Chapter 5: synchronous machines  (Page 4/5)

$\text{SCR}=\frac{O{f}^{\text{'}}}{O{f}^{\text{'}\text{'}}}$ (5.37)

$\text{SCR}=\frac{\text{AFNL}}{\text{AFSC}}$ (5.38)

Figure 5.13 Typical form of short-circuit load loss and stray load-loss curves.

$\frac{R_T}{R_T}=\frac{\text{234}\text{.}5+T}{\text{234}\text{.}5+t}$ (5.39)

$R_{a,\text{eff}}=\frac{\text{short}-\text{circuit load loss}}{(\text{short}-\text{circuit armature current}{)}^2}$ (5.40)

§5.5 Steady-State Operating Characteristics

Figure 5.14 Characteristic form of synchronous-generator compounding curves.

Figure 5.15 Capability curves of an 0.85 power factor, 0.80 short-circuit ratio,

hydrogen-cooled turbine generator. Base MVA is rated MVA at 0.5 psig hydrogen.

$\text{Apprent power}=\sqrt{P^2+Q^2}=V_a{I}_a$

Figure 5.16 Construction used for the derivation of a synchronous generator capability curve.

$P-\text{jQ}={\hat{V}}_a+{\text{jX}}_s{\hat{I}}_a$ (5.41)

${\hat{E}}_{\text{af}}={\hat{V}}_a+{\text{jX}}_s{\hat{I}}_a$ (5.42)

$P^2+(Q+\frac{V_a^2}{X_s}{)}^2=(\frac{V_a{E}_{\text{af}}}{X_s}{)}^2$ (5.43)

Figure 5.17 Typical form of synchronous-generator V curves.

§5.6 Effects of Salient Poles; Introduction to Direct-And

Quadrature-Axis Theory

§5.6.1 Flux and MMF Waves

Figure 5.18 Direct-axis air-gap fluxes in a salient-pole synchronous machine.

$E_{\mathrm{3,}a}=\sqrt{2}{V}_3\text{cos}({\mathrm{3\omega }}_{e}t+{\phi }_3)$ (5.44)

$E_{\mathrm{3,}b}=\sqrt{2}{V}_3\text{cos}(3({\omega }_e-{\text{120}}^o)+{\phi }_3)=\sqrt{2}{V}_3\text{cos}({\mathrm{3\omega }}_{e}t+{\phi }_3)$ (5.45)

$E_{\mathrm{3,}c}=\sqrt{2}{V}_3\text{cos}(3({\omega }_{e}t-{\text{120}}^o)+{\phi }_3)=\sqrt{2}{V}_3\text{cos}({\mathrm{3\omega }}_{e}t+{\phi }_3)$ (5.45)

Figure 5.19 Quadrature-axis air-gap fluxes in a salient-pole synchronous machine.

Figure 5.20 Phasor diagram of a salient-pole synchronous generator.

§5.6.2 Phasor Diagrams for Salient-Pole Machines

Figure 5.21 Phasor diagram for a synchronous generator showing

the relationship between the voltages and the currents.

$X_d=X_{\text{al}}+X_{\mathrm{\varphi d}}$ (5.46)

$X_q=X_{\text{al}}+X_{\mathrm{\varphi q}}$ (5.47)

Figure 5.22 Relationships between component voltages in a phasor diagram.

$\frac{o^{\text{'}}{a}^{\text{'}}}{\text{oa}}=\frac{b^{\text{'}}{a}^{\text{'}}}{\text{ba}}$ (5.48)

$o^{\text{'}}{a}^{\text{'}}=(\frac{b^{\text{'}}{a}^{\text{'}}}{\text{ba}})\text{oa}=\frac{\mid {\hat{I}}_q\mid X_q}{\mid {\hat{I}}_q\mid }\mid {\hat{I}}_a\mid =X_q\mid {\hat{I}}_a\mid$ (5.49)

${\hat{E}}_{\text{af}}={\hat{V}}_a+R_a{\hat{I}}_a+{\text{jX}}_d{\hat{I}}_d+{\text{jX}}_q{\hat{I}}_q$ (5.50)

5.7 Power-Angle Characteristics Of Salient-Pole Machines

• For the purposes of this discussion, it is sufficient to limit our discussion to the simple system shown in the schematic diagram of Fig.5.23a, consisting of a salient pole synchronous machine SM connected to an infinite bus of voltage ${\hat{V}}_{\text{EQ}}$ through a series impedance of reactance $X_{\text{EQ}}$ . Resistance will be neglected because it is usually small. Consider that the synchronous machine is acting as a generator. The phasor diagram is shown by the solid-line phasors in Fig.5.23b. The dashed phasors show the external reactance drop resolved into components due to ${\hat{I}}_d$ and ${\hat{I}}_q$ . The effect of the external impedance is merely to add its reactance to the reactances of the machine; the total values of the reactance between the excitation voltage ${\hat{E}}_{\text{af}}$ and the bus voltage ${\hat{V}}_{\text{EQ}}$ is therefore

$X_{\text{dT}}=X_d+X_{\text{EQ}}$ (5.50)

$X_{\text{qT}}=X_q+X_{\text{EQ}}$ (5.51)

• If the bus voltage ${\hat{V}}_{\text{EQ}}$ is resolved into components its direct-axis component $V_d=V_{\text{EQ}}\text{sin}\delta$ and quadrature-axis component $V_q=V_{\text{EQ}}\text{cos}\delta$ in phase with ${\hat{I}}_d$ and ${\hat{I}}_q$ , respectively, the power P delivered to the bus per phase (or in per unit) is

$P=I_d{V}_d+I_q{V}_q=I_d{V}_{\text{EQ}}\text{sin}\delta +I_q{V}_{\text{EQ}}\text{cos}\delta$ (5.52)

$I_d=\frac{E_{\text{af}}-V_{\text{EQ}}\text{cos}\delta }{X_{\text{dT}}}$ (5.53)

$I_q=\frac{V_{\text{EQ}}\text{sin}\delta }{X_{\text{qT}}}$ (5.54)

$P=\frac{E_{\text{af}}{V}_{\text{EQ}}}{X_{\text{dT}}}\text{sin}\delta +\frac{V_{\text{EQ}}^2(X_{\text{dT}}-X_{\text{qT}})}{{\mathrm{2X}}_{\text{dT}}{X}_{\text{qT}}}\text{sin}\mathrm{2\delta }$ (5.55)

Figure 5.23 Salient-pole synchronous machine and series impedance: (a) single-line diagram and (b) phasor diagram.

Figure 5.24 Power-angle characteristic of a salient-pole synchronous machine showing the fundamental component due to field excitation and the second-harmonic component due to reluctance torque.