Chapter 5: synchronous machines  (Page 2/5)

§5.2 Synchronous-Machine Inductances; Equivalent Circuits

Figure 5.3 Schematic diagram of a two-pole,

three-phase cylindrical-rotor synchronous machine.

• A cross-sectional sketch of a three-phase cylindrical-rotor synchronous machine is shown schematically in Fig.5.3. The figure shows a two-pole machine; alternatively, this can be considered as two poles of a multipole machine. The three-phase armature winding on the stator is of the same type used in the discussion of rotating magnetic fields in Section 4.5. Coils $a{a}^{\text{'}}$ , $b{b}^{\text{'}}$ and $c{c}^{\text{'}}$ I represent distributed windings producing sinusoidal mmf and flux-density waves in the air gap. The reference directions for the currents are shown by dots and crosses. The field winding $f{f}^{\text{'}}$ on the rotor also represents a distributed winding which produces a sinusoidal mmf and flux-density wave centered on its magnetic axis and rotating with the rotor.

• When the flux linkages with armature phases a, b, c and field winding f are expressed in terms of the inductances and currents as follows,

${\lambda }_a=L_{\text{aa}}{i}_a+L_{\text{ab}}{i}_b+L_{\text{ac}}{i}_c+L_{\text{af}}{i}_f$ (5.8)

${\lambda }_b=L_{\text{ba}}{i}_a+L_{\text{bb}}{i}_b+L_{\text{bc}}{i}_c+L_{\text{bf}}{i}_f$ (5.9)

${\lambda }_c=L_{\text{ca}}{i}_a+L_{\text{cb}}{i}_b+L_{\text{cc}}{i}_c+L_{\text{cf}}{i}_f$ (5.10)

${\lambda }_f=L_{\text{fa}}{i}_a+L_{\text{fb}}{i}_b+L_{\text{fc}}{i}_c+L_{\text{ff}}{i}_f$ (5.11)

the induced voltages can be found from Faraday's law. Here, two like subscripts denote a self-inductance, and two unlike subscripts denote a mutual inductance between the two windings. The script is used to indicate that, in general, both the self- and mutual inductances of a three-phase machine may vary with rotor angle.

§5.2.1 Rotor Self-Inductance

• With a cylindrical stator, the self-inductance of the field winding is independent of the rotor position 0m when the harmonic effects of stator slot openings are neglected.

$L_{\text{ff}}=L_{\text{ff}}=L_{\text{ff}0}+L_{\mathrm{f1}}$ (5.12)

where the italic L is used for an inductance which is independent of ${\theta }_m$ . The component $L_{\text{ff}0}$ corresponds to that portion of $L_{\text{ff}}$ due to the space-fundamental component of air-gap flux

§5.2.2 Stator-to-Rotor Mutual Inductances

• The stator-to-rotor mutual inductances vary periodically with ${\theta }_{\text{me}}$ , the electrical angle between the magnetic axes of the field winding and the armature phase a as shown in Fig.5.2 and as defined by Eq.4.54. With the space-mmf and air-gap flux distribution assumed sinusoidal, the mutual inductance between the field winding f and phase a varies as $\text{cos}{\theta }_{\text{me}}$ ; thus

$L_{\text{af}}=L_{\text{fa}}=L_{\text{af}}\text{cos}{\theta }_{\text{me}}$ (5.13)

${\theta }_{\text{me}}=\left[\frac{\text{poles}}{2}\right]{\theta }_m={\omega }_{e}t+{\delta }_{\mathrm{e0}}$ (5.14)

$L_{\text{af}}=L_{\text{fa}}=L_{\text{af}}\text{cos}({\omega }_{e}t+{\delta }_{\mathrm{e0}})$ (5.15)

§5.2.3 Stator Inductances; Synchronous Inductance

• With a cylindrical rotor, the air-gap geometry is independent of ${\theta }_m$ if the effects of rotor slots are neglected. The stator self-inductances then are constant; thus

$L_{\text{aa}}=L_{\text{bb}}=L_{\text{cc}}=L_{\text{aa}}=L_{\text{aa}0}+L_{\mathrm{a1}}$ (5.16)

§5.2.4 Equivalent Circuit

• Equivalent circuit for the synchronous machine:

• Single-phase, line-to-neutral equivalent circuits for a three-phase machine operating under balanced, three-phase conditions.

$L_s=$ effective inductance seen by phase a under steady-state, balanced three-phase

machine operating conditions.

$X_s={\omega }_e{L}_s$ : synchronous reactance

$R_a=$ armature winding resistance

$e_{\text{af}}=$ voltage induced by the field winding flux (generated voltage, internal voltage)