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| − | == | + | ==Problem 3== |
===<small>Voltage Equation Transformation</small>=== | ===<small>Voltage Equation Transformation</small>=== | ||
| − | To move from the rotor reference frame (or rotor phase variables) to the stationary reference frame, pre-multiply by <math>\mathbf{K}_r^s = \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin | + | To move from the rotor reference frame (or rotor phase variables) to the stationary reference frame, pre-multiply by <math>\mathbf{K}_r^s = \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \end{bmatrix}</math>. To do the opposite and move from the stationary reference frame to the rotor reference frame (or rotor phase variables), pre-multiply by <math>\mathbf{K}_s^r = \left(\mathbf{K}_r^s\right)^{-1}</math>. |
The voltage equation is the first to be transformed. | The voltage equation is the first to be transformed. | ||
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<math>\begin{align} | <math>\begin{align} | ||
| − | \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin | + | \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \end{bmatrix} \omega_r \begin{bmatrix} -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \\ +\sin\left(\theta_r\right) & +\cos\left(\theta_r\right) \end{bmatrix} \\ |
\mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \omega_r \begin{bmatrix} 0 & -1 \\ +1 & 0 \end{bmatrix} | \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \omega_r \begin{bmatrix} 0 & -1 \\ +1 & 0 \end{bmatrix} | ||
\end{align}</math> | \end{align}</math> | ||
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===<small>Flux Linkage Equation Transformation</small>=== | ===<small>Flux Linkage Equation Transformation</small>=== | ||
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| + | Similar steps are used for the flux linkage equation. | ||
| + | |||
| + | <math>\begin{align} | ||
| + | \vec{\lambda}_{qdr}^{s'} &= \mathbf{K}_r^s \vec{\lambda}_{12r}^{'} \\ | ||
| + | \vec{\lambda}_{qdr}^{s'} &= \mathbf{K}_r^s L_m \begin{bmatrix} +\cos\left(\theta_r\right) & -\sin\left(\theta_r\right) \\ -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \end{bmatrix} \mathbf{K}_s^s \vec{i}_{qds}^{s} \\ | ||
| + | \vec{\lambda}_{qdr}^{s'} &= L_m \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin(\left(\theta_r\right) \end{bmatrix} \begin{bmatrix} +\cos\left(\theta_r\right) & -\sin\left(\theta_r\right) \\ -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \end{bmatrix} \begin{bmatrix} +1 & 0 \\ 0 & -1 \end{bmatrix} \vec{i}_{qds}^{s} \\ | ||
| + | \vec{\lambda}_{qdr}^{s'} &= L_m \begin{bmatrix} 0 & +1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} +1 & 0 \\ 0 & -1 \end{bmatrix} \vec{i}_{qds}^{s} | ||
| + | \end{align}</math> | ||
| + | |||
| + | The flux linkage equation is finished. | ||
| + | |||
| + | <math>\begin{equation} | ||
| + | \boxed{\vec{\lambda}_{qdr}^{s'} = L_m \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \vec{i}_{qds}^{s}} | ||
| + | \end{equation}</math> | ||
| + | |||
| + | ===<small>Frequency of Balanced Circuit Variables</small>=== | ||
| + | |||
| + | The stator circuit variables have an input radial frequency of <math>\omega_e</math>. The frequency of circuit variables is the input frequency with respect to a stationary reference minus the radial speed of the arbitrary reference frame. In this case, the speed of the currents, voltages, and flux linkages in the stationary reference frame is <math>\omega_e - 0 = \boxed{\omega_e}</math>. | ||
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Latest revision as of 12:49, 16 January 2018
Answers and Discussions for
Contents
Problem 3
Voltage Equation Transformation
To move from the rotor reference frame (or rotor phase variables) to the stationary reference frame, pre-multiply by $ \mathbf{K}_r^s = \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \end{bmatrix} $. To do the opposite and move from the stationary reference frame to the rotor reference frame (or rotor phase variables), pre-multiply by $ \mathbf{K}_s^r = \left(\mathbf{K}_r^s\right)^{-1} $.
The voltage equation is the first to be transformed.
