4.3 Synchronous machine dynamic equations

Synchronous Machine Dynamics in dq0 Frame

Synchronous machines are the backbone of power systems, and their dynamic equations describe how electrical and mechanical components interact during both normal operation and disturbances. Modeling these dynamics accurately is what lets engineers predict machine behavior, design control systems, and maintain grid stability.

The dq0 reference frame is the key tool here. It transforms three-phase AC stator quantities into two DC-like components (d and q axes) plus a zero-sequence component. Working with DC quantities instead of time-varying sinusoids makes the math far more tractable and reveals the physical coupling between stator and rotor circuits directly.

Dynamic Equations Formulation

The dq0 transformation uses a rotating reference frame aligned with the rotor. The d-axis lines up with the rotor field winding, and the q-axis leads the d-axis by 90 electrical degrees in the direction of rotation. This alignment is what eliminates the time-varying inductances you'd otherwise have to deal with in the stator equations.

Stator voltage equations in the dq0 frame:

• $v_d = -R_s i_d - \omega \lambda_q + \frac{d\lambda_d}{dt}$
• $v_q = -R_s i_q + \omega \lambda_d + \frac{d\lambda_q}{dt}$
• $v_0 = -R_s i_0 + \frac{d\lambda_0}{dt}$

The $-\omega \lambda_q$ and $+\omega \lambda_d$ terms are speed voltage terms (also called rotational EMF terms). They arise from the rotation of the reference frame and represent the coupling between the d and q axes. The $\frac{d\lambda}{dt}$ terms are the transformer voltage terms, capturing flux changes within each axis.

Stator flux linkage equations:

• $\lambda_d = -L_d i_d + L_{md} i_{fd}$
• $\lambda_q = -L_q i_q$
• $\lambda_0 = -L_0 i_0$

Notice that $\lambda_d$ depends on both the stator d-axis current and the rotor field current, while $\lambda_q$ depends only on the stator q-axis current. This asymmetry reflects the physical fact that the field winding sits on the d-axis.

Rotor Equations in dq0 Frame

The rotor circuits include the field winding (fd) and damper windings on both axes (kd and kq). Because the dq0 frame rotates with the rotor, the rotor equations contain no speed voltage terms.

Rotor voltage equations:

• $v_{fd} = R_f i_{fd} + \frac{d\lambda_{fd}}{dt}$
• $v_{kd} = R_{kd} i_{kd} + \frac{d\lambda_{kd}}{dt}$
• $v_{kq} = R_{kq} i_{kq} + \frac{d\lambda_{kq}}{dt}$

For the damper windings, $v_{kd} = 0$ and $v_{kq} = 0$ since they are short-circuited bars. The field winding voltage $v_{fd}$ is the excitation voltage supplied by the exciter.

Rotor flux linkage equations:

• $\lambda_{fd} = L_{fd} i_{fd} + L_{md} i_d$
• $\lambda_{kd} = L_{kd} i_{kd} + L_{md} i_d$
• $\lambda_{kq} = L_{kq} i_{kq} + L_{mq} i_q$

These equations are used to analyze machine behavior across different timescales: steady-state (constant flux linkages), transient (field winding dynamics dominate, damper currents have decayed), and subtransient (all rotor circuits active, immediately after a disturbance).

Stator-Rotor Equation Relationship

Coupling through Mutual Inductances

The stator and rotor circuits are coupled through the mutual inductances $L_{md}$ (d-axis) and $L_{mq}$ (q-axis). These parameters represent the magnetic coupling between stator and rotor windings through the air gap.

Here's how the coupling works in each axis:

• D-axis coupling: The stator flux $\lambda_d$ depends on both $i_d$ and $i_{fd}$. Conversely, the rotor fluxes $\lambda_{fd}$ and $\lambda_{kd}$ both depend on the stator current $i_d$. So any change in stator d-axis current immediately affects the rotor d-axis circuits, and vice versa.
• Q-axis coupling: The stator flux $\lambda_q$ depends only on $i_q$, but the damper winding flux $\lambda_{kq}$ depends on both $i_{kq}$ and $i_q$ through $L_{mq}$. There is no field winding on the q-axis, so the coupling path is simpler.

This mutual coupling is what makes synchronous machine dynamics interesting and complex. A disturbance on the stator side induces transient currents in the rotor circuits, which in turn affect the stator quantities.

