where:
T0 is the initial temperature 293 K.
Thermal Capacitance VHDL-AMS Model
library ieee;
use ieee.thermal_systems.all;
entity cth is
port(
quantity c_th: in real := 1.0;
quantity t0:
in real := 293.0;
terminal th1: thermal);
end entity cth;
-- thermal capacitance (J/K)
-- initial temperature (K)
architecture beha of cth is
quantity t across h through th1 to thermal_ref;
quantity h_int : real := t0;
begin
Break
h_int => t0 * c_th;
h_int'dot == h;
t == h_int / c_th;
end beha;
Thermometer
library ieee;
-Like voltmeter
use ieee.thermal_systems.all;
entity thm is
port(
quantity temp: out real; -- temperature (K)
terminal th1 : thermal);
end entity thm;
architecture beha of thm is
quantity t across th1 to thermal_ref;
begin
t == temp;
end beha;
Thermal Source
--Like voltage source
-- ---------------------------------------------------------------------------library ieee;
use ieee.thermal_systems.all;
entity t is
port(
quantity value: in real:= 293.0;
-- temperature (K)
terminal th1: thermal);
end entity t;
architecture beha of t is
quantity t_ across h through th1 to thermal_ref;
begin
t_ == value;
end beha;
rth1
k := 1

t1
thm1
cth1
c_th := 1
θ=370K; rth1: k=1 K/W; cth1: c_th=1 Ws/K
θ=370K; rth1: k=2 K/W; cth1: c_th=1 Ws/K
θ=370K; rth1: k=1 K/W; cth1: c_th=2 Ws/K
Simple Diode without Thermal Modeling
LIBRARY IEEE;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
USE IEEE.MATH_REAL.ALL;
ENTITY D IS
GENERIC (
ISAT : REAL := 1.0e-12; -- Saturation current
VT : REAL := 35.0e-3;
-- Thermal voltage
RR : REAL := 100.0e3; -- Reverse resistance
VF : REAL := 0.8; -- Forward voltage
RB : REAL := 1.0e-3 -- Bulk Resistance
);
PORT (TERMINAL p,m : ELECTRICAL);
END ENTITY D;
ARCHITECTURE behav OF D IS
QUANTITY v ACROSS i THROUGH p TO m;
BEGIN
IF (v >= 0.0) USE
i == ISAT * ((exp(v/VT)) - 1.0);
ELSE
i == v/RR;
END USE;
END ARCHITECTURE;
ARCHITECTURE equiv OF d IS
CONSTANT vlimit : REAL := VF/(1.0 - (RB/RR));
QUANTITY v ACROSS i THROUGH p TO m;
BEGIN
IF (v < vlimit) USE
i == v/RR;
ELSE
i == (v-VF)/RB;
END USE;
END ARCHITECTURE equiv;
Model Using Basics Vs AMS
Models Usings Basics
r1
E1
r := 1k
FREQ := 1
AMPL := 4
OFF := 1
TDELAY := -0.2
4
E1.V [V]
2
0
-2
-4
-6
0
0.2
0.4
0.6
4.5m
0.8
1 t [s]
d1.i
3m
2m
1m
-0.5m
0 0.2 0.4 0.6 0.8 1 t [s]
1
d1.v
0
-0.5
-1
-1.5
-2
-3
0
0.2
0.4
0.6
0.8
1 t [s]
Simple Diode Thermal Model1 (vt is assumed constant)
LIBRARY IEEE;
USE IEEE.ENERGY_SYSTEMS.ALL;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
USE IEEE.THERMAL_SYSTEMS.ALL;
USE IEEE.MATH_REAL.ALL;
--Define K (boltzmann), Q(elec_charge)
ENTITY MyDiodeTh IS
GENERIC (
ISAT : REAL := 1.0e-12; -- Saturation current
vt : REAL := 35.0e-3; -- Thermal voltage
RR : REAL := 100.0e3; -- Reverse resistance
VF : REAL := 0.8; -- Forward voltage
