Hey guys, I am new to this forum.

I am working on a problem analyzing a potentiometer accelerometer. For the first part, I need to find the transfer function of the potentiometer.

known: y=x2-x1
dynamic system: y,x1

For the mechanical aspect of the problem, I found that the transfer function was --> Y(s)/(s^2)*X1(s) = -m/(ms^2 + bs + k)

For the electrical part, I know I need to find the transfer function E0(s)/Y(s) and multiply it with the mechanical part to form the total transfer function. I cannot figure out how to determine the transfer function though. I am assuming I should make 3 loops, where the e0 'arm' is moving along the resistor, but if I do this, i would have another loop with es as the input, would that come out to the right answer?

Any help would be appreciated, Thanks in advance!

 

Actually, now that I'm more deeply thinking about it, the final transfer function for the potentiometer should be the input voltage of the circuit over the displacement (x1) of the body right?

For loop equations I have:

R0*i1 + e0 + (R0 + R0*y)*i1 = 0
R0*i2 + (R0 + R0*y)*i2 +e0 = 0
R0*(i3 - i1) + R0(i3 + i2) + es = 0

So, the output voltage would technically cancel out and we can relate the input (es) to y right?

 

Or, as I keep thinking about it, I realize that there is no way to relate the input and displacement output without being left with an i3 term in the function. Therefore, would it be possible to analyze this problem using only 2 loops? Is the current in the top loop always the same or does it change as the eo moves?

 

This simulation should show how to approach this problem. It is not a general, algebraic solution, but it should show you what your algebraic solution should account for and an approach of how to structure a circuit that you can model algebraically:

I show a potentiometer where both the upper and lower arms are a function of acceleration. Note the terms R1=1K*(1-V(accel)) and R2=1K*V(accel). V(accel) is the independent variable (acceleration) that is the input to the simulation. The resistances of the two arms, the currents through the resistors, and the output voltage V(out) are all dependent variables. Note I selected an unrealistically low load resistance (R3=1K). In the real world, you would make R3 a factor of a hundred or thousand bigger than the arms of the accelerometer.

In addition to all the voltages and currents, I also plot the deviation from what the output would be if the load resistance R3 was infinite. (see the gray trace). Note that this voltage is shown on the left Y-axis of the bottom plot pane, while the currents are plotted on the right Y-axis. The X-axis on both plot panes is the independent variable: acceleration (represented as a voltage that goes from 1mV to 991mV).