# Sensors, MEMS and control
How tiny mechanical sensors work and how feedback makes systems behave - MEMS resonators, capacitive sensing and PID control. A first-principles learning page.
Full page with the site theme: https://krunalpanchal1995.github.io/projects/sensors-mems-control/
## Figures
*A mass-spring-damper is the model for most MEMS: a high Q gives a tall, narrow resonance, ideal for sensing small shifts.*
*Proportional control leaves an offset, integral action removes it at the cost of overshoot, and derivative action damps it.*
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A learning page: standard theory from the textbooks listed at the bottom, with figures computed from the formulas shown. It makes no claim of new research results, and the parameters are illustrative.
**Try it:** the [Bode filter instrument](https://krunalpanchal1995.github.io/electronics-lab/#bode) in Circuit Lab runs the formulas on this page.
## Intuition
A MEMS sensor is a microscopic mechanical part, usually a small mass on springs etched into silicon, whose motion is read out electrically. Shake the chip and the mass lags behind; measure how far it lags and you have measured the acceleration. Control is the other half of the story: use the measurement to push the system back towards where you want it.
## School level
- A mass on a spring bounces at a natural frequency f0 = (1 / 2π) √(k / m): stiffer springs and lighter masses ring faster.
- Friction (damping) makes the bouncing die away. Light damping gives a long ring; heavy damping gives a slow, sluggish return.
- **Feedback**: measure the output, compare it with the target, and act on the difference. A thermostat does this with heat.
## College level
**MEMS resonator.** The mass-spring-damper has natural angular frequency ω0 = √(k / m) and quality factor Q = ω0 m / b. Its response to a force at frequency ω is flat at low frequency, peaks by a factor of about Q at ω0, then falls (figure 1). A high Q gives a narrow, sharp resonance, which makes small shifts easy to detect.
**Accelerometer.** Well below resonance the proof mass moves by x = a / ω02, so sensitivity grows as the resonant frequency falls, but bandwidth shrinks: a basic trade-off. Capacitive read-out uses C = ε A / d, so a displacement Δd changes the capacitance by ΔC / C ≈ Δd / d. The mechanical noise floor (Brownian motion) corresponds to an acceleration of about √(4 kB T ω0 / (m Q)) per root hertz, so heavier masses and higher Q give quieter sensors.
**PID control.** The controller output is u = Kp e + Ki ∫ e dt + Kd de/dt, with e the error. For the plant G(s) = 1 / (s + 1)3 in figure 2:
- Proportional only (Kp = 2) settles at Kp / (1 + Kp) = 0.67 of the target, a steady-state error.
- Adding integral action removes that error, at the cost of overshoot and slower settling.
- Adding derivative action damps the overshoot.
## Advanced
- Frequency-domain design (Bode and Nyquist plots, gain and phase margin) predicts stability before building anything.
- Sensor fusion (for example, a Kalman filter combining a gyroscope with an accelerometer) beats any single sensor.
- Real controllers must handle noise, saturation of the actuator and integral wind-up.
## Where this shows up
- Phone motion sensors, airbag accelerometers, gyroscopes in drones, pressure sensors.
- Motor drives, temperature controllers and cruise control.
## Check yourself
1. A resonator has Q = 50. By roughly what factor is the response at resonance larger than at DC? *Answer: about 50.*
2. Why does a proportional-only controller on a plant with unit DC gain leave an error? *Answer: it needs a non-zero error to produce any output; with gain Kp the loop settles where the output is Kp / (1 + Kp) of the target.*
3. An accelerometer's natural frequency doubles. What happens to its low-frequency sensitivity a / ω02? *Answer: it falls to one quarter.*
## Read next
- S. D. Senturia, *Microsystem Design*.
- K. J. Åström and R. M. Murray, *Feedback Systems: An Introduction for Scientists and Engineers*.
- G. F. Franklin, J. D. Powell and A. Emami-Naeini, *Feedback Control of Dynamic Systems*.