Sparse Matrix Converter

The same AC-to-AC job as a matrix converter — but with fewer switches. The sparse matrix converter keeps the two-stage indirect layout and an imaginary DC link with no energy storage, while a clever bidirectional cell (one IGBT inside a four-diode bridge) cuts the transistor count.

Table of Contents:

Introduction

A sparse matrix converter (SMC) is a direct AC-to-AC converter — it turns a fixed three-phase supply into a three-phase output of adjustable voltage and frequency, with no bulky DC-link capacitor. It belongs to the matrix-converter family and is built on the indirect matrix converter layout: a rectifier stage and an inverter stage joined by an imaginary DC link that stores no energy.

What makes it “sparse” is the reduced number of switches. In an ordinary matrix converter every bidirectional switch needs two transistors. The sparse converter re-uses a single transistor for both current directions by wrapping it inside a small four-diode bridge. The result is a converter that is functionally equivalent to the standard matrix converter but uses fewer active devices, which lowers cost and simplifies the gate-drive circuitry.

This tutorial explains the sparse matrix converter from the ground up: its block and power-circuit diagrams, how the two stages work, its switching states and waveforms, the well-known 0.866 voltage limit, and how a modern model-predictive current controller can regulate its output current.

Block Diagram

At block level the sparse matrix converter reads left to right as a chain of five things. The three-phase supply passes through a small LC input filter, then into a rectifier stage made of bidirectional switch cells. Its output is the DC-link voltage Vdc — a pulsating but always-positive voltage on two rails that hold no stored energy (the “imaginary” DC link). That feeds an ordinary six-switch inverter stage, which produces the variable-voltage, variable-frequency output. A single modulation and control unit coordinates both stages.

Block diagram of a sparse matrix converter: three-phase supply, LC input filter, a rectifier stage of bidirectional cells, an imaginary DC link with no energy storage, a six-switch inverter stage, and a variable-voltage variable-frequency three-phase output feeding a load or motor, with one modulation and control unit driving both stages
Figure 1: Block diagram — LC filter, rectifier stage, imaginary DC link and inverter stage

Circuit Diagram

The full power circuit is shown below. This particular sparse matrix converter is built from 12 IGBTs and 30 diodes: an LC input filter, a rectifier stage of six bidirectional switch cells, an imaginary DC link, and a conventional six-switch inverter feeding a three-phase motor.

Sparse matrix converter power circuit of 12 IGBTs and 30 diodes: a three-phase star source a, b, c on the left feeds an LC input filter; the filtered phases feed a rectifier stage of six bidirectional switch cells arranged in three legs between a positive rail and a negative rail, each cell being one IGBT inside a four-diode bridge; the two rails form an imaginary DC link with no energy storage; on the right a conventional six-switch voltage-source inverter, each switch an IGBT with an anti-parallel diode, produces output phases A, B, C feeding a three-phase motor; two insets show the internal make-up of a rectifier cell and an inverter switch
Figure 2: Sparse matrix converter power circuit — LC filter, six bidirectional rectifier cells, imaginary DC link, six-switch inverter

Reading the circuit from left to right:

  • The three input phases a, b, c come from the balanced supply on the left (shown in star with neutral N).
  • An LC input filter (series inductors Lf and shunt capacitors Cf) sits between the grid and the converter. It smooths the switched currents drawn by the rectifier so that the current taken from the grid stays sinusoidal.
  • The rectifier stage connects the filtered input to the two DC rails. It has six bidirectional switch cells in three legs — one leg per input phase. In each leg an upper cell can connect its phase to the positive rail (P) and a lower cell to the negative rail (N). Crucially, each cell is a single IGBT sitting inside a bridge of four diodes, which lets that one transistor carry current in either direction — the trick that gives the “sparse” converter its low switch count.
  • The two rails in the middle are the imaginary DC link. There is no capacitor and no inductor here — the crossed-out symbol marks the storage element a normal drive would have but this converter deliberately omits. The rail-to-rail voltage is the DC-link voltage Vdc.
  • The inverter stage is an ordinary voltage-source inverter of six switches S1–S6 in three half-bridge legs. Each leg’s midpoint is one output phase (A, B, C). Because it sits on a DC rail whose polarity never reverses, each of these is just a single IGBT with one anti-parallel diode.
  • The three output phases A, B, C feed the three-phase load (typically an AC motor, shown as M).
Why “30 diodes”? Each of the six rectifier cells is one IGBT wrapped in a four-diode bridge, so the rectifier alone uses 6 IGBTs and 24 diodes. The six-switch inverter adds 6 IGBTs and 6 anti-parallel diodes. Together that is 12 IGBTs and 30 diodes — fewer transistors than the eighteen a conventional matrix converter would need.

