# Resolver Decoder

Compute motor mechanical position and speed as well as sine and cosine values of motor electrical position

**Library:**Motor Control Blockset / Sensor Decoders

## Description

The Resolver Decoder block computes the following for a resolver connected to a motor shaft:

Mechanical angular position of motor

Sine and cosine of electrical angular position of motor

Mechanical speed of motor

### Resolver Operation

A resolver uses a primary excitation input signal to generate the modulated secondary sine and cosine waveforms, which is then sampled by the ADC. The resolver utilizes one winding to two winding transformations. The sine and cosine modulation occurs in the secondary windings because of the design and construction of these windings, which places them at positions that are 90 degrees apart.

### Block algorithm

The block uses the correctly sampled and normalized version of secondary sine and cosine waveforms to demodulate the sine and cosine signals using which it determines the electrical position of the resolver. It converts this position into its mechanical equivalent according to the number of pole pairs available in the resolver. The resulting value indicates the mechanical position of the motor.

The block also uses the demodulated sine and cosine signals (sine and cosine of resolver electrical angle) to compute the motor speed as well as the sine and cosine values of the motor electrical position.

### Resolver excitation methods

The block supports the following two excitation methods:

Sinusoidal excitation — When the primary excitation signal is sinusoidal. This generates sinusoidal secondary waveforms after modulation.

Square pulse excitation — When the primary excitation signal is a square pulse signal. This generates square pulse secondary signals after modulation.

### Sinusoidal excitation

When you use sinusoidal excitation method, the block expects the secondary sine and cosine resolver signals that are sampled, by default, at the rate of 16 samples per excitation signal cycle, as shown below. It may also add a phase delay to the sampled sine and cosine signals with respect to the excitation signal.

When using sinusoidal excitation, the block does not normalize the modulated sine and cosine resolver output automatically. Normalize these modulated waveforms (within the range of [-1,1] and centred at 0) before you provide them to the block.

The block then demodulates the sampled signal. It computes the average, peak amplitude values, and the sign of the peak amplitude of a signal cycle as:

The block computes the average, peak amplitude values, and the sign of the peak amplitude of a signal cycle as

${\u212b}_{average}=\frac{1}{n}{\displaystyle \sum}_{i=0}^{n-1}(\left|{\u212b}_{i}\right|)$

${\u212b}_{peak}={\u212b}_{average}\times \frac{\pi}{2}$

$SignofPeak=Signof\text{}\left[{\displaystyle \sum}_{i=phasedelay}^{\frac{n}{2}-1+phasedelay}{\u212b}_{i}\right]$

where:

${\u212b}_{average}$ is the average amplitude value of a signal cycle

$$n$$ is the number of samples per excitation cycle

${\u212b}_{peak}$ is the peak amplitude value of a signal cycle

The block computes the electrical angular position of the resolver as

$\theta =\text{atan}2\frac{{u}_{\text{sin}\_peak}\text{}}{{u}_{\text{cos}\_peak}\text{}}$

where:

${u}_{\text{sin}\_peak}$ is the ${\u212b}_{peak}$ of the secondary sine signal

${u}_{\text{cos}\_peak}$ is the ${\u212b}_{peak}$ of the secondary cosine signal

$\theta $ is the electrical angular position of the resolver

This enables the block to demodulate and extract the sine and cosine envelopes. It uses these demodulated envelop signals to compute the block outputs.

### Square pulse excitation

When using the square pulse excitation method, the block expects the secondary sine and cosine signals sampled at the rate of one sample every pulse as shown below.

If you select **Enable input normalization** field, then after
receiving the discrete time sampled waveform, the block automatically normalizes the
waveform (within the range of [-1,1] and centred at 0). To perform normalization within the
block, ensure that both input signals have equal peak magnitudes.

It demodulates the sine and cosine signals. It uses these demodulated signals to compute the block outputs.

