Output voltage control

Ideal three-phase matrix converters are obtained by assembling nine bidirectional switches, with the turn-off capability, to allow the connection of each one of the input phases to any one of the output phases (Fig. 35.64). The states of these switches are usually represented as a nine-element matrix S (Eq. (35.171)), in which each matrix element, S_kj k, j∈{1, 2, 3}, has two possible states: S_kj=1 if the switch is closed (ON) and S_kj=0 if it is open (OFF). Only 27 switching combinations are possible (Table 35.6), as a result of the topological constraints (the input phases should never be short-circuited, and the output inductive currents should never be interrupted), which implies that the sum of all the S_kj of each one of the matrix, k rows, must always equal to 1 (Eq. (35.171)):

Table 35.6

Switching combinations and output line voltage/input current state-space vectors

Group	Name	von1	von2	von3	vo12	vo23	vo31	ii1	ii2	ii3	Vol	δo	Ii	μi
I	1 g	vi1	vi2	vi3	vi12	vi23	vi31	io1	io2	io3	vi	δi	io	μo
	2 g	vi1	vi3	vi2	−vi31	−vi23	−vi12	io1	io3	io2	−vi	−δi+4π/3	io	−μo
	3 g	vi2	vi1	vi3	–vi12	–vi31	–vi23	io2	io1	io3	−vi	−δi	io	−μo+2π/3
	4 g	vi2	vi3	vi1	vi23	vi31	vi12	io3	io1	io2	vi	δi+4 π/3	io	μo+2π/3
	5 g	vi3	vi1	vi2	vi31	vi12	vi23	io2	io3	io1	vi	δi+2 π/3	io	μo+4π/3
	6 g	vi3	vi2	vi1	–vi23	–vi12	–vi31	io3	io2	io1	−vi	−δi+2 π/3	io	−μo+4π/3
II	+1	vi1	vi2	vi2	vi12	0	–vi12	io1	−io1	0	$2/\sqrt{3}$ vi12	π/6	$2/\sqrt{3}$ io1	−π/6
	–1	vi2	vi1	vi1	–vi12	0	vi12	–io1	io1	0	$-2/\sqrt{3}$ vi12	π/6	$-2/\sqrt{3}$ io1	−π/6
	+2	vi2	vi3	vi3	vi23	0	–vi23	0	io1	−io1	$2/\sqrt{3}$ vi23	π/6	$2/\sqrt{3}$ io1	π/2
	–2	vi3	vi2	vi2	–vi23	0	vi23	0	−io1	io1	$-2/\sqrt{3}$ vi23	π/6	$-2/\sqrt{3}$ io1	π/2
	+3	vi3	vi1	vi1	vi31	0	–vi31	–io1	0	io1	$2/\sqrt{3}$ vi31	π/6	$2/\sqrt{3}$ io1	7π/6
	–3	vi1	vi3	vi3	–vi31	0	vi31	io1	0	−io1	$-2/\sqrt{3}$ vi31	π/6	$-2/\sqrt{3}$ io1	7π/6
	+4	vi2	vi1	vi2	–vi12	vi12	0	io2	−io2	0	$2/\sqrt{3}$ vi12	5π/6	$2/\sqrt{3}$ io2	−π/6
	–4	vi1	vi2	vi1	vi12	–vi12	0	–io2	io2	0	$-2/\sqrt{3}$ vi12	5π/6	$-2/\sqrt{3}$ io2	−π/6
	+5	vi3	vi2	vi3	–vi23	vi23	0	0	io2	−io2	$2/\sqrt{3}$ vi23	5π/6	$2/\sqrt{3}$ io2	π/2
	–5	vi2	vi3	vi2	vi23	–vi23	0	0	−io2	io2	$-2/\sqrt{3}$ vi23	5π/6	$-2/\sqrt{3}$ io2	π/2
	+6	vi1	vi3	vi1	–vi31	vi31	0	–io2	0	io2	$2/\sqrt{3}$ vi31	5π/6	$2/\sqrt{3}$ io2	7π/6
	–6	vi3	vi1	vi3	vi31	–vi31	0	io2	0	−io2	$-2/\sqrt{3}$ vi31	5π/6	$-2/\sqrt{3}$ io2	7π/6
	+7	vi2	vi2	vi1	0	vi12	vi12	io3	−io3	0	$2/\sqrt{3}$ vi12	3π/2	$2/\sqrt{3}$ io3	−π/6
	–7	vi1	vi1	vi2	0	vi12	vi12	–io3	io3	0	$-2/\sqrt{3}$ vi12	3π/2	$-2/\sqrt{3}$ io3	−π/6
	+8	vi3	vi3	vi2	0	–vi23	vi23	0	io3	−io3	$2/\sqrt{3}$ vi23	3π/2	$2/\sqrt{3}$ io3	π/2
	–8	vi2	vi2	vi3	0	vi23	–vi23	0	−io3	io3	$-2/\sqrt{3}$ vi23	3π/2	$-2/\sqrt{3}$ io3	π/2
	+9	vi1	vi1	vi3	0	–vi31	vi31	–io3	0	io3	$2/\sqrt{3}$ vi31	3π/2	$2/\sqrt{3}$ io3	7π/6
	–9	vi3	vi3	vi1	0	vi31	−vi31	io3	0	−io3	$-2/\sqrt{3}$ vi31	3π/2	$-2/\sqrt{3}$io3	7π/6
III	z1	vi1	vi1	vi1	0	0	0	0	0	0	0	–	0	–
	z2	vi2	vi2	vi2	0	0	0	0	0	0	0	–	0	–
	z3	vi3	vi3	vi3	0	0	0	0	0	0	0	–	0	–

