SUGAR 1.0 Outline on DC, SS and Mode Analysis Ningning Zhou Monday, Nov.1, 1999 • DC analysis algorithm and simulation example • Steady state algorithm and simulation example • 3D mode analysis example • Future work on DC and SS
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 q 0 – iterating: q n+1 = q n - [J f (q n )] -1 f(q n ) J f (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 • Test structures are fabricated by MCNC; Lb • Beam: Nominal Lb=100um, w=2um, h=2um. Measured : L=100um, w=1.74um, h=2.003um 6 • Gap plate: Lg=100um, w=10um, h=2.003um. + V • Young’s Modulus: assume 165GPa. - • Simulation was done by considering fringing-field effects; • Contact force model was used to get pull-in voltage; 22 1.8 20 1.6 18 O Experimental results Gap distance at node 6 (um) 16 1.4 Pull-in Voltages (V) Simulation results 1.2 14 12 1 0.8 10 8 0.6 6 0.4 4 0.2 2 0 40 60 80 100 120 140 160 180 200 220 240 6 6.5 7 7.5 8 8.5 9 9.5 10 Length of the beam L (um) Voltage V (v )
SUGAR 1.0 DC Analysis in Electro-Mechanical Domain 20 • Simple circuit elements was implemented 15 into SUGAR to simulate devices in Vout(V) 10 coupled electro-mechanical domain. 5 • For detailed and refined circuits analysis, 0 an integration of SPICE results may be 0 1 2 3 4 5 6 Vin (V) needed. 2 • Simulation was done by using contact 1.5 force model and without fringing-field d(um ) 1 correction. 0.5 Vdd =16V 0 0 1 2 3 4 5 6 Pull-in Vin (V) R d Vout d (um) nmos Vin Vout (V)
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) or: a n q (n) + a n-1 q (n-1) + ……+ a 2 q (2) + a 1 q (1) + a 0 = 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. The response of induced current in lower comb The response of vertical displacement of mass -11 -5 log10(magnitude) log10(magnitude) -6 -12 -7 -13 -8 -14 -9 -15 -10 2 3 4 5 6 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) 100 200 phase(degree) 50 phase(degree) 100 0 0 -50 -100 -100 -200 2 3 4 5 6 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)
SUGAR 1.0 3D Mode Analysis Example Mode 1 Mode 2 at 15454 Hz at 26983 Hz Mode 3 Mode 6 at 123010 Hz at 31112 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.
Recommend
More recommend
