C:\@files\mea\kon\gain\mea_con_gain_calc-mfe-ob0.mfe
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MEA Gain Equations from Geometry
Mechanical Expansion Amplifier (US Patent 7,707,896)
mea_con_gain_calc-mfe-ob0, mea_con_gain_calc
By James A. Kuzdrall
Intrel Service Company
Box 1247
Nashua NH 03061
Copyright 2008
Some rights reserved
http://www.intrel.com/copyright
Change Log:
12-Feb-12:
03-Aug-08:
15-Jul-08:
04-May-08:
02-May-08:
documentation; m->m0
gain correction
revise mea_dsn_gain-mfe to mea_con_gain_equations-mfe
drawings, revised text
created from isc_mea_omd-mfe
15-Jul-08
Overview
Issues addressed:
1)
2)
3)
4)
5)
What does the physical Mechanical Expansion Amplifier look like?
Show the idealized model and the gain results.
Derive an exact solution for the gain from geometry-based equations.
Show a simplified equation for high gains.
How far from correct does the simplified equation get at low gains?
Approach:
1) Key observation: the ratio of gap change to length change approaches
infinity as the gap approaches zero. Since small length changes produce much
larger gap changes, there is a mechanical (or at least, dimensional) gain.
2) Solution: The solution uses simple algebra, some elementary trigonometry,
and one differentiation.
3) The approximate solution uses a truncated Taylor series for the arctangent
of the small angle at the stop, formed by the substrate and bowing strip
Conclusion:
The approximate gain formula is in error by less than 1% for gains greater
than 3 or for len0/m0 ratios greater than 13, based on a bowing strip length=
10mm, and quiescent gap size m0= 7.9um.
12-Feb-12
mea_con_gain_calc-mfe-ob0
Approximate gain formula :
gain= dm/dlen= 3*len0/(16*m0)
(d14)
approx gain =
3 len0
16 m0
12-Feb-12
mea_con_gain_calc-mfe-ob0
Exact gain formula :
gain= dm/dlen= 2*m0^2/((4*m0^2-len0^2)*atan(2*m0/len0)+2*len0*m0)
Intrel Service Company
2/12/112
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2
dm/dlenx = gain =
(d13)
15-Jul-08
(4 m02 - len02)
2 m0
 2 m0 
 + 2 len0 m0
atan
 len0 
General Information
References:
Ref 1) This idea is recorded in patent notebook "Inventions 1980 to ..." by
James A. Kuzdrall, page 67 to 73.
Parameters:
Define terms used in equations:
arct
geometric arc of length len
len
length of bowed strip, len0+lenx
len0
distance between stops
lenx
expansion of strip (beyond len0)
m0
quiescent bowing gap
rad_arc
radius of bowing arc
theta
central angle of the arc
Units
m
m
m
m
m
m
radians
Constants:
(c45)
(kill(all), timedate())
(d0)
15-Jul-08
Sunday, February 12, 2012, 10:38am
Mechanical Definition
Figure 1 showa a Mechanical Expansion Amplifier. It comprises a bowed strip
or plate constrained by rigid, non-compliant stops. The stops may be barriers,
clamps, or adjustable barriers. A substrate holds the stops separated by a
fixed distance.
The cross section in Figure 2 below shows the bowed strip in detail. As the
strip is stimulated by an external energy, it expands. The energy may come from
radiant energy, conducted heat, magnetic field, electric field, etc. The linear
expansion causes an increase in bowing. The sections which follow calculate the
increase in bowing due to the expansion - in essence, the gain. The gain allows
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2/12/112
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us to calculate the sensitivity to an external stimulus that causes expansion.
Several types of non-contacting external instrumentation may be employed to
measures the change in the bowing gap. Laser interferometers and beam
deflection instruments use light for the measurement. Another approach forms a
capacitor between the strip and an electrode embedded in the substrate. The
capacitance can be measured in a Blumlein Bridge or as via the frequency change
of a resistance-capacitance oscillator.
15-Jul-08
Bowing Gain, Non-Compliant Supports
The geometric schematic in Figure 3 below represents the sensing strip in
cross section. It is held between two non-compliant supports, A and B. Assume
the supports are perfectly rigid. If the strip wants to expand, it must bow.
With no environmental excitation or normal excitation, the strip just fits
between the supports (stress-free), as indicated by the line AB. The width of
the strip is the normal width, len0. If the environmental excitation is
temperature, the just-fits condition occurs at some temperature tmp0.
At some higher excitation, the strip bows out, moving a distance m from the
original position at the center. The ends are still the same distance apart.
For a strip which expands with temperature, the higher excitation is
temperature, tmp.
The length of the new arc is len= len0+lenx. If the strip is responsive to
temperature, lenx is tec*len0*(tmp-tmp0), where tec is the thermal expansion
coefficent.
The radius of arct is rad_arc.
center of the arc is at C.
The arc subtends an angle of theta.
The
The table in Figure 3 shows the calculated gain for a strip that is 10
millimeters long. The gain is inversely proportional to m0 (the quiescent gap)
and directly proportional to len0 (stop separation) when the ratio of len0/m0 is
large.
