Why is the sum of the three phase voltages always zero in this model?

In a balanced three-phase system with no neutral wire (a three-wire system), the three voltages are generated 120° apart. At any instant, when one phase is at its positive peak, the other two are at negative values that precisely cancel it out. This is a consequence of the symmetry of the sine waves and Kirchhoff's Current Law, which implies the conductors carry only the differences between phases. This zero-sum condition is a fundamental property of balanced three-phase power.

What is the practical use of the Clarke transform and the space vector?

The Clarke transform simplifies the analysis and control of three-phase systems. By reducing three interdependent AC quantities to a single rotating vector in two dimensions, it becomes much easier to design controllers for devices like induction motors. This vector representation is the first step in Field-Oriented Control (FOC), a high-performance method that allows AC motors to be controlled with precision similar to DC motors, enabling variable speed drives in industrial and automotive applications.

Does the space vector's constant magnitude in the simulator reflect reality?

For an ideal, balanced three-phase system with pure sine waves, yes, the space vector magnitude is constant. This represents a perfectly rotating magnetic field in a motor or a perfectly balanced power feed. In real-world systems, imbalances, harmonics, or faults can cause the vector's magnitude and rotation speed to wobble or fluctuate, which is a key diagnostic tool for power quality analysis.

How does the α–β plane relate to the physical windings of a motor?

The two orthogonal axes (α and β) of the Clarke transform correspond to a mathematical representation of a two-phase equivalent system. Imagine a motor with two sets of stationary windings placed perpendicular to each other. The α and β components represent the instantaneous magnetic forces that would be produced in those hypothetical windings to create the same rotating magnetic field as the original three-phase windings. This abstraction is crucial for advanced control techniques.

Three-Phase AC

Three-phase alternating current (AC) systems form the backbone of modern electrical power generation, transmission, and motor drives. This simulator visualizes the core mathematical and physical principles behind a balanced three-phase system. It generates three sinusoidal voltages, typically labeled v_A, v_B, and v_C, each separated by a 120° phase shift. A fundamental constraint of a balanced three-wire system is that the instantaneous sum of these three voltages is always zero: v_A(t) + v_B(t) + v_C(t) = 0. The model then applies the Clarke transformation, a mathematical tool used in power electronics and motor control. This transformation projects the three-phase quantities from the ABC reference frame onto a stationary two-axis orthogonal coordinate system known as the α–β (alpha–beta) plane. The transformation equations are: v_α = (2/3) * [v_A - (1/2)v_B - (1/2)v_C] and v_β = (2/3) * [(√3/2)v_B - (√3/2)v_C]. For a balanced system, this simplifies significantly. The resulting vector, S = v_α + j v_β, is the Clarke space vector. The simulator shows this vector rotating in the α–β plane with a constant magnitude and angular speed directly related to the AC frequency. This visualization powerfully demonstrates how three time-varying scalar quantities can be represented as a single rotating vector. Key simplifications include assuming perfectly sinusoidal waveforms, a perfectly balanced system (equal amplitudes, exact 120° separation), and no harmonic distortion. By interacting with this model, students learn to connect the time-domain behavior of three-phase voltages to their elegant vector representation, understand the principle of instantaneous voltage summation, and grasp the foundational mathematics behind the Clarke transform used in field-oriented control of AC motors.

Who it's for: Undergraduate engineering students in power systems, electrical machines, or power electronics courses, as well as educators and professionals seeking to visualize space vector concepts.

Key terms

Three-Phase AC
Clarke Transformation
Space Vector
Alpha-Beta (α–β) Frame
Phase Shift (120°)
Balanced System
Sinusoidal Waveform
Instantaneous Voltage Sum

Live graphs

How it works

Three sinusoidal phases separated by 120° (2π/3 rad) model a balanced three-wire or Y-connected source. Instantaneous voltages sum to zero. In the α–β (Clarke) plane the equivalent space vector has constant magnitude V₀ and rotates at angular frequency ω — a compact picture of why three-phase motors can produce smooth rotating fields.

Key equations

v_A = V₀ cos ωt , v_B = V₀ cos(ωt − 2π/3) , v_C = V₀ cos(ωt − 4π/3)

v_α = (2/3)(v_A − v_B/2 − v_C/2) , v_β = (√3/3)(v_B − v_C)

Frequently asked questions

Why is the sum of the three phase voltages always zero in this model?: In a balanced three-phase system with no neutral wire (a three-wire system), the three voltages are generated 120° apart. At any instant, when one phase is at its positive peak, the other two are at negative values that precisely cancel it out. This is a consequence of the symmetry of the sine waves and Kirchhoff's Current Law, which implies the conductors carry only the differences between phases. This zero-sum condition is a fundamental property of balanced three-phase power.
What is the practical use of the Clarke transform and the space vector?: The Clarke transform simplifies the analysis and control of three-phase systems. By reducing three interdependent AC quantities to a single rotating vector in two dimensions, it becomes much easier to design controllers for devices like induction motors. This vector representation is the first step in Field-Oriented Control (FOC), a high-performance method that allows AC motors to be controlled with precision similar to DC motors, enabling variable speed drives in industrial and automotive applications.
Does the space vector's constant magnitude in the simulator reflect reality?: For an ideal, balanced three-phase system with pure sine waves, yes, the space vector magnitude is constant. This represents a perfectly rotating magnetic field in a motor or a perfectly balanced power feed. In real-world systems, imbalances, harmonics, or faults can cause the vector's magnitude and rotation speed to wobble or fluctuate, which is a key diagnostic tool for power quality analysis.
How does the α–β plane relate to the physical windings of a motor?: The two orthogonal axes (α and β) of the Clarke transform correspond to a mathematical representation of a two-phase equivalent system. Imagine a motor with two sets of stationary windings placed perpendicular to each other. The α and β components represent the instantaneous magnetic forces that would be produced in those hypothetical windings to create the same rotating magnetic field as the original three-phase windings. This abstraction is crucial for advanced control techniques.

Other simulators in this category — or see all 56.

View category →

NewSchool

Hall Effect

U_H = R_H I B / t; R_H sign for e⁻ vs holes; B, I, n, t.

Launch Simulator

NewSchool

Photoelectric Effect

E = hc/λ vs φ, K_max, V_s; I vs λ and vs intensity (model).

Launch Simulator

NewSchool

Particle in E and B Fields

Lorentz force: cyclotron motion, E×B drift, trajectories in sim units.

Launch Simulator

FeaturedSchool

Circuit Builder

Drag and drop components: battery, resistor, bulb, switch. See current flow.

Launch Simulator

School

Ohm's Law

Adjust voltage and resistance. See current change with interactive V-I graph.

Launch Simulator

School

Series & Parallel

Compare total resistance and current distribution side by side.

Launch Simulator

Three-Phase AC

Live graphs

How it works

Key equations

Frequently asked questions

More from Electricity & Magnetism

Hall Effect

Photoelectric Effect

Particle in E and B Fields

Circuit Builder

Ohm's Law

Series & Parallel

Three-Phase AC

Live graphs

How it works

Key equations

Frequently asked questions