Two kinds of switch

The two insets under the circuit show why the stages use different devices. A rectifier cell must carry current in both directions, so it is a bidirectional switch — here made from a four-diode bridge with a single IGBT across its DC terminals. Whichever way the phase current wants to flow, it finds a path through two of the diodes and the one transistor. An inverter switch, by contrast, sits on a rail whose polarity never reverses, so an ordinary single IGBT with one anti-parallel diode is enough, exactly as in any voltage-source inverter.

Working Principle

The sparse matrix converter works exactly like the indirect matrix converter it is based on: the control problem is decoupled into a rectifier part and an inverter part, and the two solutions are combined.

Rectifier side. At every instant the rectifier selects two input phases — the most positive and the most negative — and clamps them to the two DC rails. This makes the DC-link voltage Vdc as large and as positive as possible, and, by choosing how long each pair is used over a switching period, it shapes the input current so it is sinusoidal and in phase with the input voltage for near-unity power factor. Because the rails carry no stored charge, the rectifier is always switched between the largest and the second-largest line-to-line voltages.

Inverter side. The six-switch inverter chops that DC-link voltage with ordinary space-vector PWM. By varying the fraction of each switching period that each output leg is tied to the positive or negative rail, it builds three output phase voltages whose local averages trace smooth sine waves — at whatever amplitude and frequency the load needs, independent of the supply.

A detail that matters for the sparse converter is timing between the stages. Because there is no DC-link capacitor, the rectifier cannot be switched while current is flowing in the rails without risking a short between input phases. So the inverter is first put into a free-wheeling (zero) state that drives the rail current to zero, and only then does the rectifier change over. This zero-current commutation keeps the switching safe and low-loss.

Modes of Operation (Switching States)

A matrix converter does not have “modes” in the way a chopper does; it steps rapidly between allowed switching states. Thanks to the indirect layout, the states of the two stages can be listed separately. (These are described in words only — the circuit itself is the one in Figure 2.)

Rectifier-stage states

The rectifier obeys one hard rule: the two DC rails must always have a current path, and two input phases must never be short-circuited. That leaves it connecting the positive rail to one input phase and the negative rail to a different one. Counting the useful combinations gives nine allowed states: six “active” states (each connects a definite pair of input lines across the DC link) and three “zero” states (both rails tied to the same phase, so Vdc collapses to zero). To keep the DC-link voltage as high as possible, the rectifier only uses the pair with the largest available voltage, which is why Vdc follows the upper envelope of the line-to-line voltages.

Inverter-stage states

The inverter is an ordinary three-leg voltage-source bridge, so the two switches in a leg are complementary (never both on — that would short the DC link; never both off — that would open the inductive output). With three legs each choosing “top” or “bottom”, there are 2 × 2 × 2 = eight allowed states: six “active” vectors (100, 110, 010, 011, 001, 101) that apply a real voltage to the load, and two “zero” vectors (000 and 111) that apply zero line-to-line voltage. These eight states are exactly the corners and centre of the space-vector hexagon shown later.

Commutation & combining the stages

In each switching period the modulator picks a rectifier state (which sets Vdc and the input-current direction) and a sequence of inverter states (which set the output voltage), and dwells in each for a calculated time. The overall behaviour is the product of the two. A key practical point is the commutation sequence: the inverter is switched into a zero (free-wheeling) state so the DC-link current is zero before the rectifier changes its state. Because a full switching cycle of the inverter fits inside each half of a rectifier pulse, the inverter effectively switches at twice the rectifier rate.

9 rectifier states × 8 inverter states. Multiplying them out reproduces the same set of input-to-output connections a matrix converter reaches — but each stage is modulated with textbook rectifier and inverter PWM, and the input stage always changes over at zero DC-link current.

Waveforms & Explanation

The clearest way to understand the sparse matrix converter is to follow the voltage across the two stages. Start with the DC-link voltage. The lower panel below is the three input phase voltages; the upper panel is Vdc, formed at every instant as the largest minus the smallest input voltage — because the rectifier always ties the most-positive input to the +rail and the most-negative to the −rail. The strip underneath names which phase is clamped to each rail in every 60° interval.

Top panel: the DC-link voltage V_dc, an always-positive six-pulse waveform formed as the most positive input phase minus the most negative, rippling six times per input cycle between 1.5 and about 1.73 times the phase peak. Bottom panel: the three input phase voltages va, vb, vc at 60 Hz with a strip naming which phase is clamped to the positive and negative rails in each 60 degree interval
Figure 3: The DC-link voltage Vdc — always positive, six pulses per input cycle
  • It never goes negative. Because the rectifier always chooses the most-positive and most-negative phases, Vdc is a positive, pulsating voltage — just what the inverter needs on its rails.
  • It ripples six times per input cycle. As the “most positive” and “most negative” roles hand over between phases, Vdc traces the upper envelope of the line-to-line voltages, dipping to 1.5 and peaking near √3 ≈ 1.73 times the phase peak.
  • There is no smoothing capacitor. A normal drive would flatten this ripple with a big capacitor; here the inverter’s modulation is made to account for the ripple instead.

