0.3 Chapter 4: introduction to rotating machines  (Page 7/9)

$F_{\text{agl}}=F_{\text{max}}[\frac{1}{2}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)+\frac{1}{2}\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t)]$ (4.20)

$F_{\text{agl}}^{+}=F_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)$ (4.21)

$F_{\text{agl}}^{-}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t)$ (4.22)

• $F_{\text{agl}}$ travels in the $+{\theta }_a$ direction and $F_{\text{agl}}$ travels in the $-{\theta }_a$ direction.
• This decomposition is shown graphically in Fig. 4.19(b) and in a phasor representation in Fig. 4.19(c).

Figure 4.19 Single-phase-winding space-fundamental air-gap mmf: (a) mmf distribution of a

single-phase winding at various times; (b) total mmf $F_{\text{agl}}$ decomposed into two traveling waves F and $F^{+}$ ; (c) phasor decomposition of F.

§4.5.2 MMF Wave of a Polyphase Winding

• We are to study the mmf distribution of three-phase windings such as those found on the stator of three-phase induction and synchronous machines.

In a three-phase machine, the windings of the individual phases are displaced from each other by 120 electrical degrees in space around the air-gap circumference as shown in Fig.4.20 in which the concentrated full-pitch coils may be considered to represent distributed windings.

• Under balanced three-phase conditions, the excitation currents (Fig. 4.20) are

$i_a=I_m\text{cos}{\omega }_{e}t$ (4.23)

$i_b=I_m\text{cos}({\omega }_{e}t-{\text{120}}^o)$ (4.24)

$i_c=I_m\text{cos}({\omega }_{e}t+{\text{120}}^o)$ (4.25)

Figure 4.20 Simplified two-pole three-phase stator winding.

Figure 4.21 Instantaneous phase currents under balanced three-phase conditions.

• The mmf of phase a has been shown to be

$F_{\mathrm{a1}}=F_{\mathrm{a1}}^{+}+F_{\mathrm{a1}}^{-}$ (4.26)

$F_{\mathrm{a1}}^{+}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)$ (4.27)

$F_{\mathrm{a1}}^{-}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t)$ (4.28)

$F_{\text{max}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{\text{poles}})I_m$ (4.29)

• Similarly, for phases b and c

$F_{\mathrm{b1}}=F_{\mathrm{b1}}^{+}+F_{\mathrm{b1}}^{-}$ (4.30)

$F_{\mathrm{b1}}^{+}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)$ (4.31)

$F_{\mathrm{b1}}^{-}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t+{\text{120}}^o)$ (4.32)

$F_{\mathrm{c1}}=F_{\mathrm{c1}}^{+}+F_{\mathrm{c1}}^{-}$ (4.33)

$F_{\mathrm{c1}}^{+}=\frac{1}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)$ (4.34)

$F_{\mathrm{c1}}^{-}=F_{\text{max}}\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t-{\text{120}}^o)$ (4.35)

• The total mmf is the sum

$F({\theta }_{\text{ae}},t)=F_{\mathrm{a1}}+F_{\mathrm{b1}}+F_{\mathrm{c1}}$ (4.36)

It can be performed in terms of the positive- and negative- traveling waves.

$\begin{array}{}{F}^{-}({\theta }_{\text{ae}},t)=F_{\mathrm{a1}}^{-}+F_{\mathrm{b1}}^{-}+F_{\mathrm{c1}}^{-}\\ =\frac{1}{2}{F}_{\text{max}}[\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t)+\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t-{\text{120}}^o)+\text{cos}({\theta }_{\text{ae}}+{\omega }_{e}t+{\text{120}}^o)]\\ =0\end{array}$

$\begin{array}{}{F}^{+}({\theta }_{\text{ae}},t)=F_{\mathrm{a1}}^{+}+F_{\mathrm{b1}}^{+}+F_{\mathrm{c1}}^{+}\\ =\frac{3}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)\end{array}$ (4.37)

• The result of displacing the three windings by ${\text{120}}^o$ in space phase and displacing the winding currents by ${\text{120}}^o$ in time phase is a single positive-traveling mmf wave

$\begin{array}{}F({\theta }_{\text{ae}},t)=\frac{3}{2}{F}_{\text{max}}\text{cos}({\theta }_{\text{ae}}-{\omega }_{e}t)\\ \frac{3}{2}{F}_{\text{max}}\text{cos}((\frac{\text{poles}}{2}){\theta }_a-{\omega }_{e}t)\end{array}$ (4.38)

• Under balanced three-phase conditions, the three-phase winding produces an air-gap mmf wave which rotates at synchronous angular velocity ${\omega }_s$ (rad/sec)

${\omega }_s=(\frac{2}{\text{poles}}){\omega }_e$ (4.39)

${\omega }_c$ : angular velocity of the applied electrical excitation (rad/sec)

• : synchronous speed

$f_e={\omega }_e/(\mathrm{2\pi })$ : applied electrical frequency

$n_s=(\frac{\text{120}}{\text{poles}})f_{e}r\text{/min}$ (4.40)

• A polyphase winding exicted by balanced polyphase currents produces a rotating mmf wave.

• It is the interaction of this magnetic flux wave with that of the rotor which produces torque.
• Constant torque is produced when rotor-produced magnetic flux rotates in synchronism with that of the stator.

§4.5.3 Graphical Analysis of Polyphase MMF

• For balanced three-phase currents, the production of a rotating mmf can also be shown graphically.

• Refer to Fig. 4.22.

• As time passes, the resultant mmf wave retains its sinusoidal form and amplitude but rotates progressively around the air gap.
• The net result is an mmf wave of constant amplitude rotating at uniform angular velocity.