Outline on DC, SS and Mode Analysis Ningning Zhou Monday, Nov.1, - - PowerPoint PPT Presentation

SUGAR 1.0

Outline on DC, SS and Mode Analysis

• DC analysis algorithm and simulation example
• Steady state algorithm and simulation example
• 3D mode analysis example
• Future work on DC and SS

Ningning Zhou Monday, Nov.1, 1999

SUGAR 1.0

DC Algorithm Finding the equilibrium point of the system

• Newton-Raphson method solving nonlinear equation system f(q) = 0,

q is the equilibrium position vector. – starting from an initial guess q0 – iterating: q n+1 = q n - [Jf(q n)] -1 f(q n) Jf(q n): Jacobian matrix of f(q n); – until: || q n+1 - q n || [ tolerance

• Modified Newton-Raphson methods are required to improve the

convergence properties. For example, to find the equilibrium position after pull- in, simple-limiting algorithm is used with contact force model.

SUGAR 1.0

DC Simulation Example

40 60 80 100 120 140 160 180 200 220 240 2 4 6 8 10 12 14 16 18 20 22

Pull-in Voltages (V) Length of the beam L (um) O Experimental results Simulation results

• Test structures are fabricated by MCNC;
• Beam: Nominal Lb=100um, w=2um, h=2um. Measured :

L=100um, w=1.74um, h=2.003um

• Gap plate: Lg=100um, w=10um, h=2.003um.
• Young’s Modulus: assume 165GPa.
• Simulation was done by considering fringing-field effects;
• Contact force model was used to get pull-in voltage;

6 6.5 7 7.5 8 8.5 9 9.5 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Voltage V (v) Gap distance at node 6 (um)

• +

V

Lb 6

SUGAR 1.0

• Simple circuit elements was implemented

into SUGAR to simulate devices in coupled electro-mechanical domain.

• For detailed and refined circuits analysis,

an integration of SPICE results may be needed.

• Simulation was done by using contact

force model and without fringing-field correction.

Vout (V) d (um)

1 2 3 4 5 6 5 10 15 20 1 2 3 4 5 6 0.5 1 1.5 2

Vout(V) d(um) Pull-in

Vin (V) Vin (V)

DC Analysis in Electro-Mechanical Domain

Vdd =16V R

Vin

Vout

d

nmos

SUGAR 1.0

Steady State Algorithm

Finding the sinusoidal response of the system

• Linearizing the system at a DC equilibrium point.
• Solving linear ordinary differential equation

x(t)’ = A x(t) + B u(t) q(t) = C x(t) + D u(t)

• r: an q(n) + an-1 q(n-1) + ……+ a2 q(2) + a1 q(1) + a0 = u(t)

u(t) is the sinusoidal input;

SUGAR 1.0

Steady State Simulation Example

• Simulation of a linear multiple mode

resonator by Reid Brennen. Sugar results matches his measurements within 5%.

• A 2D electrostatic comb model is used

in simulation.

10

2

10

3

10

4

10

5

10

6

• 10
• 9
• 8
• 7
• 6
• 5

Frequency (Hz) log10(magnitude) 10

2

10

3

10

4

10

5

10

6

• 200
• 100

100 200 Frequency (Hz) phase(degree)

The response of vertical displacement of mass

10

2

10

3

10

4

10

5

10

6

• 15
• 14
• 13
• 12
• 11

Frequency (Hz) log10(magnitude) 10

2

10

3

10

4

10

5

10

6

• 100
• 50

50 100 Frequency (Hz) phase(degree)

The response of induced current in lower comb

SUGAR 1.0

3D Mode Analysis Example

Mode 3 at 31112 Hz Mode 2 at 26983 Hz Mode 6 Mode 1 at 15454 Hz at 123010 Hz

SUGAR 1.0

Future Work on DC and SS

• Make it more designer friendly.
• Implement more element models into SUGAR to expand its

simulation ability, such as 2D and 3D rigid plate mass model, 3D gap model, refined anchor model etc.

• Add other domains such as thermal into SUGAR.
• Refine the DC and SS solver to make simulation more robust

and faster.

• Add more analysis types such as sensitivity analysis and noise

analysis.