The inverter then synthesises the output voltage as a rotating space vector. The six active inverter states point to the corners of a hexagon and the two zero states sit at its centre; any reference voltage inside the hexagon is built by time-sharing the two active vectors that straddle it (weights dα and dβ) plus a zero vector. The largest undistorted output is the circle inscribed in the hexagon:

Output-voltage space-vector hexagon: six active vectors V1 to V6 at the corners labelled with switching states 100, 110, 010, 011, 001, 101; two zero vectors 000 and 111 at the centre; a dashed inscribed circle marking the largest undistorted reference amplitude; and a reference vector V_ref at angle theta synthesised from its two neighbouring active vectors weighted by duty ratios d-alpha and d-beta
Figure 4: Output-voltage space-vector hexagon — six active vectors, two zero vectors and the reference circle

Because the load is inductive it filters the fast voltage steps into smooth currents. The three output line currents come out as clean, balanced sine waves at the chosen output frequency, 120° apart, with only a small switching ripple:

Three balanced output line currents i_A, i_B, i_C as smooth sine waves 120 degrees apart at the chosen output frequency, each with a small high-frequency switching ripple because the inductive load filters the chopped output voltage
Figure 5: Balanced sinusoidal output currents at the chosen output frequency

Finally, the payoff on the input side. Because the rectifier stage actively shapes the current it draws (helped by the LC filter), the three input line currents are also sinusoidal — at the supply frequency and, if the control chooses, in phase with the input voltages for near-unity power factor:

Three balanced input line currents i_a, i_b, i_c as smooth 60 Hz sine waves 120 degrees apart, in phase with the input voltages for near unity power factor, with a small switching ripple
Figure 6: Sinusoidal input line currents — near-unity input power factor

Predictive Current Control

The switching states above tell the converter what it can do; a controller must decide which state to apply next. A powerful modern choice is model-predictive control (MPC). The idea is intuitive: use a mathematical model of the load to predict the output current one step into the future for each possible switching state, then pick the state that brings the current closest to its reference. Fast digital signal processors make this practical to run every sampling period.

System diagram of a sparse matrix converter driven by a model-predictive current controller: along the top the power path runs from the three-phase grid through the LC input filter and the sparse matrix converter to a three-phase R-L load where the load current is measured; along the bottom the control loop feeds the measured current into a predictive model that computes the next-step current, then a cost function that squares the difference between the reference current and the prediction and finds its minimum, then the optimal switching state and space-vector modulation whose gate signals drive the converter
Figure 7: Sparse matrix converter with a model-predictive current controller

Assume the load is inductive-resistive (R–L), as most machine windings are. Its three-phase voltage and current are related by a simple differential equation, with RL the load resistance, LL the load inductance and iABC the load current:

vABC = RL·iABC + LL·(diABC/dt)

Transforming this into a rotating d–q frame turns the three-phase equation into two decoupled scalar ones for the real and imaginary current components id and iq. Sampling at intervals Ts and approximating the derivative gives a compact discrete-time prediction of the next current from the present measurement and the applied voltage:

id(k+1) = a·id(k) + b·vd(k)
iq(k+1) = a·iq(k) + b·vq(k)
with  a = e−RLTs/LL  and  b = (1 − e−RLTs/LL) / RL

For every allowed switching state the controller computes the voltage that state would apply, uses the equations above to predict the resulting current i(k+1), and scores it with a cost function — the squared distance between the predicted current and the reference current i*:

J(k) = | i* − i(k+1) |²

The state with the smallest cost is the one applied in the next period. Because the cost is a simple quadratic in the output voltage vector, the ideal reference voltage that would drive the current straight to its target can even be written in closed form; if that ideal voltage lies outside what the converter can produce (beyond the 0.866 circle), the controller clamps it to the boundary of the reachable region. A matching predictive model of the input filter lets the same scheme keep the input current sinusoidal as well.

Why engineers like it here. The method is easy to understand, handles the converter’s constraints naturally (it simply never proposes a forbidden switching state), needs no separate PI-loop tuning, and gives a very fast dynamic response — the output current can follow a step change in its reference in a fraction of a millisecond, much quicker than a classical linear controller.

Transfer Ratio & Modulation

Like every matrix converter, the sparse converter only ever connects input to output — it never boosts — so the output voltage is always smaller than the input. For undistorted sinusoidal operation the largest ratio of output to input voltage magnitude is:

qmax = √3 / 2 ≈ 0.866  (about 86.6% of the input voltage)

This ceiling is exactly the radius of the circle inscribed in the space-vector hexagon: push the reference beyond it and the inverter runs out of DC-link voltage, so the output stops following the reference and distorts. The limit is a property of the topology, so the sparse, indirect and direct matrix converters all share it. In return you get sinusoidal input and output currents, adjustable input power factor and (with fully bidirectional cells) four-quadrant power flow — all with no DC-link capacitor.