Dynamic Behavior Analysis

The coupled stator-rotor equations let you:

• Investigate the machine's response to sudden disturbances (faults, load steps, voltage dips) by tracking how currents and flux linkages evolve in both stator and rotor circuits simultaneously
• Design control strategies such as excitation systems and power system stabilizers (PSS) that act on rotor-side quantities to improve stator-side performance
• Optimize operating conditions (efficiency, power factor, reactive power output) by understanding how adjusting field excitation affects the balance between d-axis and q-axis quantities

Saliency Impact on Synchronous Machines

Saliency and Inductance Difference

Saliency refers to the difference between the d-axis and q-axis inductances ($L_d \neq L_q$). It arises from the physical geometry of the rotor.

• In a salient-pole machine (common in hydro generators), the rotor poles protrude, creating a non-uniform air gap. The air gap is smaller along the d-axis (under the poles) and larger along the q-axis (between the poles). A smaller air gap means lower reluctance and higher inductance, so typically $L_d > L_q$ for salient-pole machines.
• In a cylindrical rotor machine (common in turbo generators), the air gap is essentially uniform, so $L_d \approx L_q$ and saliency effects are negligible.

Reluctance Torque

Saliency introduces an additional torque component called reluctance torque. The electromagnetic torque of a salient-pole machine has two parts:

$T_e = T_{excitation} + T_{reluctance}$

The reluctance torque is proportional to $(L_d - L_q)$ and to the product $i_d i_q$. In terms of the power-angle curve, the total power output can be written as:

$P = \frac{V_t E_{fd}}{X_d} \sin \delta + \frac{V_t^2 (X_d - X_q)}{2 X_d X_q} \sin 2\delta$

The first term is the power due to field excitation. The second term is the reluctance power, which peaks at $\delta = 45°$ and depends entirely on the saliency difference. For cylindrical rotor machines, $X_d = X_q$, so the reluctance term vanishes.

Impact on Machine Characteristics

Saliency affects several aspects of machine performance:

• Transient and subtransient reactances: The d-axis and q-axis transient reactances ($X_d'$, $X_q'$) and subtransient reactances ($X_d''$, $X_q''$) differ in salient-pole machines, which matters for fault current calculations and stability studies.
• Power-angle curve shape: The $\sin 2\delta$ reluctance term shifts the peak of the power-angle curve to an angle less than 90°, which affects the steady-state stability limit.
• Excitation requirements: The difference between $L_d$ and $L_q$ influences how much field current is needed to achieve a given reactive power output.
• Damping during oscillations: Reluctance torque can provide additional synchronizing torque, which contributes to the machine's ability to remain stable during power system oscillations.

Dynamic Response Analysis of Synchronous Machines

Numerical Solution of Dynamic Equations

The synchronous machine dynamic equations form a set of nonlinear ordinary differential equations that generally require numerical methods to solve. The process follows these steps:

1. Establish the steady-state operating point. Before any disturbance, solve the algebraic steady-state equations to find initial values for all state variables (currents, flux linkages, rotor angle, rotor speed).
2. Define the disturbance. This could be a step change in mechanical input power, a sudden load change, a three-phase fault, or a change in terminal voltage.
3. Integrate the differential equations. Use a numerical method such as Runge-Kutta (4th order) or the trapezoidal rule to advance the state variables forward in time. Euler's method works conceptually but is rarely accurate enough for practical studies.
4. Extract time-domain waveforms. The solution gives you the evolution of stator currents ($i_d$, $i_q$), rotor currents ($i_{fd}$, $i_{kd}$, $i_{kq}$), flux linkages, rotor angle ($\delta$), and rotor speed ($\omega$) over time.

The time step size matters: too large and the solution becomes inaccurate or unstable, too small and computation time increases unnecessarily. For electromechanical transients, time steps on the order of 1-10 ms are typical.

Response Analysis and Control Design

Once you have the time-domain response, the analysis proceeds as follows:

• Assess stability: Does the rotor angle settle to a new equilibrium, or does it diverge? A diverging $\delta$ indicates loss of synchronism.
• Identify oscillatory modes: Look for dominant oscillation frequencies and their damping. Eigenvalue analysis of the linearized system provides damping ratios and natural frequencies directly, complementing the time-domain results.
• Design controllers: Excitation systems regulate terminal voltage and can improve transient stability. A power system stabilizer (PSS) adds a supplementary signal to the exciter to provide damping torque for low-frequency oscillations (typically 0.2-2 Hz).
• Tune controller parameters: Adjust gains and time constants to achieve adequate damping (typically a damping ratio above 0.05 for all oscillatory modes) while maintaining acceptable response speed and robustness.
• Validate through simulation: Test the controller across multiple operating conditions and disturbance scenarios. Sensitivity analysis confirms that the design performs well despite uncertainties in machine parameters or system configuration.