resistance-inductance-estimation.md

Field	Value
Title	Electrical Parameters Identification — R and L
Type	theory
Status	approved
Version	1.0.0
Component	service-electrical-ident
Date	2025-01-01

Overview

FOC performance depends directly on accurate stator resistance $R_s$ and inductance $L_s$. These parameters are used to:

1. Design the PI current controller gains ($K_p = L_s \omega_{bw}$, $K_i = R_s \omega_{bw}$).
2. Implement feed-forward decoupling of the dq cross-coupling terms.
3. Estimate motor temperature from measured $R_s$ (since $R_s \propto T$).

This identification procedure applies a DC voltage step to the motor aligned on the d-axis and measures the current transient. Alignment to the d-axis suppresses the back-EMF (which only appears on the q-axis), yielding a clean RL circuit step response. Resistance is derived from the DC steady-state, and inductance from the measured time constant.

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Prerequisites

Symbol	Meaning	Unit
$R_s$	Stator resistance per phase	Ω
$L_s$	Stator inductance (d-axis, $L_d$)	H
$\tau$	Electrical time constant = $L_s / R_s$	s
$V_{step}$	Applied step voltage (d-axis)	V
$I_{ss}$	Steady-state current = $V_{step} / R_s$	A
$I_\tau$	Current at time $\tau$: $I_{ss} \cdot (1 - e^{-1})$	A
$f_s$	Sampling frequency	Hz
$T_s$	Sampling period = $1/f_s$	s
$N_{avg}$	Moving average filter length	samples
$N_{buf}$	Total sample buffer size	samples

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Mathematical Foundation

1. d-Axis Alignment and Back-EMF Suppression

Before applying the voltage step, the motor is aligned to $\theta_e = 0$ (rotor d-axis aligned with stator $\alpha$-axis). In this condition:

• The back-EMF is $e_\alpha = -\psi_f \omega_e \sin(0) = 0$.
• The q-axis current is forced to zero: $i_q = 0$.
• Only the d-axis RL circuit is excited.

The stator d-axis circuit model reduces to:

$$v_d = R_s\, i_d + L_s \frac{di_d}{dt}$$

This is a first-order linear system driven by a unit step of amplitude $V_{step}$.

2. RL Step Response

For a step input $v_d(t) = V_{step} \cdot u(t)$ with zero initial conditions ($i_d(0) = 0$):

$$\boxed{i_d(t) = \frac{V_{step}}{R_s}\!\left(1 - e^{-t/\tau}\right)}, \qquad \tau = \frac{L_s}{R_s}$$

Key properties of this response:

• At $t = \tau$: $i_d(\tau) = I_{ss}(1 - e^{-1}) \approx 0.6321 \cdot I_{ss}$
• At $t = 5\tau$: $i_d(5\tau) \approx 0.9933 \cdot I_{ss}$ (essentially settled)
• Slope at $t = 0$: $\left.\frac{di_d}{dt}\right|_{t=0} = \frac{V_{step}}{L_s}$

See: documentation/theory/images/rl_step_response.svg (generated by documentation/tools/generate_plots.gp)

3. Resistance Estimation

Once the current has fully settled to steady state (after $5\tau$):

$$\boxed{R_s = \frac{V_{step}}{I_{ss}}}$$

where $I_{ss}$ is measured from the mean of the last 10% of the sample buffer (to average out noise).

Noise considerations: $R_s$ estimation is straightforward but requires:

• Accurate current sensor calibration (ADC offset error directly biases $I_{ss}$).
• Stable $V_{step}$ (DC bus voltage variation during measurement corrupts the result).
• Sufficient settling time in the buffer ($N_{buf} \geq 5\tau / T_s$ samples).