### Computation of motor mechanical position

When using either sinusoidal or square pulse excitation, the block uses the demodulated waveforms to compute the electrical position of the resolver. This resolver position may vary using either a positive ramp (for clockwise rotation) or a negative ramp (for anti-clockwise rotation).

To correctly detect the wrap-around (from 0 PU to 1 PU or from 1 PU to 0 PU), the block measures the difference between the two consecutive samples. Because the input signal frequency is always less than ½ of sampling frequency, a difference less than -0.5 PU indicates a positive ramp whereas a difference that is less than +0.5 PU indicates a negative ramp.

After identifying the resolver electrical position direction, the block uses the number of resolver pole pairs to compute the mechanical position of the resolver (and the motor) by extrapolating the ramp signal.

For example, for a 3-pole pair resolver, the block extrapolates the ramp to achieve a magnitude that is 3 times the original magnitude. It then performs a division by 3 to obtain the resolver mechanical position as shown below.

### Computation of sine and cosine of electrical motor position

When using either sinusoidal or square pulse excitation, the block generates the demodulated sine and cosine waveforms (sine and cosine of resolver electrical position). From these demodulated waveforms, the block computes sine and cosine of motor electrical position using an arithmetic computation according to the ratio of motor and resolver pole pairs. To accommodate different possible ratios, the block utilizes unit blocks, which in turn use a generic binary-coded decimal (BCD) based structure.

For example, if the ratio of motor and resolver pole pairs (*n*) is 5,
the following image indicates the BCD based algorithm that eventually generates sine and
cosine of motor electrical position.

Therefore, *n* = number of motor pole pairs/ number of resolver pole
pairs = 5.