$\mathbf{S}=\left[\begin{array}{lll}{S}_{11}\hfill & S_{12}\hfill & S_{13}\hfill \\ S_{21}\hfill & S_{22}\hfill & S_{23}\hfill \\ S_{31}\hfill & S_{32}\hfill & S_{33}\hfill \end{array}\right]{\displaystyle \sum _{j=1}^3{S}_{kj}=1k,j\in \left\{1,2,3\right\}}$

Based on matrix S, output voltages related to input neutral $v_{o{n}_1}$, $v_{o{n}_2}$, and $v_{o{n}_3}$ and line voltages $v_{o_{12}}$, $v_{o_{23}}$, and $v_{o_{31}}$ can be expressed in terms of matrix converter input phase voltages $v_{i_1}$, $v_{i_2}$, and $v_{i_3}$. Input currents $i_{i_1}$, $i_{i_2}$, and $i_{i_3}$ can be expressed as a function of output currents $i_{o_1}$, $i_{o_2}$, and $i_{o_3}$:

$\begin{array}{l}\left[\begin{array}{l}{v}_{o{n}_1}\hfill \\ v_{o{n}_2}\hfill \\ v_{o{n}_3}\hfill \end{array}\right]=\mathbf{S}\left[\begin{array}{l}{v}_{i_1}\hfill \\ v_{i_2}\hfill \\ v_{i_3}\hfill \end{array}\right]\\ \left[\begin{array}{l}{v}_{o_{12}}\hfill \\ v_{o_{23}}\hfill \\ v_{o_{31}}\hfill \end{array}\right]=\left[\begin{array}{lll}{S}_{11}-S_{21}\hfill & S_{12}-S_{22}\hfill & S_{13}-S_{23}\hfill \\ S_{21}-S_{31}\hfill & S_{22}-S_{32}\hfill & S_{23}-S_{33}\hfill \\ S_{31}-S_{11}\hfill & S_{32}-S_{12}\hfill & S_{33}-S_{13}\hfill \end{array}\right]\left[\begin{array}{l}{v}_{i_1}\hfill \\ v_{i_2}\hfill \\ v_{i_3}\hfill \end{array}\right]\\ \left[\begin{array}{l}{i}_{i_1}\hfill \\ i_{i_2}\hfill \\ i_{i_3}\hfill \end{array}\right]={\mathbf{S}}^T\left[\begin{array}{l}{i}_{o_1}\hfill \\ i_{o_2}\hfill \\ i_{o_3}\hfill \end{array}\right]\end{array}$

The application of Concordia transformation [X_α,β,0]^T=C^T [X_1,2,3]^T to the output line-to-line voltages of Eq. (35.172) results in the output voltage vector:

