1- sensor Derivation
When measuring a potentiometer with a volt meter, there will be some error. The percent error is derived in the book. Use my notes to fill in the missing algebra.
When using a capacitive sensor, one must compare the two cases: case 1 when the plates move apart and case 2 when the plates move together. In this combined sensor, the capacitive change is proportional to "x". Use my notes to fill in the algebra
2- optical Sensor
Research a "quadrature encoder". In your own notes, explain how one can get 4 times the accuracy of an encoder using this method.
Full Answer Section
Therefore, the percentage error in V_out is equal to the percentage error in x.
Capacitive Sensor
Case 1: Plates Move Apart
- Capacitance decreases as the distance between the plates increases.
- The change in capacitance is proportional to
-x.
Case 2: Plates Move Together
- Capacitance increases as the distance between the plates decreases.
- The change in capacitance is proportional to
x.
Combined Sensor
By combining the two cases, we can obtain a capacitive sensor that is proportional to x over a wider range. The overall change in capacitance is given by:
ΔC = C_initial + kx
where C_initial is the initial capacitance, k is a constant, and x is the displacement.
2. Quadrature Encoder
A quadrature encoder is a rotary encoder that produces two digital output signals that are quadrature in phase. This means that the two signals are 90 degrees out of phase with each other. By analyzing the phase relationship between the two signals, it is possible to determine the direction and distance of rotation.
How to Achieve 4x Accuracy:
- Count the Rising Edges: Count the number of rising edges on one of the output signals. This gives you a measure of the number of pulses generated by the encoder.
- Determine Direction: Analyze the phase relationship between the two output signals to determine the direction of rotation.
- Calculate Position: Multiply the number of pulses by the resolution of the encoder to determine the angular position.
- Interpolate Between Pulses: To improve accuracy, interpolate between pulses by analyzing the phase relationship between the two signals at the rising edge of one of the signals. This can effectively quadruple the resolution of the encoder.
By using quadrature encoding, it is possible to achieve a much higher accuracy and resolution compared to a single-phase encoder.
Sample Answer
1. Sensor Derivation
Potentiometer Error
Given:
V_out is the output voltage of the potentiometer.V_in is the input voltage to the potentiometer.x is the position of the wiper.L is the total length of the potentiometer.
Formula:
V_out = V_in * (x / L)
Error Calculation:
The error in the measurement is due to the uncertainty in the values of V_in, x, and L. Assuming the uncertainties in V_in and L are negligible compared to the uncertainty in x, we can calculate the relative error in V_out as:
relative error in V_out = relative error in x