Sparse, Very Sparse & Ultra Sparse

The “sparse” idea can be pushed to different degrees, trading switch count against capability. The three members of the family are all derived from the indirect matrix converter:

VariantRectifier switchesKey trait
Sparse (SMC)Fewer transistors than a full matrix converter, still fully bidirectionalFull four-quadrant operation; a direct simplification of the indirect converter’s input stage
Very sparse (VSMC)One transistor per cell inside a four-diode bridge (the 12-IGBT / 30-diode circuit above)Lowest transistor count while keeping bidirectional current; the version drawn on this page
Ultra sparse (USMC)Fewest transistors of allSimplest and cheapest, but power flow is one-directional and the input displacement angle is limited

All three keep the same headline features — no DC-link storage, sinusoidal input and output, and the 0.866 voltage ceiling — and differ only in how far the rectifier’s switch count is reduced and what that costs in flexibility.

Advantages & Disadvantages

Advantages

  • Fewer switches. A single transistor per bidirectional cell cuts the transistor count versus a conventional matrix converter, lowering cost and simplifying the gate drives.
  • No DC-link capacitor. Like every matrix converter it removes the bulky, failure-prone electrolytic capacitor, giving a compact, long-life converter.
  • Simpler, safer commutation. The input stage changes over while the DC-link current is zero, avoiding the risky multi-step commutation of the direct matrix converter.
  • Sinusoidal input and output currents with adjustable, near-unity input power factor.

Disadvantages

  • Voltage transfer ratio limited to 0.866 — the output cannot reach the full input voltage.
  • More diodes. The four-diode bridge per cell adds conduction losses (current passes through two diodes plus the transistor at all times) and raises the diode count.
  • Pulsating DC link. With no smoothing capacitor, the modulation must actively account for the six-times ripple in Vdc.
  • Sensitive to supply disturbances and usually needs a protective clamp circuit, since there is no capacitor to ride through dips or absorb spikes.

Applications

  • Compact AC motor drives where removing the DC-link capacitor saves size, weight and service cost.
  • Aerospace and defence actuators and drives, which benefit from dropping heavy, temperature-sensitive electrolytic capacitors.
  • Wind and other renewable generation needing clean, bidirectional grid interfacing.
  • Regenerative and four-quadrant drives for lifts, cranes and traction that feed energy back on braking.
  • High-performance servo drives where a fast predictive current loop gives quick, accurate torque control.

Frequently Asked Questions – FAQs

It is a direct AC-to-AC converter based on the indirect matrix converter — a rectifier stage and a six-switch inverter joined by an imaginary DC link with no energy storage — but built with a reduced number of switches. Each bidirectional rectifier cell uses a single IGBT inside a four-diode bridge, so it does the same job as a full matrix converter with fewer transistors.

The version shown here uses 12 IGBTs and 30 diodes: the rectifier has six bidirectional cells (one IGBT and four diodes each, so 6 IGBTs and 24 diodes) and the inverter is a conventional six-switch bridge (6 IGBTs and 6 anti-parallel diodes). That is fewer transistors than the eighteen a conventional matrix converter needs.

The sparse matrix converter is derived from the indirect matrix converter and is functionally equivalent to it. The difference is in the rectifier: instead of building each bidirectional switch from two transistors, the sparse converter uses a single transistor inside a four-diode bridge, which reduces the transistor count. Both use an imaginary DC link and share the 0.866 voltage limit.

It is the voltage the rectifier presents across the two rails at each instant: the most-positive input phase minus the most-negative one. It is always positive and ripples six times per input cycle, moving between about 1.5 and 1.73 times the input phase peak. There is no capacitor to smooth it, so the inverter modulation accounts for the ripple.

It is a control method that uses a model of the load to predict the output current one step ahead for every allowed switching state, then applies the state that minimises a cost function — the squared error between the predicted current and the reference. It handles the converter’s constraints naturally, needs no PI tuning and gives a very fast dynamic response.

For undistorted sinusoidal output it is √3/2, about 0.866 (86.6%). The output can never exceed the input voltage; this limit is the radius of the circle inscribed in the space-vector hexagon and is the same for the sparse, very sparse, ultra sparse, indirect and direct matrix converters.

They are variants that reduce the rectifier switch count further. The very sparse matrix converter uses one transistor per cell inside a four-diode bridge (the 12-IGBT circuit on this page) and keeps bidirectional current. The ultra sparse matrix converter uses the fewest transistors of all but only supports one-directional power flow and a limited input displacement angle.

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