The block uses the PHF injection technique. PHF injection needs initial estimation to start the algorithm.

If you use only one initial estimation, θ_{i_est}, (for PHF) for all possible actual rotor positions, the block algorithm might not work accurately for certain actual rotor positions when the motor has low saliency. These ambiguous positions are:

To make PHF work for motors with low saliency, the block picks different initial estimation for different actual rotor positions.

The block picks the best possible initial estimation from these three alternatives:

Therefore, the block sequentially injects three high-frequency voltage signals (that have the same peak amplitude V_phf and the same frequency f_h) across the preceding 3 θ_{i_est} values and measures the resulting i_q currents.

The following equations describe the injected voltages in the estimated d-q reference frame:

The following equations describe the injected voltages in the estimated α-β reference fame:

The following equations describe the injected voltages in the actual d-q reference frame:

We can derive the following equation for the resulting i_q current using the IPMSM motor dynamic equations:

Therefore, $i_{q\_peak}\propto \left|sin\left(2({\theta }_{actual}-{\theta }_{i\_est})\right)\right|$

, where

θ_{i_est} — Initial estimation of rotor position (which can be either 0, 2π/3, or -2π/3) (in degrees, per-unit, or radians).

θ_actual — Actual rotor position (in degrees, per-unit, or radians)

θ_err = θ_actual-θ_{i_est} (in degrees, per-unit, or radians).

L_d and L_q are the d and q axis inductances of the IPMSM.

V_{d_est} and V_{q_est} are the voltages injected across the d and q axes of the estimated d-q reference frame.

V_d and V_q are the voltages injected across the d and q axes of the actual d-q reference frame.

V_α and V_β are the voltages injected across the α and β axes of the stationary α-β reference frame.

With

i_q1 = i_{q_peak} for θ_{i_est} = 0

i_q2 = i_{q_peak} for θ_{i_est} = 2π/3

i_q3 = i_{q_peak} for θ_{i_est} = -2π/3

For θ_{i_est} = 0, when you vary θ_actual from 0 to 2π, $i_{q1}\propto \left|sin 2({\theta }_{actual}-0)\right|$ is the maximum value between the i_q1, i_q2, and i_q3 current peak values when the rotor lies in the following four highlighted regions.

If θ_{i_est} = 0 is used for all θ_actual = 0 to 2π, the algorithm fails for following ambiguous regions (when motor saliency is low):

Because these ambiguous regions are not present in the preceding four regions, you can eliminate the regions where the algorithm might fail.

For θ_{i_est} = (2π/3), when you vary θ_actual from 0 to 2π, $i_{q2}\propto \left|sin 2({\theta }_{actual}-\frac{2\pi }{3})\right|$ is the maximum value between the i_q1, i_q2, and i_q3 current peak values when the rotor lies in the following four highlighted regions.

If θ_{i_est} = (2π/3) is used for all θ_actual = 0 to 2π, the algorithm fails for following ambiguous regions (when motor saliency is low):

Because these ambiguous regions are not present in the preceding four regions, you can eliminate the regions where the algorithm might fail.

For θ_{i_est} = -(2π/3), when you vary θ_actual from 0 to 2π, $i_{q3}\propto \left|sin 2({\theta }_{actual}-\left(-\frac{2\pi }{3}\right))\right|$ is the maximum value between i_q1, i_q2, and i_q3 current peak values when the rotor lies in the following four highlighted regions.

If θ_{i_est} = -(2π/3) is used for all θ_actual = 0 to 2π, the algorithm fails for following ambiguous regions (when motor saliency is low):

Because these ambiguous regions are not present in the preceding four regions, you can eliminate the regions where the algorithm might fail.

Therefore, using this approach you have three sets of regions corresponding to three different initial estimates. These three sets or regions cover all the motor sectors where the rotor can lie:

In part A, the block picks the θ_{i_est} corresponding to the maximum i_q peak current value between i_q1, i_q2, and i_q3 for Part B.

After determining the best possible initial estimate θ_{i_est}, the block injects a sinusoidal high-frequency voltage (with peak amplitude V_phf and frequency f_h) along the finalized (θ_est | t = 0) = θ_{i_est} and reads the q-axis current response of the motor as described in Part A. However, in addition, it applies signal processing routines on the q-axis current response to make corrections to θ_est using a PI controller in a closed-loop configuration as shown below:

The following steps describe the signal processing routines used by the block:

From the output of the preceding 3 steps, we can derive the following error signal:

Because we can approximate sin(θ) to θ (when θ is close to zero), we can derive the following approximated error signal after considering that θerr is close to zero:

Therefore, using the preceding result, we can simplify model 1 as shown below:

The following term represents the transfer function (simplified) of the entire model:

where, k_p and k_i proportional and integral gains of the PI controller.

Therefore, when the block executes, the estimated position is initially θ_{i_est}, but it rises steadily (dynamically according to G, k_p, and k_i gains) to saturate at a certain angle (with respect to a-axis) such that Error_signal is close to zero.

The approximation used earlier introduces a limitation in the PHF method due to which the algorithm might compute the rotor position with an ambiguity of π. If the rotor lies in the range $${\theta }_{actual}=\left(0,\frac{\pi }{2}\right]\cup \left[\frac{3\pi }{2},2\pi \right)$$ electrical radians (region 1), the estimated position is accurate (π compensation is not required).

If the rotor lies in the range $${\theta }_{actual}=\left(\frac{\pi }{2},\frac{3\pi }{2}\right)$$ electrical radians (region 2), the estimated position shows an ambiguity of π (π compensation is required).

Therefore, the block uses the dual-pulse method to determine if the estimated position needs π compensation.

The block injects two very short duration voltage pulses (with the same width and magnitude) in the following positions:

Because pulse width is very short, the motor does not run, and the rotor remains stationary after pulse injection.

The interaction between the resulting stator magnetic flux and the rotor permanent magnets results in two current impulses along the d-axis of the rotor that rise and fall quickly.

Because the stator core is saturated, it shows nonlinear behaviour. A small L_d results in higher current I_d, and a high L_d results in smaller current I_d. Therefore, the I_d current impulses generated by Pulse 1 and Pulse 2 show different peak values.

The block computes the difference between the peak values of the two current impulses ΔI_d to determine if the position estimated in Part A needs π compensation.

ΔI_d = |I_d1| - |I_d2|

PHF injection benefits use cases that require the rotor to remain stationary or that require position estimation without starting the motor. The PHF injection technique also benefits use cases where you need to avoid open loop runs to estimate position (before transitioning to closed-loop speed control) or where you require the motor to start directly in the closed-loop mode. The block algorithm in stage 1 algorithm addresses these use cases by estimating the position while keeping the rotor stationary and avoiding an open-loop run.