Capacitive Fixed-Fixed Beam RF MEMS SPST Switch

Generated from RF_MEMS.wxm with rfMaxima 0.2.5 on Maxima 5.46.0. All results were computed by executing the notebook; none is transcribed.

/* Put rfMaxima.mac in the same directory as this notebook, or add its
   directory to file_search_maxima. */
load("rfMaxima.mac")$

(See "RF MEMS Theory, Design and Technology", by G. M. Rebeiz, Wiley, 2003)

Model Parameters

Dimensions

g_0: nominal gap height [m]
l: beam length [m]
t: beam thickness [m]
t_d: dielectric thickness [m]
w: beam width [m]
W: electrode width [m]

g_0:2e-6$
l:300e-6$
t:0.5e-6$
t_d:0.2e-6$
w:80e-6$
W:100e-6$

A: electrode area [m^2]

A:W*w;
8.0×10−98.0 \times 10^{-9}

Material Parameters

Beam (Al, Au, Cu, UNCD):
%nu: Poisson ratio of the beam material []. The Poisson ratio of Au is 0.44.
%rho: mass density of the beam material [kg/m^3]. The mass density of Au is 19200 kg/m^3.
%sigma: residual stress of the beam material [Pa]. Assume 20 to 160 MPa.
E: Young's modulus of the beam material [Pa]. The Young's modulus of Au is 78 GPa.

Dielectric (silicon dioxide, silicon nitride):
%epsilon_0: permittivity of vacuum [F/m]
%epsilon_r: dielectric's dielectric constant []. The dielectric constant of Si_3N_4 is 6-9.

Gas (air, nitrogen):
%lambda: mean-free path, distance covered by a molecule in a gas between successive collisions [m]
%mu: viscosity coefficient, measure of resistance of a gas to changes in the transport of molecular momentum [kg/(m.s)]

%rho:19.2e3$
E:78e9$
%nu:0.44$
%sigma:100e6$

%epsilon_0:8.854e-12$
%epsilon_r:7.6$

%lambda:1.5e-7$
%mu:1.845e-5$

Electromechanical Properties & Figures of Merit

Static Properties & Figures of Merit

m: effective beam mass [kg]
k: distributed force spring constant, incl. residual stress [N/m]
k_s: stretching coefficient [N/m^3]
V_p: pull-in voltage [V]
V_bias: bias voltage [V]
f_m_0: mechanical resonant frequency [Hz]
Q_m: mechanical quality factor []
b: damping coefficient [N.s/m]
H(%i*%omega):=X(%i*%omega)/F(%i*%omega), mechanical transfer function [m/N]

m:0.4*%rho*l*t*w;
k_1:(32*E*t^3*w)/l^3*(27/49);
k_2:(8*t*w*(1-%nu)*%sigma)/l*(3/5);
k:k_1+k_2;
k_s:float((t*w*%pi^4*E)/(64*l^3));
V_p:sqrt((8*k*(g_0+t_d/%epsilon_r)^3)/(27*A*%epsilon_0));
V_bias:40;
t_s:3.67*V_p/(V_bias*sqrt(k/m));
f_m_0:float(sqrt(k/m)/(2*%pi));
/* TO VERIFY against Rebeiz: this expression uses the BEAM width w and l/2,
   while the electrostatic area above uses the ELECTRODE width W. */
Q_m:sqrt(k/k_1)*(sqrt(E*%rho)*t^2*g_0^3)/(%mu*(w*l/2)^2);
b:k/(2*%pi*f_m_0*Q_m);
H(f):=1/(k*(1-(f/f_m_0)^2+%i*(f/(Q_m*f_m_0))));
9.216000000000001×10−119.216000000000001 \times 10^{-11}0.50938775510204080.509387755102040835.8400000000000135.8400000000000136.3493877551020536.349387755102051.7587752547806×10+111.7587752547806 \times 10^{+11}35.5678490389319935.5678490389319940405.196205974264681×10−65.196205974264681 \times 10^{-6}99953.3717247939499953.371724793940.24608964990192570.24608964990192577.388840767766875×10−4π{{7.388840767766875 \times 10^{-4}}\over{\pi}}H(f):=1k (1−(ffm,0)2+i (fQm fm,0))H\left(f\right):={{1}\over{k\,\left(1-\left({{f}\over{\mathit{f}_{m,0}}}\right)^2+i\,\left({{f}\over{\mathit{Q}_{m}\,\mathit{f}_{m,0}}}\right)\right)}}
wxplot2d([ev(20*log10(abs(H(f))))],
    [f,1,1e7],
    [logx],
    [legend,false],
    [xlabel,"Frequency [Hz]"],
    [ylabel,"20*log10(Abs(H(f)))"],
    [style,[lines,3,11]],
    [gnuplot_preamble,"set grid"])$
Plot generated by RF_MEMS.wxm

