Hi Designer's Guide Community ,

since this is my first post, I wanted to say a few words about myself. My name is Daniel and I do my PHD at the university of Freiburg in germany. I am working on sensor readout electronics for MEMS sensors, especially Lorentz force magnetometers. The problem I am having right now is bugging me for quite some time and until now I could get around it by linearizing. This is no longer possible as I want to simulate a circuit that tunes the resonance frequency of a MEMS structure. I created a model for a 1D Lorentz force magnetometer (torsional) a while ago and implemented the drive and sense capacitances as follows:

     Cs1 = Cs*(d/(d-ls*Theta(theta))); //Sensecap1
     Cs2 = Cs*(d/(d+ls*Theta(theta))); //Sensecap2
     
     I(Csense1)<+ddt(Cs1*V(Csense1));
     I(Csense2)<+ddt(Cs2*V(Csense2));

       Tcs1 = ((pow(V(Csense1),2))/(d-ls*Theta(theta))*Cs1*ls*0.5; //torque of Cs1
       Tcs2 = ((pow(V(Csense2),2))/(d+ls*Theta(theta))*Cs2*ls*0.5; //torque of Cs2


This leads to some strange phenomena, e.g. if it is driven via the sensecaps at the resonance frequency (I use Cadence Virtuoso and Spectre to simulate), the phase between input voltage and the output (angle in this case) is not 90° but either around 10° or around 180°. I found this article: http://www.designers-guide.org/modeling/varactors.pdf and changed the capacitances accordingly:

     Cs1 = Cs*(d/(d-ls*Theta(theta))); //sensecap1
     vCs1 = V(Csense1);
     qCs1 = Cs1*vCs1;
     I(Csense1) <+ ddt(qCs1);

     Cs2 = Cs*(d/(d+ls*Theta(theta))); //sensecap2
     vCs2 = V(Csense2);
     qCs2 = Cs2*vCs2;
     I(Csense2) <+ ddt(qCs2);

     Tcs1 = (pow(vCs1,2))/(d-ls*Theta(theta))*Cs1*ls*0.5; //torque of Cs1
     Tcs2 = (pow(vCs2,2))/(d+ls*Theta(theta))*Cs2*ls*0.5; //torque of Cs2      

The result is even worse. With the first implementation, if the simulator is set to euler-method only, this phase error did not occur. Now it does occur whether euler-only is set or not. I am really sure, that the problem comes from the nonlinear capacitance, as also mentionned in the article. If I use a linear capacitance then everything is working correctly. I tried many different implementations for the torque, or the capacitances but so far nothing has changed the result. I attached the complete model.

I hope someone knows how to solve this. Many thanks in advance .

Best regards,
Daniel

Mmh ok, I thought the first implementation is not charge conservative and that this could be a reason for the unexpected behavior of the model. The device itself is a stiff plate that is suspended by two beam springs with two electrode pairs below the plate on the substrate and a coil on top of the plate. If a current is put on the coil in the presence of an in-plane magnetic field a Lorentz force is acting on the plate. The counterpart to the electrodes on the substrate is an electrode on the bottom of the plate which is set to a constant bias voltage. Therefore if an AC driving voltage is applied to the bottom electrodes there is always a DC- and an AC-voltage across the capacitances. The idea is now to use the electrostatic and dynamic forces from the capacitances to control the movement of the plate. The Tcs lines give the torque caused by electrostatic and dynamic forces that act on the device if an external voltage is put on the capacitors.

The underlying differential equation looks somewhat like that:

Iyy*(d^2/dt^2(Theta)+(w0/Q)*d/dt(Theta)+w0^2*Theta) =Tcs1-Tcs2

where Theta is the tilting angle of the plate, Iyy the moment of inertia of the plate, w0 the resonance frequency and Q the quality factor.

In the model it is implemented like this:      

//differential equation variables

Omega(theta_v)<+ddt(Theta(theta));  //angular velocity
dOmega(theta_a)<+ddt(Omega(theta_v)); //angular acceleration

Theta(theta) <+ (-Tau(TLor)+Tcs1-Tcs2+Tcd1-Tcd2-Iyy*(omega/Q)*Omega(theta_v)-Iyy*dOmega(theta_a)
)/(Iyy*pow(omega,2));

There are other torques involved (the device has two pairs of capacitances for driving the structure (Tcd1/2) and sensing its movement and the Lorentz force causes another torque (Tau(TLor))) but either way what is causing the problems is the torques coming from the capacitors. The capacitance itself can be seen as linear but the differential equation becomes nonlinear with Tcs = constant*V^2*1/(d+ls*Theta)^2, where d is the air gap in idle state and ls the distance to the rotational axis. If I make a linearization here like d>>ls*theta => Tcs = constant*V^2*1/d^2 the problem does not occur. I want to realize a control scheme that changes the resonance frequency by adjusting the bias voltage on the capacitances. This is only possible if I keep to the nonlinear variant of the model. This problem also occurs only if the plate is moving. In a control scheme that keeps the plate in idle mode it works fine.


As to the phase, I think there is a misunderstanding here. To test the model I put an AC sine voltage of 100 mV at around the resonance frequency across the capacitances and used the angle as an output signal. The test is therefore open-loop and the phase between driving voltage and angle should go towards -90° if the driving frequency is close to the resonance frequency as the model forms a system with one conjugate complex pole pair. In a closed-loop system the phase around the loop has to match 0° to operate in resonance, as you wrote.

Best regards,
Daniel

PS: I attached a figure of a cross section of the structure (simplified).

I am referring to the resonance frequency of the mechanical structure. I found a solution to the problem. It is not a modelling issue. The parameters that I used for the modelling give a very low spring stiffness and this leads to a nonlinear phase/frequency relationship (nonlinear springsoftening effect) that also happens in reality. If I put a higher value for the spring stiffness (Iyy*w0^2) then the problem does not occur. The parameters for the model come from a project partner and thus it is a design issue of this specific sensor.

Thanks a lot for your help. It made me second guess my assumptions quite a bit .

Best regards,
Daniel