The purpose of this project is to dynamically visualise the Clarke Transform and the Park Transform (animation).
A much heavier project to dynamically visualise the symmetrical components of three-phase inputs (a_{\pm}, b_{\pm}, c_{\pm}, Zero), the Clarke Transform (\alpha_{\pm}, \beta_{\pm}, Zero) and the Park Transform (d_{\pm}, q_{\pm}, Zero) is on the way.
\omega = 2 \pi f
\left[\begin{matrix} a \\ b \\ c \end{matrix}\right] = \left[\begin{matrix} \cos(n \cdot \omega t) \\ \cos[n \cdot (\omega t - \frac{2}{3} \pi)] \\ \cos[n \cdot (\omega t + \frac{2}{3} \pi)] \end{matrix}\right]
where, n \geqslant 0
Note
f is the system's base frequency.
\omega is the corresponding angular velocity.
n is the harmonic order.
t is the time.
When n = 0 + 3k, k = 0, 1, 2, 3, \cdots , the three-phase inputs are said to be the Zero Sequences.
When n = 1 + 3k, k = 0, 1, 2, 3, \cdots , the three-phase inputs are said to be the Positive Sequences.
When n = 2 + 3k, k = 0, 1, 2, 3, \cdots , the three-phase inputs are said to be the Negative Sequences.
\left[\begin{matrix} \alpha \\ \beta \\ Zero \end{matrix}\right] = \frac{2}{3} \left[\begin{matrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix}\right] \left[\begin{matrix} a \\ b \\ c \end{matrix}\right]
Note
For Zero Sequences \alpha and \beta components are zero.
For Positive Sequences \alpha leads \beta by 90^{\circ}.
For Negative Sequences \alpha lags \beta by 90^{\circ}.
The Clarke Transform does not alter the frequency. I.e., if the three-phase input frequency is 50 Hz, the frequencies of the Clarke components would still be 50 Hz.
Interharmonics are not of any sequence because they are not evenly space by 120^{\circ}.
\left[\begin{matrix} d \\ q \\ Zero \end{matrix}\right] = \left[\begin{matrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{matrix}\right] \left[\begin{matrix} \alpha \\ \beta \\ Zero \end{matrix}\right]
Note
\theta is usually provided by the Phase-Locked Loop (PLL), which
is usually locked on the fundamental. However, it is possible for the PLL
to locked on to other harmonics. When the PLL is locked on, the d
and q components are DC.
\theta has a huge impact on the outputs of the Park Transform.
Usually, there are the following:
For Zero Sequences d and q components are zero.
For Positive Sequences d leads q by 90^{\circ} (if they are not DC).
For Negative Sequences d lags q by 90^{\circ} (if they are not DC).
The Park Transform does not alter the phase difference between the components. I.e., if \alpha leads \beta, then d would still lead q.
However, the Park Transform changes the frequencies. This change is related to the sequence of the three-phase inputs.
For Positive Sequences, the frequencies of the d and q components would be reduced by 1 order of the base frequency.
For Negative Sequences, the frequencies of the d and q components would be increased by 1 order of the base frequency.
- Examples :
It is recommended to stop the animation before making changes to the input fields. Otherwise the UI may not register the focus.
Input Harmonic Oder :
The order of harmonic to be analysed. Should be a positive read number (unsigned float) .
Input PLL Oder :
The order of the PLL. Positive number means anti-clockwise rotation. Negative number means clockwise rotation. The value of the number means how many times the base frequency the PLL frequency is. Should be a real number (signed float).
Samples :
The number of samples to be taken within one base period. Should be unsigned int.
FPS :
Only applied when saving video. NOT applied in real time. Should be unsigned int.
Base Freq :
Base frequency of the system, i.e., 50 or 60. This can be any non-zero positive number (unsigned float, non-zero).
FFmpeg path :
Path of the FFmpeg binary (string).
Note
Zero Sequences are not plotted since their \alpha, \beta, d and q components are zero. Also, they need 3D coordinates.
The input fields, the buttons and the corresponding labels are hidden in the saved videos.
FFmpeg is a popular multi-media codec and it is free to download and use.
The following information would change dynamically with changes made to the user configurations.
Note that these pieces of information would only be refreshed when the "Play" button is clicked.
Information for the input harmonic :
These pieces of information are displayed in the left top corner, inside the red box.
They include: the harmonic frequency, the sequence of the input harmonic, the rotational direction of this input harmonic.
Information for the PLL :
These pieces of information are displayed in the left top middle corner, inside the blue box.
They include: frequency of the PLL and the rotational direction of the PLL.
Information for the Clarke Transform :
These pieces of information are displayed in the right top corner.
They include: frequency of the Clarke components and the phase relation between them.
Information for the Park Transform :
These pieces of information are displayed in the right middle corner.
They include: frequency of the Park components and the phase relation between them.
(2) PLL locked on to the fundamental while the input is the 2nd harmonic
(3) PLL locked on to the fundamental while the input is the 1.3 times harmonic (interharmonic)