BCD code for *n* = `5`

is ```
[0 0 1 0
1]
```

or `[2`

. The block configures the algorithm for this
sequence as shown below.^{4} 2^{3}
2^{2} 2^{1}
2^{0}]

The block supports values of *n* (integers) ranging from 1 to
31.

**Note**

The block optimizes the computation according to the ratio n. For example, if
*n* = `5`

, the block only computes
`sin_2th`

and `cos_2th`

twice by utilizing the
following two subsystems only.

Similarly, whenever a switch does not bypass the signals, it terminates them as shown below to ensure minimal and optimized code generation.

### Computation of motor speed

When using either sinusoidal or square pulse excitation, the block generates the demodulated sine and cosine waveforms (sine and cosine of resolver electrical position). From these waveforms, the block computes the motor mechanical speed using the algorithm shown below.

At first, it computes the difference between the two samples of these discrete time sampled signals and adds them:

In the ideal scenario, when using continuous time signals:

$\frac{d}{dt}\mathrm{sin}\theta ={\omega}_{r}\mathrm{cos}{\omega}_{r}{}_{}t$

$\frac{d}{dt}\mathrm{cos}\theta =-{\omega}_{r}{}_{}\mathrm{sin}{\omega}_{r}t$

$\mathrm{cos}\theta \frac{d}{dt}\mathrm{sin}\theta -\mathrm{sin}\theta \frac{d}{dt}\mathrm{cos}\theta ={\omega}_{r}{\mathrm{cos}}^{2}\theta +{\omega}_{r}{\mathrm{sin}}^{2}\theta ={\omega}_{r}{}_{}$

$\theta ={\omega}_{r}{}_{}t$

Where, *ω _{r}* is constant with respect to
time.

Because we use discrete time sampled signals:

$\mathrm{cos}\theta \frac{\Delta \mathrm{sin}\theta}{\Delta t}-\mathrm{sin}\theta \frac{\Delta \mathrm{cos}\theta}{\Delta t}$

$\mathrm{sin}\theta =\mathrm{sin}{\omega}_{r}{T}_{s}n$

$\mathrm{cos}\theta =\mathrm{cos}{\omega}_{r}{T}_{s}n$

$\frac{\Delta \mathrm{sin}\theta}{\Delta t}=\frac{\mathrm{sin}{\omega}_{r}{T}_{s}n-\mathrm{sin}\left[{\omega}_{r}{T}_{s}\left(n-1\right)\right]}{{T}_{s}}$

$\frac{\Delta \mathrm{cos}\theta}{\Delta t}=\frac{\mathrm{cos}{\omega}_{r}{T}_{s}n-\mathrm{cos}\left[{\omega}_{r}{T}_{s}(n-1)\right]}{{T}_{s}}$

$\mathrm{cos}\theta \frac{\Delta \mathrm{sin}\theta}{\Delta t}-\mathrm{sin}\theta \frac{\Delta \mathrm{cos}\theta}{\Delta t}=\frac{\mathrm{sin}{\omega}_{r}{T}_{s}}{{T}_{s}}$

$$\Delta t={T}_{s}$$

$$\frac{\mathrm{cos}\theta \cdot \Delta \mathrm{sin}\theta}{{T}_{s}}-\frac{\mathrm{sin}\theta \cdot \Delta \mathrm{cos}\theta}{{T}_{s}}=\frac{\mathrm{sin}{\omega}_{r}{T}_{s}}{{T}_{s}}$$

$\mathrm{cos}\theta \cdot \Delta \mathrm{sin}\theta -\mathrm{sin}\theta \cdot \Delta \mathrm{cos}\theta =\mathrm{sin}{\omega}_{r}{T}_{s}$

If *ωxT _{s}* is very small, then

*sinω*approximately equals to

_{r}T_{s}*ω*. This is

_{r}T_{s}`speed x time`

, therefore, this term indicates position difference for a
sample duration `Δθ`

.$\mathrm{cos}\theta \cdot \Delta \mathrm{sin}\theta -\mathrm{sin}\theta \cdot \Delta \mathrm{cos}\theta =\Delta \theta =\mathrm{sin}{\omega}_{r}{T}_{s}\approx {\omega}_{r}{T}_{s}$

${\omega}_{r}=\frac{1}{{T}_{s}}{\mathrm{sin}}^{-1}\Delta \theta $

${\omega}_{r}\approx \frac{\Delta \theta}{{T}_{s}}$

To obtain a more accurate speed, we use the Taylor series expansion of
*sin ^{-1}* to add the following compensation of $$\frac{\Delta {\theta}^{3}}{6}$$.

${\omega}_{r}=\frac{1}{{T}_{s}}{\mathrm{sin}}^{-1}\Delta \theta $

$\Rightarrow {\omega}_{r}\approx \frac{1}{{T}_{s}}\left[\Delta \theta +\frac{\Delta {\theta}^{3}}{6}\right]$

Because the error is negligible, the preceding
*ω _{r}* value can be considered accurate. Therefore,
we can state:

${\omega}_{r}=\frac{1}{{T}_{s}}\left[\Delta \theta +\frac{\Delta {\theta}^{3}}{6}\right]$

${\omega}_{r}=\left(\frac{g}{{T}_{s}}\right)\left[\Delta \theta +\frac{\Delta {\theta}^{3}}{6}\right]$

$$\omega =\frac{{\omega}_{r}}{{P}_{r}}$$

$\omega =\left(\frac{g}{{T}_{s}Pr}\right)\left[\Delta \theta +\frac{\Delta {\theta}^{3}}{6}\right]$

Where, *P _{r}* is the number of pole pairs of
resolver.

The block multiplies the gain of $\left(\frac{g}{{T}_{s}Pr}\right)$ separately to avoid data overflow when using fixed point data type.

Here, $\frac{\Delta {\theta}^{3}}{6}$ is the approximate compensation that the block applies to the computed speed as shown below to obtain more accurate version of the actual mechanical speed.

Where:

*g*is the speed conversion factor.*θ*is the electrical position of the resolver.*ω*is the electrical speed of the resolver._{r}*Δθ*is the resolver electrical position difference per sample.*ω*is the mechanical speed of the resolver (or motor).*T*is the sampling time of the sine and cosine envelop signals._{s}

**Note**

The block inputs should have identical amplitude and data types (either signed fixed or floating point).

## Ports

### Input

### Output

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2020a**