Frequency warping when integrators are replaced with backward-euler and forward-euler integration Resonant controllers are used in the power industry. The transfer function
is
$G_{res}(s) = \frac{K_i s}{s^2 +2\omega_c s+ \omega_o^2}$
The &quot;textbook&quot; discrete implementation is depicted in the image The first integrator is a forward Euler and the second one a backward integrator. It is not possible to use Tustin integrators because this would create algebraic loops. Note that $\omega_o$ is not a constant, it will vary slowly and will be fed by a PLL or FLL.
Here's my question. How can one characterize the frequency warping?
In one paper, they evaluate frequency warping by applying the bilinear transform to $G_{res}(s)$, I feel like this is wrong.
To the extent this helps, here are some interesting magnitude and phase plots of the backward Euler (here using the method of impulse invariance mapping which for 1/s has the same result as backward Euler but as Matt L points out in the comments is not generally the case) vs Tustin (method of Bilinear Transform mapping) integrators. Notice the Euler on the left has dominant phase distortion while the Tustin on the right has more dominant amplitude distortion (no phase distortion from an analog integrator at all!).

The analog frequency response for a pure integrator as $1/s$ is -20 dB/decade in magnitude ($1/f$) and -90° for phase.