4. Inductance Estimation — Time-Constant Method

The time constant $\tau = L_s / R_s$ is found by identifying the sample index $n_\tau$ where the current first reaches the 63.2% threshold:

$$i_d[n_\tau] \geq 0.6321 \cdot I_{ss}$$

The time constant is then:

$$\tau = n_\tau \cdot T_s$$

and the inductance:

$$\boxed{L_s = R_s \cdot \tau = R_s \cdot n_\tau \cdot T_s}$$

Moving Average Filter Correction

A causal moving average filter of length $N_{avg}$ is applied to the raw current samples before threshold detection:

$$\bar{i}[n] = \frac{1}{N_{avg}} \sum_{k=0}^{N_{avg}-1} i[n-k]$$

This FIR filter introduces a lag of $(N_{avg} - 1)/2$ samples. The threshold index is corrected:

$$n_\tau^{corrected} = n_\tau - \left\lfloor \frac{N_{avg} - 1}{2} \right\rfloor - 1$$

Without this correction, $\tau$ is overestimated by the filter group delay, leading to an overestimate of $L_s$.

Filter trade-off: larger $N_{avg}$ reduces noise variance $\propto 1/N_{avg}$ but increases the index correction and requires a corresponding increase in buffer size $N_{buf}$.

5. Pole Pair Estimation

The number of electrical cycles per mechanical revolution equals the number of pole pairs $p$. During the multi-step alignment sweep, the motor is driven through exactly 12 electrical steps over one full electrical revolution ($2\pi$electrical). The encoder counts$C_{mech}$per step multiplied by$N_{steps} = 12$gives the total encoder counts per full electrical cycle. Dividing by the known encoder counts per mechanical revolution$C_{rev}$ gives:

$$p = \frac{C_{rev}}{12 \cdot C_{per\_step}} \quad \text{(integer, rounded)}$$

or equivalently, the total electrical angle traversed over the full 12-step sweep spans exactly $2\pi electrical = \2\pi/p$ mechanical, so:

$$p = \frac{2\pi}{\Delta\theta_{mech,total}}$$

where $\Delta\theta_{mech,total}$ is the measured total mechanical rotation during the sweep.

6. Complete Identification Sequence

1. Drive alignment: rotate field through N_steps to θ_e = 0.
   Record encoder offset θ_offset (see alignment theory).

2. Apply step voltage V_step on d-axis (i_q* = 0, v_d = V_step).

3. Sample i_d at f_s = 10 kHz for N_buf samples.

4. Apply moving average filter (length N_avg = 5) to samples.

5. Find steady-state current I_ss from mean of last 10% of buffer.

6. Compute R_s = V_step / I_ss.

7. Find first index n_τ where i_d[n] ≥ 0.6321 · I_ss.

8. Correct for filter delay: n_τ_corr = n_τ − ⌊(N_avg−1)/2⌋ − 1.

9. Compute τ = n_τ_corr · T_s, L_s = R_s · τ [H].
   Express L_s in mH: L_s_mH = R_s · n_τ_corr / f_s · 1000.

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Block Diagrams

Electrical Identification Signal Flow

graph TD
    A[Set θ_e = 0\nAlign rotor to d-axis] --> B[Apply V_step\non d-axis\ni_q* = 0]
    B --> C[Sample i_d\nf_s = 10 kHz\nN_buf samples]
    C --> D[Moving Average\nFilter\nN_avg = 5]
    D --> E[Find I_ss\nmean of last 10%]
    D --> F[Find n_τ\nfirst sample ≥ 63.2% I_ss]
    E --> G[R_s = V_step / I_ss]
    F --> H[n_τ_corr = n_τ − delay]
    G --> I[L_s = R_s · n_τ_corr · T_s]
    H --> I

RL Step Response — ASCII Approximation

i_d (normalised: I_ss = 1.0)
  │
1.0├───────────────────────────────── I_ss = V/R (steady state)
   │                       ──────────
0.86├──────────────────────/
   │                     /
0.63├────────────────────/ ← i(τ) = 0.632·I_ss
   │                   /
   │                  /
0.39├─────────────────/
   │               /
   │             /
   │           /
0.0├───────────
   └───────────────────────────────── samples (n·T_s)
       0      τ/T_s   2τ/T_s  5τ/T_s
              ↑
          n_τ (63.2% crossing) — filter delay correction applied

