$I_a=A\text{ sin}\left(\omega t\right)$ 

$I_b=A\text{ sin}(\omega t-\frac{2\pi }{3})$ 

$I_c=A\text{ sin}(\omega t-\frac{4\pi }{3})$ 

$V_a=V\text{ sin}\left(\omega t\right)$ 

$V_b=V\text{ sin}(\omega t-\frac{2\pi }{3})$ 

$V_c=V\text{ sin}(\omega t-\frac{4\pi }{3})$ 

This chapter presents a recap of the fundamental operating principle of the three-phase rotating electrical machine model.

As illustrated in the animation, the magnetic field of the rotor sweeps through the electric field of the stator within the core of the generator. This process induces a voltage in the stator coils and thus gives rise to a phase current.

In the three-phase machine model, the generator is designed to produce three individual phase voltages that are each offset by 120° ( i.e.: $\frac{2\pi }{3}\text{rads}$ ). 

Figure 1 presents the induced voltage for Phase A in the time frame.

Note that for simplicity this waveform has a voltage amplitude of 1 Volt, starts at time = 0 seconds and is an ideal 50 Hz sine wave. This waveform can be described by the following formula:

$V_a=V\text{ sin}\left(\omega t\right)$ 

where:

$\omega =2\pi f$ 

Figure 1: Phase A Voltage (Time Frame)

Figure 2 includes the induced voltage for Phase B in the time frame.

Note that Vb is offset from Va by 2π/3 rads. This second waveform can be described by the following formula:

$V_b=\text{ sin}(\omega t-\frac{2\pi }{3})$ 

where:

$\omega =2\pi f$ 

Figure 2: Phase A & B Voltages (Time Frame)

Figure 3 includes the induced voltage for Phase C in the time frame.

Note that Vc is offset from Va by 4π/3 rads. This second waveform can be described by the following formula:

$V_c=\text{ sin}(\omega t-\frac{4\pi }{3})$ 

where:

$\omega =2\pi f$ 

Figure 3: Phase A, B & C Voltages (Time Frame)

In the previous chapter, we saw how Va, Vb and Vc would be typically represented in the time frame. At any given time each phase voltage can also be represented purely as a magnitude value on a single axis graph, i.e., at t = 3 ms as shown on Figure 4:

$V_a=\text{ sin}\left(\text{2 ∗ 3.14 ∗ 50 ∗ 0.003}\right)=0.81$

Figure 4: Phase A (Phase Space)

All phase voltages could therefore be illustrated in a vector space representation simultaneously if we rotate each axis by 120° (i.e.: 2π/3 rads) and set the zero crossing point of each axis at the centre of the abc frame.

In the following pages, examples of three phase voltage values at various times have been plotted in both the time frame and in the abc frame.

Figure 5: The abc Frame

At any given time, the resultant vector (V ̅) is of constant magnitude and sweeps around the circle of the abc frame as illustrated in the animation.

Note that in this animation, time (t) is increasing.

Representation in abc frame allows us to easily visualise the vector of the resultant voltage.

However, this representation remains complex due to the requirement for any analyses to be done using three axes. In the next chapter we shall see how Va, Vb and Vc might be represented in a dual axis reference frame (the αβ frame).

Original image: Mathworks

A given vector can be expressed as:

Rectangular Form: $\stackrel{\text{‾}}{Z}=a+jb$ 

Polar Form: $\stackrel{\text{‾}}{Z}=r\text{ cos }\theta +jr\text{ sin }\theta$ 

Exponential Form: $\stackrel{\text{‾}}{Z}={re}^{j\theta }$ 

where r is the magnitude of the vector $\stackrel{\text{‾}}{Z}$.

In this figure, the new αβ frame is superimposed upon the abc frame. It should be noted that the centre point of the αβ frame is at the centre point of the abc frame and that for simplicity the alpha (α) axis is aligned with the a axis.

Any given vector can therefore be written as a vector sum of each of its [α and β] components or its [a, b and c] components (and these are equal) such that:

$\overline{R_{\alpha \beta }}=r_{\alpha }+{jr}_{\beta }$       $\overline{R_{\text{abc}}}=r_a{e}^{j0}+r_b{e}^{j\frac{2\pi }{3}}+r_c{e}^{j\frac{4\pi }{3}}$ 

thus:

$\overline{R_{\alpha \beta }}=\overline{R_{\text{abc}}}$ 

Figure 6: The αβ Frame

$V_{\alpha }=(V_a+V_b(-\frac{1}{2})+V_c(-\frac{1}{2}\left)\right)$ 

$V_{\beta }=\left(r_b\right(\frac{\sqrt{3}}{2})+r_c(-\frac{\sqrt{3}}{2}\left)\right)$ 

This animation illustrates the behaviour of Vα and Vβ as they vary with time (t).

The Clarke Transform converts the individual components of a three-phase system (the abc frame) to two components defined in a new orthogonal stationary system (the αβ frame). The αβ frame representation is a simplification in that the behaviour of the three-phase system is now fully described using only two variables, Vα and Vβ each of which vary with time (t).

In the next chapter, we shall see how Vα and Vβ can be represented in a rotating dual-axes reference frame (the dq frame) such that the resultant components are constant rather than variable with regards to time.

Original image

Now we can say that the for any given vector in phase space, the reference frame is 'arbitrary', i.e., it does not matter that the alpha (α) axis of the αβ frame happened to be aligned with the a axis of the abc frame. The alignment of these frames was selected because it simplified the arithmetic and was visually appealing (in that the αβ frame appears to us like a typical xy frame). However, we could have referenced the resultant vector to any other frame (centred at zero) and defined the relationship between this new frame and the αβ frame by the angle rho (ρ) as illustrated:

As we have seen in the previous chapter that this xy frame relates to the αβ frame by the following relationship:

$\overline{R_{xy}}=\overline{R_{\alpha \beta }}.e^{jp}$ 

where:

$\overline{R_{xy}}=R_x+{jr}_y$ 

and

$\overline{R_{\alpha \beta }}.e^{jp}=(r_{\alpha }+r_{\beta }).(\text{ cos }\rho -j\text{ sin }\rho )$ 

Figure 7: An arbitrary offset reference frame

$V_x=(V_{\alpha }\text{ cos }\rho +V_{\beta }\text{ sin }\rho )$ 

$V_y=(-V_{\alpha }\text{ sin }\rho +V_{\beta }\text{ cos }\rho )$ 

As this animation illustrates, when we set rho (ρ) to the synchronous speed; Vd and Vq are constant with regards to time (t) as the frame itself is rotating.

The Park transform can therefore be expressed as:

$\left[\begin{array}{c}{V}_d\\ V_q\end{array}\right]=\left[\begin{array}{cc}\text{cos }\rho & \text{sin }\rho \\ -\text{sin }\rho & \text{cos }\rho \end{array}\right]\left[\begin{array}{c}{V}_{\alpha }\\ V_{\beta }\end{array}\right]$ 

where:

$\rho =\omega t$ 

The Park transform therefore converts the two time-variant components in the αβ frame to an orthogonal rotating reference frame (dq) and in doing so simplifies computations by producing two time-invariant components.

Original image