$\begin{array}{l}{\mathbf{v}}_{\mathbf{o}{\mathbf{l}}_{\mathbf{\alpha }\mathbf{\beta }}}=\left[\begin{array}{l}{v}_{o{l}_{\alpha }}\\ v_{o{l}_{\beta }}\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{lll}1\hfill & -\frac{1}{2}\hfill & -\frac{1}{2}\hfill \\ \hfill 0\hfill & \hfill \frac{\sqrt{3}}{2}\hfill & \hfill -\frac{\sqrt{3}}{2}\hfill \end{array}\right]\left[\begin{array}{lll}{S}_{11}-S_{21}\hfill & S_{12}-S_{22}\hfill & S_{13}-S_{23}\hfill \\ S_{21}-S_{31}\hfill & S_{22}-S_{32}\hfill & S_{23}-S_{33}\hfill \\ S_{31}-S_{11}\hfill & S_{32}-S_{12}\hfill & S_{33}-S_{13}\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_{i_1}\hfill \\ \hfill v_{i_2}\hfill \\ \hfill v_{i_3}\hfill \end{array}\right]\\ =\left[\begin{array}{cc}\hfill {\rho }_{v_{\alpha \alpha }}\hfill & \hfill {\rho }_{v_{\alpha \beta }}\hfill \\ \hfill {\rho }_{v_{\beta \alpha }}\hfill & \hfill {\rho }_{v_{\beta \beta }}\hfill \end{array}\right]\left[\begin{array}{l}{v}_{i{c}_{\alpha }}\\ v_{i{c}_{\beta }}\end{array}\right]\end{array}$

where v_icαβ is the input filter capacitor voltage and ${\rho }_{v_{\alpha \alpha }}$, ${\rho }_{v_{\alpha \beta }}$, ${\rho }_{v_{\beta \alpha }}$, and ${\rho }_{v_{\beta \beta }}$ are functions of the ON/OFF state of the nine S_kj switches:

$\begin{array}{l}\left[\begin{array}{cc}{\rho }_{v_{\alpha \alpha }}& {\rho }_{v_{\alpha \beta }}\\ {\rho }_{v_{\beta \alpha }}& {\rho }_{v_{\beta \beta }}\end{array}\right]=\frac{1}{2}[\begin{array}{c}\left(S_{11}-S_{21}-S_{12}+S_{22}\right)\\ \sqrt{3}\left(S_{11}+S_{21}-2S_{31}-S_{12}-S_{22}+2S_{32}\right)\end{array}\\ \begin{array}{c}\sqrt{3}\left(S_{11}-S_{21}+S_{12}-S_{22}\right)\\ \left(S_{11}+S_{21}-2S_{31}+S_{12}+S_{22}+2S_{32}\right)\end{array}]\end{array}$

The average value $\overline{v_{o{l}_{\mathrm{\alpha }\beta }}}$ of the output line-to-line voltages, in αβ coordinates, during one switching period, is the output variable to be controlled (since v_olαβ is discontinuous) [23,50]:

$\overline{v_{o{l}_{\alpha \beta }}}=\frac{1}{T_s}\underset{n{T}_s}{\overset{\left(n+1\right)T_s}{\int }}{v}_{o{l}_{\alpha \beta }}\mathrm{d}t$

Considering the control goal $\overline{v_{o{l}_{\mathrm{\alpha }\beta }}}=\overline{v_{o{l}_{\mathrm{\alpha }\beta ref}}}$, the sliding surface S(e_αβ, t)(k_αβ>0) is

$S\left(e_{\alpha \beta },t\right)=\frac{k_{\alpha \beta }}{T}\underset{0}{\overset{T}{\int }}\left(v_{o{l}_{\alpha {\beta }_{ref}}}-v_{o{l}_{\alpha \beta }}\right)\mathrm{d}t=0$

The first derivative of Eq. (35.176) is

$\stackrel{.}{S}\left(e_{\alpha \beta },t\right)=k_{\alpha }\left(v_{o{l}_{\mathit{\alpha \beta ref}}}-v_{o{l}_{\alpha \beta }}\right)$

As the sliding-mode stability is guaranteed if S_αβ(e_αβ, t) ${\stackrel{.}{S}}_{\alpha \beta }\left(e_{\alpha \beta },t\right)<0$, the criterion to choose the output line voltages state-space vectors is

$\begin{array}{l}{S}_{\alpha \beta }\left(e_{\alpha \beta },t\right)<0\Rightarrow {\stackrel{.}{S}}_{\alpha \beta }\left(e_{\alpha \beta },t\right)>0\Rightarrow v_{o{l}_{\alpha \beta }}<v_{o{l}_{\alpha \beta ref}}\\ S_{\alpha \beta }\left(e_{\alpha \beta },t\right)>0\Rightarrow {\stackrel{.}{S}}_{\alpha \beta }\left(e_{\alpha \beta },t\right)<0\Rightarrow v_{o{l}_{\alpha \beta }}>v_{o{l}_{\alpha \beta ref}}\end{array}$

This implies that the sliding mode is reached only when the vector applied to the converter has the desired amplitude and phase.