RB : REAL := 1.0e-3 -- Bulk Resistance
);
PORT (TERMINAL anode,cathode : ELECTRICAL;
TERMINAL junction: THERMAL);
END ENTITY MyDiodeTh;
ARCHITECTURE behav OF mydiodeth IS
QUANTITY vd ACROSS id THROUGH anode TO cathode;
QUANTITY temp ACROSS power THROUGH thermal_ref TO junction;
BEGIN
IF (vd >= 0.0) USE
id == ISAT * ((exp(vd/vt)) - 1.0);
ELSE
id == vd/RR;
END USE;
power == vd*id;
END ARCHITECTURE behav;
mydiodeth1
r1
rth1
thm1
MyDiodeTh
r := 1k
k := 1.0
E1
cth1
c_th := 1.0
Simple Diode Thermal Model2 (vt depends on temp)
LIBRARY IEEE;
USE IEEE.ENERGY_SYSTEMS.ALL;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
USE IEEE.THERMAL_SYSTEMS.ALL;
USE IEEE.MATH_REAL.ALL;
--Define K (boltzmann), Q(elec_charge)
ENTITY MyDiodeTh IS
GENERIC (
ISAT : REAL := 1.0e-12; -- Saturation current
RR : REAL := 100.0e3; -- Reverse resistance
VF : REAL := 0.8; -- Forward voltage
RB : REAL := 1.0e-3 -- Bulk Resistance
);
PORT (TERMINAL anode,cathode : ELECTRICAL;
TERMINAL junction: THERMAL);
END ENTITY MyDiodeTh;
ARCHITECTURE behav OF mydiodeth IS
QUANTITY vd ACROSS id THROUGH anode TO cathode;
QUANTITY temp ACROSS power THROUGH thermal_ref TO junction;
QUANTITY vt : voltage;
-- thermal voltage
BEGIN
IF (vd >= 0.0) USE
id == ISAT * ((exp(vd/vt)) - 1.0);
ELSE
id == vd/RR;
END USE;
vt == temp * K/Q;
power == vd*id;
END ARCHITECTURE behav;
-- thermal voltage is temp dependend
Simple Diode Thermal Model3 (vt depends on temp & id depends on id)
LIBRARY IEEE;
USE IEEE.ENERGY_SYSTEMS.ALL;
USE IEEE.ELECTRICAL_SYSTEMS.ALL;
USE IEEE.THERMAL_SYSTEMS.ALL;
USE IEEE.MATH_REAL.ALL;
--Define K (boltzmann), Q(elec_charge)
ENTITY DiodeTh IS
GENERIC (iss : REAL := 1.0e-4;
n, af : REAL := 1.0;
tt, cj0, vj, rs, kf : REAL :=0.0);
PORT (TERMINAL anode, cathode : ELECTRICAL;
TERMINAL junction: THERMAL);
END ENTITY DiodeTh;
ARCHITECTURE Level0 OF diodeth IS
QUANTITY vd ACROSS id, ic THROUGH anode TO cathode;
QUANTITY temp ACROSS power THROUGH thermal_ref TO junction;
QUANTITY qc : charge;
QUANTITY vt : voltage;
-- thermal voltagge
BEGIN
qc == tt*id - 2.0*cj0 * sqrt(vj**2 -vj*vd+1.0e-20);
zero error
ic == qc'DOT;
id == iss*(exp((vd-rs*id)/(n*vt)) - 1.0);
vt == temp * K/Q;
power == vd*id;
END ARCHITECTURE Level0;
--1.0e-20 is added to prevent division by
Note: When the diode is forward bias vd >0 (<0 reverse polarity), the solver oscillate.
The solution is incorrect.
REFERENCES
1. “Analog and Mixed-Signal Modeling Using the VHDL-AMS Language”,
E.C. Beaverton et.al., 36th Design Automation Conference, New Orleans,
June 21-25, 1999.
2. “Simulation System SIMPLORER VHDL-AMS Tutorial,” English
Edition, © 2003 Ansoft Corporation.
3. Simulation System SIMPLORER® v.6.0 User Manual, English Edition, ©
1996-2002, Ansoft Corporation.