Technology / Technical guide
Three-Phase Space Vectors & Ideal Inverter States
Three instantaneous phase quantities can be represented through stationary vector coordinates with a declared scaling convention. An ideal three-leg two-level inverter then produces a finite set of voltage vectors. Space-vector modulation concerns their time average; it is not the same subject as rotating d–q coordinates or switching frequency.
Define the vector scale before interpreting its magnitude.
The familiar six-vector geometry belongs to a specific ideal inverter model, not every converter topology or voltage convention.
01 / Stationary map
Coordinates are a representation, not additional phases.
Use a stationary α–β representation with the declared amplitude-preserving convention: vα = (2/3)(va − vb/2 − vc/2), and vβ = (√3/3)(vb − vc).
Use a stationary α–β representation with the declared amplitude-preserving convention: vα = (2/3)(va − vb/2 − vc/2), and vβ = (√3/3)(vb − vc). The three voltages share a common reference. A third zero-sequence coordinate is needed to retain their shared component.
Adding the same voltage to all three leaves these α–β expressions unchanged. That algebraic cancellation does not mean the shared voltage has no physical consequence. Alternative power-preserving scaling changes the vector magnitude; the chosen factor must remain attached to any comparison.
02 / Ideal state model
Three binary leg choices give eight combinations.
Consider only an ideal three-leg two-level model with fixed DC rail difference Vdc.
Consider only an ideal three-leg two-level model with fixed DC rail difference Vdc. Let Sa, Sb and Sc be binary pole states relative to the lower rail, so the pole voltages in this model are Vdc Sa, Vdc Sb and Vdc Sc.
Each leg has one selected ideal state, giving 2³ = 8 combinations. Real device overlap, dead time, current commutation and voltage drops are excluded from that geometrical model. Counting its states does not provide an actual switch command or establish MARSTON hardware.
03 / Vector geometry
Six combinations give active vectors under the stated map.
Applying the stated α–β map gives six nonzero vectors separated by 60°.
Applying the stated α–β map gives six nonzero vectors separated by 60°. Their magnitude is 2Vdc/3 in this amplitude-preserving convention. MathWorks’ space-vector explanation illustrates the six active patterns and two zero patterns.
A different normalisation can report another magnitude for the same represented physical pole voltages. Comparing diagrams without their scale can therefore produce an apparent discrepancy even when their states match. Peak, RMS and line-voltage quantities must also retain their separate definitions.
04 / Zero vectors
Zero differential vector does not mean every voltage or current is zero.
When all three ideal pole states match, their α–β vector is zero.
When all three ideal pole states match, their α–β vector is zero. The two matching-state combinations place the poles at different common potentials while giving zero phase differences. They are equivalent for this differential vector, but not identical in every reference-dependent quantity.
An inductive current can continue during such a state, depending on the circuit. A null voltage vector therefore does not prove zero phase current, zero stored energy or a de-energised assembly. The vector diagram represents selected voltage information rather than the complete electrical state.
05 / Time average
A continuous reference and instantaneous states are different.
Over a defined interval, the average vector is the time-weighted average of the vectors present in that interval.
Over a defined interval, the average vector is the time-weighted average of the vectors present in that interval. Convex combinations of the ideal states occupy a bounded region of the vector plane. The instantaneous switching waveform need not equal that average at each moment.
A modulation method assigns time among states to represent a requested average under its assumptions. This conceptual role differs from the rotating coordinate transform and from the carrier-frequency distinction. No state order or dwell-time prescription is supplied here.
06 / Model scope
Geometry does not disclose an implemented controller.
The vector model omits loss, timing error, parasitic coupling and connected-network response.
The vector model omits loss, timing error, parasitic coupling and connected-network response. Those can affect the relationship between the requested average and delivered waveform, as the dead-time guide explains for one mechanism.
No MARSTON two-level inverter, SVM, FOC, vector scaling or modulation capability is assigned. This article explains a declared ideal representation without gate sequences, duty calculations, controller settings or a claim of actual voltage utilisation or waveform quality.
Related technical guides: Inverter Dead Time: Commutation Interval & Waveform Error · Three-Phase Imbalance & Symmetrical Components · Converter Instantaneous Power, Storage & Dissipation.
General background: MathWorks: stationary α–β coordinate transformation · MathWorks: readable space-vector explanation and ideal state geometry. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Stationary coordinates | Amplitude-preserving α–β map | Shared voltage needs a separate zero-sequence term. |
| Pole states | Sa, Sb, Sc each 0 or 1 | Ideal three-leg two-level fixed-rail model. |
| State count | 2³ = 8 | One selected binary state per leg. |
| Active magnitude | 2Vdc/3 | Only the declared amplitude-preserving scale. |
| Average vector | Time-weighted represented vectors | Not the instantaneous waveform or full circuit state. |
Engineering questions.
Is a space-vector magnitude always scaled the same way?
No. Amplitude-preserving and power-preserving conventions differ.
Do all converter topologies have eight states?
No. Eight belongs to the stated ideal three-leg two-level model.
Does a zero vector mean zero phase current?
No. Existing inductive current can continue depending on the circuit.
Are both zero states identical in every voltage reference?
No. Their differential vector matches while common potentials differ.
Is SVM the same as a d–q transformation?
No. Modulation and coordinate representation have different roles.
Does this disclose MARSTON switching logic?
No actual topology, state sequence or modulation implementation is assigned.
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Three instantaneous phase quantities can be represented through stationary vector coordinates with a declared scaling convention. Share the model and applicable operating or design information for an engineering discussion.