Signal Processing

Base Concept

Calculation & data processing done below order,

Remove the DC offset in all signal.
RMS Amplitude of voltage, current signal.
Calculate RMS Voltage, Current
With modulation method, calculate the phase
Now derive the complex impedance value

Reference from TI TIDA-060029

Amplitude & Phase calculation

RMS value - Voltage & Current

RMS(Root Mean Square) is the square root of the mean square of a set of numbers or a continuous function $$ x_\text{RMS} = \sqrt{ \frac{1}{n} \left( {x_1}^2 + {x_2}^2 + \cdots + {x_n}^2 \right) }. $$

Phase calculate option 1 - Goertzel algorithm

The Goertzel algorithm provides an efficient method for computing a single DFT (Discrete Fourier Transform) term at a specific frequency. This implementation is particularly useful for embedded systems and microcontrollers where computational resources are limited.

The implementation follows these steps:

Calculates the angular frequency (omega) and coefficient
Processes all samples using a second-order filter
Computes the magnitude of the real and imaginary components

Computational Efficiency

Time Complexity: O(N)
More efficient than FFT for single frequency analysis
Requires minimal memory overhead

Usage Example

The Goertzel algorithm efficiently computes a single DFT term for a specific target frequency. Here's the implementation:

float goertzel(float* data, int N, float target_freq, float sampling_freq) {
  float omega = (2.0 * M_PI * target_freq) / sampling_freq;
  float coeff = 2.0 * cos(omega);

  float q0 = 0;
  float q1 = 0;
  float q2 = 0;

  // Process all samples
  for (int i = 0; i < N; i++) {
    q0 = coeff * q1 - q2 + data[i];
    q2 = q1;
    q1 = q0;
  }

  // Calculate magnitude
  float real = (q1 - q2 * cos(omega));
  float imag = (q2 * sin(omega));
  float magnitude = sqrt(real*real + imag*imag);

  return magnitude;
}

The algorithm uses less computational resources than FFT when analyzing a single frequency component.

Phase calculate option 2 - Synchronous Demodulation

The voltages are acquired using a two channel differential ADC and processed in the following form. Followed below steps

Modulation of the signal by multiplying the signal with a unity magnitude square wave of 0 degree phase and taking the average of the resulting signal ($V_1$).
Modulation of the signal by multiplying the signal with a unity magnitude square wave of 90 degree phase and taking the average of the resulting signal ($V_2$).

$$ \alpha = \tan^{-1} (\frac{V_2}{V_1}) $$ phase ‘$\alpha$’ of any signal is estimated with respect to the reference signal.