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Numerical Properties

Property	Value / Condition
Sampling rate	$f_s = 10\ \text{kHz}$, $T_s = 100\ \mu\text{s}$
Filter length	$N_{avg} = 5$ samples
Filter group delay	$(N_{avg}-1)/2 = 2$ samples = $200\ \mu\text{s}$
Threshold	$0.6321 \cdot I_{ss} (i.e. \1 - e^{-1}$)
$R_s$ range	Nominally $0.1\ \Omega to \50\ \Omega$ (ADC current range)
$L_s$ resolution	$R_s \cdot T_s$ (one sample step) = depends on $R_s$
$L_s$ min detectable	Approx. $R_s \cdot 2 T_s$ (due to filter delay correction)
Trigger voltage	$V_{step}$ must be small enough to avoid magnetic saturation

Sensitivity Analysis

Source of Error	Effect on $R_s$	Effect on $L_s$
ADC current offset	Directly biases $I_{ss}$	Indirect via $R_s$ error
$V_{dc}$ variation	Biases $V_{step}$	Indirect via $R_s$ error
Thermal drift in $R_s$	Measurement valid at $T_{meas}$ only	—
Filter delay not corrected	—	$L_s$ overestimated by $N_{avg}/2$ steps
Insufficient buffer	$I_{ss}$ underestimated	$\tau$ underestimated
Magnetic saturation	$R_s$ underestimated	$L_s$ underestimated (nonlinear)

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Worked Example

Motor: $V_{step} = 2\ \text{V}$, $R_s = 1.2\ \Omega$, $L_s = 0.6\ \text{mH}$, $f_s = 10\ \text{kHz}$, $N_{avg} = 5$.

Expected results:

$I_{ss} = \frac{2}{1.2} \approx 1.667\ \text{A}$

$\tau = \frac{L_s}{R_s} = \frac{0.6 \times 10^{-3}}{1.2} = 0.5\ \text{ms} = 5\ T_s$

At the 63.2% threshold: $i_d[n_\tau] \geq 0.6321 \times 1.667 = 1.054\ \text{A}$

The raw crossing occurs at $n_\tau = 5$. Filter delay correction: $n_\tau^{corr} = 5 - 2 - 1 = 2$.

$L_{s,\mathrm{mH}} = 1.2 \times 2 \times 100 \times 10^{-6} \times 1000 = 0.24\ \text{mH}$

The example shows that a very short $\tau$ (5 samples) combined with a 5-tap filter and a 2-sample delay correction can yield significant estimation error. In practice $\tau$ should be at least 15–20 samples for accurate identification.

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Limitations & Assumptions

• Assumes: The rotor is aligned to the d-axis ($\theta_e = 0$, $\dot\theta = 0$). Any rotor motion during identification overlays a back-EMF on the d-axis current.
• Assumes: $L_d \approx L_q$ (surface-mounted PMSM). For interior PMSM, the step must be repeated on both axes if the design requires both $L_d$ and $L_q$.
• Assumes: Magnetic linearity (no saturation). The identification current $I_d^{step}$ must be kept below the saturation current.
• Does not handle: Temperature-dependent $R_s$ variation during operation.
• Does not handle: Identification at running speed where back-EMF cannot be zeroed by alignment alone.

References

1. Rauf, A. et al. — "Online Identification of PMSM Parameters Based on Extended Kalman Filter", IEEE Transactions on Industrial Electronics, 2019.
2. Underwood, S.J. & Husain, I. — "Online Parameter Estimation and Adaptive Control of PMSM", IEEE Transactions on Industrial Electronics, 2010.
3. Texas Instruments Application Report SPRABV5 — Motor Control in Embedded Applications, 2018.