According to Table 35.6, the six vectors of group I have fixed amplitude but time-varying phase, the 18 vectors of group II have variable amplitude, and the vectors of group III are null. Therefore, from the load viewpoint, the 18 highest amplitude vectors (6 vectors from group I and 12 vectors from group II) and one null vector are suitable to guarantee the sliding-mode stability.

Therefore, if two three-level comparators (C_αβ∈{−1, 0, 1}) are used to quantize the deviations of Eq. (35.178) from zero, the nine output voltage error combinations (3³) are not enough to guarantee the choice of all the 19 available vectors. The extra vectors may be used to control the input power factor. As an example, if the output voltage error is quantized as C_α=1 and C_β=1, at sector V_i1 (Fig. 35.65), the vectors −3, +1, or 1 g might be used to control the output voltage. The final choice would depend on the input current error.

Output current control

For some applications, it is more useful to control the output currents, which implies some changes on the previously designed controllers. From Fig. 35.64, the output currents can be obtained as a function of the phase voltages applied to the load:

$\left\{\begin{array}{l}\frac{d{i}_{o_{\alpha }}}{dt}=-\frac{R}{L}{i}_{o_{\alpha }}+\frac{1}{L}{v}_{o_{\alpha }}+\frac{1}{L}{E}_{\alpha }\\ \frac{d{i}_{o_{\beta }}}{dt}=-\frac{R}{L}{i}_{o_{\beta }}+\frac{1}{L}{v}_{o_{\beta }}+\frac{1}{L}{E}_{\beta }\end{array}\right.$

As the output currents are state variables, with a strong relative degree of 1, the sliding surfaces should depend directly on their errors:

$S_{i{o}_{\alpha \beta }}\left(e_{i{o}_{\alpha \beta }},t\right)=k_{i_o}\left(i_{o_{\mathit{\alpha \beta ref}}}-i_{o_{\alpha \beta }}\right)$

The state-space vectors should be chosen to guarantee the stability criterion $S_{i{o}_{\alpha \beta }}\left(e_{i{o}_{\alpha \beta }},t\right){\stackrel{.}{S}}_{i{o}_{\alpha \beta }}\left(e_{i{o}_{\alpha \beta }},t\right)<0$, according to the following:

(a) If $S_{i{o}_{\alpha \beta }}\left(e_{i{o}_{\alpha \beta }},t\right)<0$, then $i_{o_{\alpha \beta }}>i_{o_{\mathit{\alpha \beta ref}}}$ and the output current must decrease $i_{o_{\alpha \beta }}\downarrow$. This means that the chosen output voltage $v_{o_{\alpha \beta }}$ phase vector should be low enough to guarantee $\mathrm{d}{i}_{o_{\alpha \beta }}/\mathrm{d}t<0$.

(b) If $S_{i{o}_{\alpha \beta }}\left(e_{i{o}_{\alpha \beta }},t\right)>0$, then $i_{o_{\alpha \beta }}<i_{o_{\mathit{\alpha \beta ref}}}$ and the output current must increase $i_{o_{\alpha \beta }}\uparrow$. This means that the chosen output voltage $v_{o_{\alpha \beta }}$ phase vector should be high enough to guarantee $\mathrm{d}{i}_{o_{\alpha \beta }}/\mathrm{d}t>0$.

The output voltage phase vectors can be obtained from the output voltage line vectors of Table 35.6, using the well-known relations between three-phase line and phase voltages:

${\mathbf{V}}_{\mathbf{o}}={\left[\begin{array}{ccc}\hfill v_{o_1}\hfill & \hfill v_{o_2}\hfill & \hfill v_{o_3}\hfill \end{array}\right]}^T={\left[\begin{array}{ccc}\hfill \frac{2v_{o_{12}}+v_{o_{23}}}{3}\hfill & \hfill \frac{2v_{o_{23}}+v_{o_{31}}}{3}\hfill & \hfill \frac{2v_{o_{31}}+v_{o_{12}}}{3}\hfill \end{array}\right]}^T$

As expected, the resultant output voltage phase vectors will have an amplitude $\sqrt{3}$ lower than the voltage line vectors of Table 35.6 and a lagging phase of π/6. As an example, from Fig. 35.65B, this will result in a $-\pi /6$ rotation of all the voltage vectors represented.