$ \begin{align} \vec{v}_{qdr}^{s'} &= \mathbf{K}_r^s \vec{v}_{12r}^{'} \\ \vec{v}_{qdr}^{s'} &= \mathbf{K}_r^s \mathbf{r}_r^{'} \left(\mathbf{K}_s^r \vec{i}_{qdr}^{s'}\right) + \mathbf{K}_r^s \mathit{p} \left(\mathbf{K}_s^r \vec{\lambda}_{qdr}^{s'}\right) \\ \vec{v}_{qdr}^{s'} &= r_r^{'} \cancelto{\mathbf{I}_2}{\mathbf{K}_r^s \mathbf{I}_2 \mathbf{K}_s^r} \vec{i}_{qdr}^{s'} + \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) \vec{\lambda}_{qdr}^{s'} + \cancelto{\mathbf{I}_2}{\mathbf{K}_r^s \mathbf{K}_s^r} \mathit{p}\vec{\lambda}_{qdr}^{s'} \end{align} $
The property that $ \mathbf{r}_r^{'} = r_r^{'} \mathbf{I}_2 $ was exploited with the commutative nature of scalar multiplication to simplify the first term. The Product Rule is applied for $ \mathit{p} \left(\mathbf{K}_s^r \vec{\lambda}_{qdr}^{s'}\right) $ to produce two terms, the latter of which also simplifies due to $ \mathbf{K}_r^s $ and $ \mathbf{K}_s^r $ being matrix inverses. A detour is needed to simplify the middle term. Using the First Fundamental Theorem of Calculus, $ \mathit{p}\theta_r(t) = \mathit{p} \int_{\tau = 0}^{t} \omega_r(\tau) \, d\tau + \mathit{p}\theta_r(0) = \omega_r(t) $.
$ \begin{align} \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \end{bmatrix} \omega_r \begin{bmatrix} -\cos\left(\theta_r\right) & +\sin\left(\theta_r\right) \\ +\sin\left(\theta_r\right) & +\cos\left(\theta_r\right) \end{bmatrix} \\ \mathbf{K}_r^s \left(\mathit{p} \mathbf{K}_s^r\right) &= \omega_r \begin{bmatrix} 0 & -1 \\ +1 & 0 \end{bmatrix} \end{align} $
The voltage equation is finished.
$ \begin{equation} \boxed{\vec{v}_{qdr}^{s'} = r_r^{'} \vec{i}_{qdr}^{s'} + \omega_r \begin{bmatrix} 0 & -1 \\ +1 & 0 \end{bmatrix} \vec{\lambda}_{qdr}^{s'} + \mathit{p}\vec{\lambda}_{qdr}^{s'}} \end{equation} $
Flux Linkage Equation Transformation
Similar steps are used for the flux linkage equation.
$ \begin{align} \vec{\lambda}_{qdr}^{s'} &= \mathbf{K}_r^s \vec{\lambda}_{12r}^{'} \\ \vec{\lambda}_{qdr}^{s'} &= \mathbf{K}_r^s L_m \begin{bmatrix} +\cos\left(\theta_r\right) & -\sin\left(\theta_r\right) \\ -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \end{bmatrix} \mathbf{K}_s^s \vec{i}_{qds}^{s} \\ \vec{\lambda}_{qdr}^{s'} &= L_m \begin{bmatrix} -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \\ -\cos\left(\theta_r\right) & +\sin(\left(\theta_r\right) \end{bmatrix} \begin{bmatrix} +\cos\left(\theta_r\right) & -\sin\left(\theta_r\right) \\ -\sin\left(\theta_r\right) & -\cos\left(\theta_r\right) \end{bmatrix} \begin{bmatrix} +1 & 0 \\ 0 & -1 \end{bmatrix} \vec{i}_{qds}^{s} \\ \vec{\lambda}_{qdr}^{s'} &= L_m \begin{bmatrix} 0 & +1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} +1 & 0 \\ 0 & -1 \end{bmatrix} \vec{i}_{qds}^{s} \end{align} $
The flux linkage equation is finished.
$ \begin{equation} \boxed{\vec{\lambda}_{qdr}^{s'} = L_m \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \vec{i}_{qds}^{s}} \end{equation} $
Frequency of Balanced Circuit Variables
The stator circuit variables have an input radial frequency of $ \omega_e $. The frequency of circuit variables is the input frequency with respect to a stationary reference minus the radial speed of the arbitrary reference frame. In this case, the speed of the currents, voltages, and flux linkages in the stationary reference frame is $ \omega_e - 0 = \boxed{\omega_e} $.
