I am developing the signal conditioning of the conductive polymer which gives the resistance change when the force/strain is applied to the sensor. To develop a signal conditioning, first the transfer function of the sensor should be defined. so how can i calculate my transfer function in terms of strain.
Electrical – transfer function of the sensor
sensortransfer function
Related Solutions
see correction in the last line:
I don`t know if you are asking for the following analyses. Nevertheless, here it comes:
gm=Transistor transconductance; Vo=opamp output voltage; Acl=opamp closed-loop gain; k=feedback factor
Iout=VGS*gm
VGS=Vo-V1
V1=Iout*R1
Vo=Vin*Acl
Acl=Ao/(1+kAo)
k=gmR1/(1+gmR1) >> source follower
This gives (for infinite Ao): Acl=(1+gmR1)/gm*R1.
Now you can combine all the equations - starting at the top (simply insert the succeeding expressions):
The result is: Iout=Vin/R1
EDIT: In case, the real and frequency-dependent open-loop gain Ao(s) is to be considered, we arrive at the following expression (same set of equations, however, without setting Ao to infinite):
CORRECTION: There was a computational error (1/gm was missing in the denominator)
Iout=Vin/[R1 + (R1+1/gm)/Ao(s)]=Ao(s)Vin/[1/gm + R1 + R1*Ao(s)]
T(s)=Iout/Vin=Ao(s)/[1/gm + R1 + Ao(s)*R1]
This is just answering one part of your question:
I don't quite understand the last part. How does he calculate X and ϕ
This is just applying the trigonometric identity $$\sin\left(\alpha + \beta\right)=\sin\alpha\cos\beta + \cos\alpha\sin\beta$$ with \$\alpha=\omega{}t\$ and \$\beta=\phi\$.
Using this identity on the r.h.s. gives $$X\sin\left(\omega{}t+\phi\right)=X\left(\sin{}\omega{}t\cos\phi+\cos\omega{}t\sin\phi\right)$$ so our equation becomes $$\frac{A}{\omega}\sin\left(\omega{}t\right)+B\cos\left(\omega{}t\right)=X\left(\sin{}\omega{}t\cos\phi+\cos\omega{}t\sin\phi\right)$$
Which we can break into two parts, $$\frac{A}{\omega}\sin\left(\omega{}t\right)=X\cos\phi\sin{}\omega{}t$$ and $$B\cos\left(\omega{}t\right)=X\sin\phi\cos\omega{}t$$
So, $$\frac{A}{\omega} = X\cos\phi$$ and $$B=X\sin\phi$$
From there you should be able to get the conclusions from your source.
Best Answer
For something like this, you don't calculate it, you measure it.
You need to set up some mechanical system to allow you to vary the strain accurately, perhaps clamps and a ruler. Then you need some system to measure the resistance, perhaps a DMM. Plot resistance against strain. Over a small range, it can be approximated by a linear relation. Over a large range, I don't expect that will be possible.
While you are measuring, you will need to see the effect of various confounding factors that might make your transfer function less accurate. The effect probably has hysteresis (sensitivity to strain history), temperature sensitivity, and time (creep). Once you know the sensitivity to these, you'll have an idea of what accuracy you can expect from the system.