Using the last entry as an example of the table's use, consider an 80mm
strip that is bowed to 1 micron center gap. If the strip expands by 1.0
picometer, it produces a bowing change of 1.0e-12*1.5e4= 15e-9 or 15 nanometers,
a measurable displacement.
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An Analytic Solution for the Bowing Gain
Write 4 equations relating the elements of the schematic above:
1. Definitional relationship between central angle and arc length:
arct= rad_arc*theta
2. Invoke the Pathagorean Theorem for the right triangle:
rad_arc^2= (rad_arc-m)^2+(len0/2)^2
3. Find a formula for theta that has
Start with the identity tan(x/2)=
Substitute the segment ratios for
the rad_arc terms cancel, leaving
theta= 4*atan(2*m/len0)
no unknowns in the trig arguments.
(1-cos(x))/sin(x).
the cosine and sine. Serendipituously,
tan(theta/4)= 2*m/len0:
4. Express the arc length in terms of its expansion from its former length,
the cord AB:
arct= len0+lenx
Show the segment substitution of step 3 explictly:
(c1)
( a:tan(theta/4)= (1-(rad_arc-m)/rad_arc)/((len0/2)/rad_arc),
allsolve:false, solvetrigwarn:false, b:solve(a,theta)[1] )
 2m 

 len0 
θ = 4 atan
(d1)
Use these 4 equations to eliminate these parameters: arct, theta, and radt.
len is known from the amplifier construction. Solve the remaining expression
for dsp:
(c2)
( "lenx(m)"= eq1: eliminate([
arct= rad_arc*theta,
rad_arc^2= (rad_arc-m)^2+(len0/2)^2,
theta= 4*atan(2*m/len0),
arct= len0+lenx],
[arct,theta,rad_arc,lenx])[1])
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(d2)
lenx(m) =
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 2m 
 - 2 len0 m
len0 
(4 m2 + len02) atan
2m
The algebra gets us the above expression for the expansion, lenx(m), in terms
of the unknown m. (len0 is a known.) The gain is the change in m for a small
change in lenx or dm/dlenx. First solve for the inverse of the gain we want:
(c3)
"dlenx/dm"= eq2: ratsimp(diff(eq1,m,1))
(d3)
dlenx/dm =
 2m 
 + 2 len0 m
len0 
(4 m2 - len02) atan
2m
2
The gain expression we want is the reciprocal of the above:
(c4)
"dm/dlenx = gain"=gain1: 1/eq2
dm/dlenx = gain =
(d4)
2m
2
 2m 
 + 2 len0 m
len0 
(4 m2 - len02) atan
The "m" terms on the right hand side of the equation are constants, the
differential terms having been gathered on the left side. Since the right side
m determines the gain, it is disirable to hold it constant. In traditional
amplifier terms, the constant m value would be the bias point. It might be
established by adding a constant (DC) offset or by some form of feedback.
The bias or quiescent value for m is defined as m0. It is used hereafter to
differentiate the bias point (m0) from variations from the bias point (dm).
(c5)
"dm/dlenx = gain"=gain:ev(gain1, m:m0)
2
dm/dlenx = gain =
(d5)
03-Aug-08
(4 m02 - len02)
2 m0
 2 m0 
 + 2 len0 m0
atan
 len0 
Approximate gain when len0 >> m0 (high gain)
In the applications of interest, len0 >> m0. Can the gain expression be
simplified on that basis?
a) replace atan() with z to simplify temporarily
b) find the Taylor series in x for atan(x) out to the 3rd power term
c) substitute the truncated Taylor series for the original atan(), now z
d) replace the dummy variable x with the original 2*m0/len0
e) use ratsimp to get over one denominator polynomial
Preserve the complete expression as gain_full
f) denominator has m0*len0^4 and m0^3; by comparison, m0^3 is zero
Use this simpler expression for t_gain.
(c6)
(a:substitute(z,atan(2*m0/len0),gain), b:taylor(atan(x),x,0,3),
c:substitute(2*m0/len0,x,b), d:substitute(c,z,a), e:ratsimp(d),
gain_full:gain, "approx gain"=gain_approx:substitute(0,m0^3,e) )
(d6)
approx gain =
03-Aug-08
3 len0
16 m0
How close is the approximation?
What is the error introduced by the approximation for typical bowing strip
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sizes? Find the len0/m0 ratio below which the gain error from the simple
formula is greater than 1%: (by cut-and-try experimentation)
(c7)
(l:10.0b-3, z:792.9980b-6, a:ev(gain_approx, len0:l, m0:z),
b:ev(gain_full, len0:l, m0:z),
c:bfloat((b-a)/a),
concat("For less than ",floor(100*sfloat(c)),"% error: Make gain > ",
floor(a*10)/10.0, " (len0/m0 > ", ceil(l/z),"), based on len0= ",
floor(l*1000),"mm, m0= ", floor(z*1.0e6)/100.0,"um"))
(d7)
For less than 1% error: Make gain > 2.3 (len0/m0 > 13), based on len0= 10mm, m0= 7.92um
Conclude: Use the simple gain formula for len0/m0 ratios greater than 13
or gains greater than 3.
(c6)
Intrel Service Company
2/12/112
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