## Real Time Response of VHM

With the ability to predict the dynamic response of the VHM in generator mode, it is now possible to examine the difficulties in extracting power. It has previously been demonstrated that the machine has a large inductance (^0.5H). With a time domain model the instantaneous values of phase flux linkage, current and theoretical open circuit emf can be recorded.

Figure 4.61: Equivalent circuit of VHM in short circuit model

Figure 4.61: Equivalent circuit of VHM in short circuit

Figure 4.61 shows an equivalent circuit model of the VHM superimposed on the preferred model (modelE, Section 4.3.2.6) connected in short circuit. Using the phasor diagram on the right hand side of the diagram it is possible to establish relationships between the three circuit elements. All the waveforms are assumed to be sinusoidal with the same angular velocity ra, which allows the maximum value of wave forms to be used to calculate relative phase differences. As such the elements shown in the equivalent circuit and on the phasor diagram can be substituted with the flux linkages according to (4.32) to make a triangle of identical aspect ratio.

Qualitatively, (4.32) states that Eopen, which represents the open circuit emf, is a function of the flux driven by the magnets only, the voltage dropped across the series inductance is a function of the current driven flux whereas the voltage dropped across the resistor is a function of the net flux in the machine. Figure 4.62 shows the timevariation of these three elements at a constant speed of 0.8 m/s.

- net flux linkage - V, | ||||||||||

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Figure 4.62: Flux flow in VHM during short circuit

The power factor can now be found using (4.33), incorporating (4.32) and geometry from Figure 4.61.

vpm 0.8976

It is clear from Figure 4.62 that the magnet driven flux and current driven flux are almost in anti-phase, giving a very small net flux flow. The collapse of the terminal voltage when current flows through the machine is thus explained. The very low power factor calculated in (4.33) confirms this.

A: normal generator B: VHM

Figure 4.63: Phasor diagram of power factor

Figure 4.63: Phasor diagram of power factor

The relationship between the three flux components and the generator characteristics of the VHM can be illustrated by use of phasor diagrams. In Figure 4.63 voltage phasor diagrams are drawn above the equivalent airgap flux phasor diagrams. Figure 4.63A shows a typical pattern for an electrical machine. The magnetic flux, yPM, leads and is pointing left. Lagging 90° behind this is the open circuit emf. Due to the inductance of the coil, the current, I, lags the voltage by an angle The flux field driven by the coil current, is in phase with the current and the resultant flux, which defines terminal voltage V, is shown. Figure 4.63B shows the equivalent diagram for a VHM, with a larger lag in current. The magnitude of the ytot phasor has considerably decreased by virtue of the flux due to the magnets and current almost opposing each other.

Figure 4.64 shows the phasor diagrams of the VHM with a corrected power factor and hence no phase difference between Eopen and I. In order to achieve this, the terminals of the generator have been connected to a load which has a current leading the voltage by an angle The resultant flux, and hence terminal voltage, of the machine is seen to be large. Operating in this manner, referred to as unity power factor, can thus be deduced to require y and yPM to have a 90° phase difference. It is common practice to use a capacitor to perform this function.

open | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

' V | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

1 |
-V |
Figure 4.64: Phasor diagram of corrected power factor Figure 4.64: Phasor diagram of corrected power factor Figure 4.65: Capacitor assisted excitation of the VHM Figure 4.65 shows the VHM model superimposed on a simple equivalent circuit for the case of capacitor assisted excitation. Also shown are two phasor diagrams, one corresponding to the circuit current and one to the voltage. The approximations used in (4.32) can again be used for E, IgqLg and IgRg in triangles where all sides have a dependency on ra. In model E the terms 'E' and 'IGraL' do not exist, with the output of the model being current, IG. The term 'ILRL' is hence not related to any one group of flux linkage in the machine and so it is not possible to use an approximation of this form. Peak resultant voltages must instead be used in this geometric triangle. Appendix B shows how the vector geometry of Figure 4.65 can be used to calculate ^ and hence power factor. It could also be used to calculate ideal values of RL and C to give unity power factor (^=0). A simple method for selecting a capacitor approximately equal to this value states that the reactive power of the capacitor is intended to cancel out the reactive power of the inductor. In order to do this the reactance of these two elements must be forced to be equal in magnitude but in anti-phase, ( 4.34). For a constant speed of 0.5 m/s, the 24 mm cyclic nature of the VHM gives ra a value of 130 rad/s. Assuming the inductance of the prototype is around 0.4 H, see Figure 4.27, and using available sizes of capacitor, ( 4.34) implies a value of 150 ^.F is suitable. Figure 4.66 shows the components of the flux linkage within the VHM being loaded in this manner. The net flux linkage within the machine is clearly dominated by the current. The magnet driven flux and current driven flux are close to being 90° out of phase and so the power factor of the machine has effectively been reduced. The method detailed in Appendix B is used to calculate the power factor of the machine in this state and found to be 0.81. Figure 4.66: Flux linkages in the VHM with a capacitive load of 150 ^F, resistive load of 630 Q at a constant velocity of 0.5 m/s velocity Figure 4.66: Flux linkages in the VHM with a capacitive load of 150 ^F, resistive load of 630 Q at a constant velocity of 0.5 m/s velocity
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