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Model rotary transformer that measures motor rotation angle
The Resolver block models a generic resolver, which consists of a rotary transformer that couples an AC voltage applied to the primary winding to two secondary windings. These secondary windings are physically oriented at 90 degrees to each other. As the rotor angle changes, the relative coupling between the primary and the two secondary windings varies. In the Resolver block model, the first secondary winding is oriented such that peak coupling occurs when the rotor is at zero degrees, and therefore the second secondary winding has minimum coupling when the rotor is at zero degrees.
Without loss of generality, it is assumed that the transformer between primary and rotor circuit is ideal with a ratio of 1:1. This results in the rotor current and voltage being equivalent to the primary current and voltage.
You have two options for defining the block equations:
Omit the dynamics by neglecting the transformer inductive terms. This model is only valid if the sensor is driven by a sine wave because any DC component on the primary side will pass to the output side.
Include the inductive terms, thereby capturing voltage amplitude loss and phase differences. This model is valid for any input waveform. Within this option, you can either specify the inductances and the peak coupling coefficient directly, or specify the transformation ratio and measured impedances, in which case the block uses these values to determine the inductive terms.
The equations are based on the superposition of two ideal transformers, both with coupling coefficients that depend on rotor angle. The two ideal transformers have a common primary winding. See the Simscape™ Ideal Transformer block reference page for more information on modeling ideal transformers. The equations are:
K_{x} = R cos(N Θ)
K_{y} = R sin(N Θ)
v_{x} = K_{x}v_{p}
v_{y} = K_{y}v_{p}
i_{p} = –K_{x}i_{x} – K_{y}i_{y}
where:
v_{p} and i_{p} are the rotor (or equivalently primary) voltage and current, respectively.
v_{x} and i_{x} are the first secondary voltage and current, respectively.
v_{y} and i_{y} are the second secondary voltage and current, respectively.
K_{x} is the coupling coefficient for the first secondary winding.
K_{y} is the coupling coefficient for the second secondary winding.
R is the transformation ratio.
N is the number of pole pairs.
Θ is the rotor angle.
The equations are based on the superposition of two mutual inductors, both with coupling coefficients that depend on rotor angle. The two mutual inductors have a common primary winding. See the Simscape Mutual Inductor block reference page for more information on modeling mutual inductors. The equations are:
$${v}_{p}={R}_{p}{i}_{p}+{L}_{p}\frac{d{i}_{p}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\left(\mathrm{cos}\left(N\theta \right)\frac{d{i}_{x}}{dt}+\mathrm{sin}\left(N\theta \right)\frac{d{i}_{y}}{dt}\right)$$
$${v}_{x}={R}_{s}{i}_{x}+{L}_{s}\frac{d{i}_{x}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\mathrm{cos}\left(N\theta \right)\frac{d{i}_{p}}{dt}$$
$${v}_{y}={R}_{s}{i}_{y}+{L}_{s}\frac{d{i}_{y}}{dt}+\sqrt{{L}_{p}{L}_{s}}k\mathrm{sin}\left(N\theta \right)\frac{d{i}_{p}}{dt}$$
where:
v_{p} and i_{p} are the rotor (or equivalently primary) voltage and current, respectively.
v_{x} and i_{x} are the first secondary voltage and current, respectively.
v_{y} and i_{y} are the second secondary voltage and current, respectively.
R_{p} is the rotor (or primary) resistance.
L_{p} is the rotor (or primary) inductance.
R_{s} is the stator (or secondary) resistance.
L_{s} is the stator (or secondary) inductance.
N is the number of pole pairs.
k is the coefficient of coupling.
Θ is the rotor angle.
It is assumed that coupling between the two secondary windings is zero.
Datasheets typically do not quote the coefficient of coupling and inductance parameters, but instead give the transformation ratio R and measured impedances. If you select Specify transformation ratio and measured impedances for the Parameterization parameter, then the values you provide are used to determine values for the equation coefficients, as defined above.
The model is based on the following assumptions:
The resolver draws no torque between the mechanical rotational ports R and C.
The transformer between primary and rotor circuit is ideal with a ratio of 1:1.
The coupling between the two secondary windings is zero.
Select one of the following methods for block parameterization:
Specify transformation ratio and omit dynamics — Provide values for transformation ratio, number of pole pairs, and initial rotor angle only. This model neglects the transformer inductive terms, and is only valid if the sensor is driven by a sine wave. The equations are based on the superposition of two ideal transformers, both with coupling coefficients that depend on rotor angle. For more information, see Equations when Omitting Dynamics. This is the default option.
Specify transformation ratio and measured impedances — Provide additional values to determine the transformer inductive terms, to model the voltage amplitude loss and phase differences. This model is valid for any input waveform. The equations are based on the superposition of two mutual inductors, both with coupling coefficients that depend on rotor angle. For more information, see Equations when Including Dynamics.
Specify equation parameters directly — Model the dynamics, but provide values for rotor and stator inductances and the peak coefficient of coupling, instead of transformation ratio and measured impedances. For more information, see Equations when Including Dynamics. This model is valid for any input waveform.
The ratio between peak output voltage and peak input voltage assuming negligible secondary voltage drop due to resistance and inductance. This parameter is only visible when you select Specify transformation ratio and omit dynamics or Specify transformation ratio and measured impedances for the Parameterization parameter. If you select Specify transformation ratio and measured impedances for the Parameterization parameter, then the transformation ratio takes into account the voltage drop due to primary winding resistance. The default value is 0.5.
This is the rotor (or equivalently the primary) ohmic resistance. This parameter is only visible when you select Specify transformation ratio and measured impedances or Specify equation parameters directly for the Parameterization parameter. The default value is 70 Ω.
This is the secondary winding ohmic resistance. It is assumed that both secondaries have the same resistance. This parameter is only visible when you select Specify transformation ratio and measured impedances or Specify equation parameters directly for the Parameterization parameter. The default value is 180 Ω.
This is the rotor (or equivalently the primary) reactance with the secondary windings open-circuit. This parameter is only visible when you select Specify transformation ratio and measured impedances for the Parameterization parameter. The default value is 100 Ω.
This is the stator (or equivalently the secondary) reactance with the primary winding open-circuit. This parameter is only visible when you select Specify transformation ratio and measured impedances for the Parameterization parameter. The default value is 300 Ω.
This is the frequency of the sinusoidal source used when measuring the reactances. This parameter is only visible when you select Specify transformation ratio and measured impedances for the Parameterization parameter. The default value is 10 kHz.
This is the rotor (or equivalently the primary) inductance L_{p}. This parameter is only visible when you select Specify equation parameters directly for the Parameterization parameter. The default value is 0.0016 H.
This is the stator (or equivalently the secondary) inductance L_{s}. This parameter is only visible when you select Specify equation parameters directly for the Parameterization parameter. The default value is 0.0048 H.
This is the peak coefficient of coupling between the primary and secondary windings. The parameter value should be greater than zero and less than one. This parameter is only visible when you select Specify equation parameters directly for the Parameterization parameter. The default value is 0.35.
The number of pole pairs on the rotor. The default value is 1.
The initial angle of the rotor, Θ. The default value is 0 degrees.