Dynamic Properties & Figures of Merit

kill(t);
done\mathbf{done}

Q_m: variable mechanical quality factor []
b: variable damping coefficient [N.s/m]

/* Veijola slip-flow correction to the squeeze-film quality factor.
   The published coefficient is 9.638, not 9.9638 (T. Veijola et al.,
   "Equivalent-circuit model of the squeezed gas film in a silicon
   accelerometer", Sensors and Actuators A 48, 1995, pp. 239-248).
   TO VERIFY: the (1-(z/g_0)^2)^(3/2) gap dependence should be checked
   against Rebeiz; squeeze-film damping scales as 1/(g_0-z)^3. */
Q_m(z):=Q_m*(1+9.638*(%lambda/(g_0-z))^1.159)*(1-(z/g_0)^2)^(3/2);
Qm(z):=Qm (1+9.638 (λg0−z)1.159) (1−(zg0)2)32\mathit{Q}_{m}\left(z\right):=\mathit{Q}_{m}\,\left(1+9.638\,\left({{\lambda}\over{g_{0}-z}}\right)^{1.159}\right)\,\left(1-\left({{z}\over{g_{0}}}\right)^2\right)^{{{3}\over{2}}}
/* 0.2.4 used f_0 here, which is not defined in this notebook. */
b(z):=k/(2*%pi*f_m_0*Q_m(z));
b(z):=k2 π fm,0 Qm(z)b\left(z\right):={{k}\over{2\,\pi\,\mathit{f}_{m,0}\,\mathit{Q}_{m}\left(z\right)}}

F_e: electrostatic force [N]

/* Electrostatic force on the beam at displacement z, with the dielectric
   included through the effective gap g_0 + t_d/%epsilon_r.  V_C is the
   voltage across the actuation electrode; 0.2.4 never assigned it, and the
   unused time argument has been dropped. */
V_C:V_bias$
F_e(z):=(%epsilon_0*A*V_C^2)/2*(1/(g_0+t_d/%epsilon_r-z)^2);
Fe(z):=ϵ0 A VC22 (1(g0+tdϵr−z)2)\mathit{F}_{e}\left(z\right):={{\epsilon_{0}\,A\,\mathit{V}_{C}^2}\over{2}}\,\left({{1}\over{\left(g_{0}+{{\mathit{t}_{d}}\over{\epsilon_{r}}}-z\right)^2}}\right)

F_VdW: Van der Waals forces [N]

c_1:10e-80;
c_2:10e-75;
1.0×10−791.0 \times 10^{-79}1.0×10−741.0 \times 10^{-74}
F_VdW(z):=c_1*A/(g_0-z)^3-c_2*A/(g_0-z)^10;
FVdW(z):=c1 A(g0−z)3−c2 A(g0−z)10\mathit{F}_{VdW}\left(z\right):={{c_{1}\,A}\over{\left(g_{0}-z\right)^3}}-{{c_{2}\,A}\over{\left(g_{0}-z\right)^{10}}}

Equation of motion of the beam:

m z'' + b z' + k z = F_e(z)

F_e goes as 1/(g_0 + t_d/%epsilon_r - z)^2, so the equation is NON-LINEAR in z and neither ode2 nor contrib_ode can solve it in closed form. (0.2.4 called contrib_ode on it, which simply returns the equation unsolved, and did so with an unassigned V_C.) Integrate it numerically with rk instead.

Contact is not modelled: once z reaches g_0 the beam has touched the dielectric and the simulation past that point is not physical.

b_0:k/(2*%pi*f_m_0*Q_m)$
PullIn:rk([v,(-b_0*v-k*z+F_e(z))/m],[z,v],[0,0],[t,0,5e-6,5e-9])$
wxplot2d([discrete,makelist([p[1]*1e6,min(p[2],g_0)*1e6],p,PullIn)],
    [legend,false],
    [xlabel,"Time [us]"],
    [ylabel,"Beam displacement [um]"],
    [style,[lines,3,2]],
    [gnuplot_preamble,"set grid"])$
Plot generated by RF_MEMS.wxm
/* Pull-in voltage and the applied bias, for reference */
float(V_p);
float(V_bias);
35.5678490389319935.5678490389319940.040.0

RF Properties & Figures of Merit

Small Signal Properties & Figures of Merit

C_u: up-state capacitance [F]
C_d: down-state capacitance [F]
f: RF frequency [GHz]
R_s: RF beam resistance [Ohm]

C_u:1.4*%epsilon_0*A/(g_0+t_d/%epsilon_r);
C_d:0.65*%epsilon_0*A/(t_d/%epsilon_r);
f:10e9$
R_s:1;
4.893847272727273×10−144.893847272727273 \times 10^{-14}1.7495504×10−121.7495504 \times 10^{-12}11

C_r: capacitance ratio []
Q: RF quality factor []

C_r:C_d/C_u;
Q:float(1/(2*%pi*f*R_s*C_u));
35.7500000000000135.75000000000001325.2143645324685325.2143645324685