Synchronous machines, such as synchronous generators and motors, play a crucial role in power systems. One of the key parameters that determine their performance is reactance, specifically direct axis reactance (Xd) and quadrature axis reactance (Xq). These reactances affect the machine’s behavior under different operating conditions, including voltage regulation, stability, and fault response.
This topic explains Xd and Xq in synchronous machines, their significance, how they are measured, and their impact on machine performance.
1. Understanding Reactance in Synchronous Machines
A. What Is Reactance?
Reactance is the opposition to alternating current (AC) flow caused by inductance in an electrical machine. In synchronous machines, reactance varies depending on the rotor’s position relative to the stator field, leading to two important reactance values:
- Direct Axis Reactance (Xd)
- Quadrature Axis Reactance (Xq)
B. Importance of Xd and Xq
- Helps in voltage regulation of synchronous generators.
- Affects the stability and transient response of the machine.
- Determines short-circuit and fault current behavior.
2. Direct Axis Reactance (Xd)
A. Definition of Xd
Direct axis reactance (Xd) is the reactance measured when the stator and rotor magnetic fields are aligned. This occurs when the rotor’s field poles are directly in line with the stator’s armature field.
B. Types of Xd
Xd has three variations depending on the time frame of analysis:
- Subtransient Reactance (Xd”) – Represents the reactance in the first few cycles after a sudden fault. It is the lowest reactance and results in high short-circuit currents.
- Transient Reactance (Xd’) – Reactance a few milliseconds after a fault, when damper windings have influenced the field current.
- Synchronous Reactance (Xd) – The steady-state reactance after all transients have died out. It is the highest reactance value.
C. Effects of Xd on Performance
- Higher Xd leads to better voltage regulation but reduces short-circuit current capability.
- Lower Xd allows higher fault currents, which can stress the protection system.
3. Quadrature Axis Reactance (Xq)
A. Definition of Xq
Quadrature axis reactance (Xq) is the reactance measured when the rotor field is 90 degrees out of phase with the stator field. In this case, the rotor poles do not directly oppose the stator field, leading to a different magnetic reluctance and thus a different reactance.
B. Xq Compared to Xd
- Xq is usually lower than Xd because the quadrature axis has a different magnetic path than the direct axis.
- The difference between Xd and Xq is more noticeable in salient pole machines, where the rotor shape influences magnetic reluctance.
C. Effects of Xq on Performance
- Affects the power factor and stability of synchronous machines.
- Important in salient pole machines where the difference between Xd and Xq is significant.
- Plays a role in torque production and machine control.
4. Difference Between Xd and Xq
| Parameter | Direct Axis Reactance (Xd) | Quadrature Axis Reactance (Xq) |
|---|---|---|
| Definition | Reactance when stator and rotor fields align | Reactance when rotor field is 90° out of phase |
| Value | Usually higher | Usually lower |
| Impact on Stability | Affects voltage regulation and fault currents | Affects power factor and dynamic performance |
| Salient Pole Machines | Xd >> Xq due to rotor shape | Xq < Xd due to lower reluctance |
5. Measurement of Xd and Xq
A. Open Circuit and Short Circuit Tests
Xd and Xq are measured using open-circuit and short-circuit tests on synchronous machines.
-
Open-Circuit Test
- The generator is run at rated speed.
- Field current is gradually increased, and terminal voltage is recorded.
- This helps determine the open-circuit characteristic (OCC) and calculate Xd.
-
Short-Circuit Test
- The generator terminals are short-circuited through an ammeter.
- Field current is varied, and short-circuit current is recorded.
- This test helps determine the short-circuit characteristic (SCC) and estimate Xd and Xq.
B. Standstill Frequency Response Test
- This test measures impedance at different rotor positions to determine Xd and Xq more accurately.
- It is used for precise machine modeling in power systems.
6. Xd and Xq in Salient Pole vs. Cylindrical Rotor Machines
A. Salient Pole Machines
- Used in hydroelectric and low-speed synchronous machines.
- Have unequal Xd and Xq because the rotor shape creates different magnetic reluctance.
- The power-angle equation includes both Xd and Xq, making machine behavior more complex.
B. Cylindrical Rotor Machines
- Used in high-speed generators like steam turbines.
- Have nearly equal Xd and Xq because the rotor is symmetrical.
- The power equation simplifies, and stability analysis is easier.
7. Impact of Xd and Xq on Power System Stability
A. Transient Stability
- During faults or sudden load changes, Xd and Xq influence how quickly a generator recovers.
- Higher Xd improves voltage stability but limits fault current.
- Lower Xq improves dynamic performance and reactive power capability.
B. Steady-State Stability
- The difference between Xd and Xq determines how power is transferred in salient pole machines.
- For cylindrical rotors, stability is mainly dependent on Xd.
C. Fault Response
- Xd’ and Xd” impact short-circuit currents, affecting protection system design.
- Proper selection of Xd and Xq values helps prevent voltage collapse and instability.
Xd and Xq are critical parameters in synchronous machines, affecting performance, stability, and fault response.
- Xd is the direct axis reactance, influencing voltage regulation and fault current behavior.
- Xq is the quadrature axis reactance, affecting power factor and torque production.
- The difference between Xd and Xq is significant in salient pole machines but less in cylindrical rotor machines.
Understanding Xd and Xq is essential for power system engineers to design reliable and stable electrical networks. Proper measurement and analysis help in optimizing generator performance, improving fault protection, and ensuring stable power generation.