Therefore, if two three-level comparators (C_αβ∈{−1,0,1}) are used to quantize the deviations of Eq. (35.180) from zero and if the output current error is quantized as C_α=1 and C_β=1, at sector V_i1 (Fig. 35.65), the vectors +9 or −7 might be used to control the output current. The final choice would depend on the input current error.

Input power factor control

Assuming that the source is a balanced sinusoidal three-phase voltage supply with frequency ω_i, the switched state-space model equations of the converter input filter are obtained in 123 coordinates:

$\left\{\begin{array}{l}\frac{\mathrm{d}{i}_{l_1}}{\mathrm{d}t}=\frac{1}{3l}{v}_{c23}+\frac{2}{3l}{v}_{c31}+\frac{1}{l}{v}_{i{g}_1}\\ \frac{\mathrm{d}{i}_{l_2}}{\mathrm{d}t}=-\frac{2}{3l}{v}_{c23}-\frac{1}{3l}{v}_{c31}+\frac{1}{l}{v}_{i{g}_2}\\ \frac{\mathrm{d}{v}_{c23}}{\mathrm{d}t}=\frac{1}{3C}{i}_{l_1}+\frac{2}{3C}{i}_{l_2}-\frac{1}{3Cr}{v}_{c23}+\frac{1}{3Cr}{v}_{i{g}_1}+\frac{2}{3Cr}{v}_{i{g}_2}\\ -\frac{1}{3C}\left(S_{11}-S_{31}+2S_{12}-2S_{32}\right)i_{o_1}\\ -\frac{1}{3C}\left(S_{21}-S_{31}+2S_{22}-2S_{32}\right)i_{o_2}\\ \frac{\mathrm{d}{v}_{c31}}{\mathrm{d}t}=-\frac{2}{3C}{i}_{l_1}-\frac{1}{3C}{i}_{l_2}-\frac{1}{3Cr}{v}_{c31}-\frac{2}{3Cr}{v}_{i{g}_1}-\frac{1}{3Cr}{v}_{i{g}_2}\\ +\frac{1}{3C}\left(2S_{11}-2S_{31}+S_{12}-S_{32}\right)i_{o_1}\\ +\frac{1}{3C}\left(2S_{21}-2S_{31}+S_{22}-S_{32}\right)i_{o_2}\end{array}\right.$

To control the input power factor, a reference frame synchronous with input voltage v_ig1 may be used applying the Blondel-Park transformation to the matrix converter switched state-space model (Eq. (35.183)), where ${\rho }_{i_{dd}}$, ${\rho }_{i_{dq}}$, ${\rho }_{i_{qd}}$, and ${\rho }_{i_{qq}}$ are functions of the on/off states of the nine S_kj switches:

$\left\{\begin{array}{l}\frac{\mathrm{d}{i}_{l_d}}{\mathrm{d}t}={\omega }_i{i}_{lq}-\frac{1}{2l}{v}_{c_d}-\frac{1}{2\sqrt{3}l}{v}_{c_q}+\frac{1}{l}{v}_{i{g}_d}\\ \frac{\mathrm{d}{i}_{l_q}}{\mathrm{d}t}=-{\omega }_i{i}_{l_d}+\frac{1}{2\sqrt{3}l}{v}_{c_d}-\frac{1}{2l}{v}_{c_q}+\frac{1}{l}{v}_{i{g}_q}\\ \frac{\mathrm{d}{v}_{c_d}}{\mathrm{d}t}=\frac{1}{2C}{i}_{l_d}-\frac{1}{2\sqrt{3}C}{i}_{l_q}-\frac{1}{3Cr}{v}_{c_d}+{\omega }_i{v}_{c_q}+\frac{-{\rho }_{i_{dd}}+\left({\rho }_{i_{qd}}/\sqrt{3}\right)}{2C}{i}_{o_d}\\ +\frac{-{\rho }_{i_{dq}}+\left({\rho }_{i_{qq}}/\sqrt{3}\right)}{2C}{i}_{o_q}+\frac{1}{2Cr}{v}_{i{g}_d}-\frac{1}{2\sqrt{3}Cr}{v}_{i{g}_q}\\ \frac{\mathrm{d}{v}_{c_q}}{\mathrm{d}t}=\frac{1}{2\sqrt{3}C}{i}_{l_d}+\frac{1}{2C}{i}_{l_q}-{\omega }_i{v}_{c_d}-\frac{1}{3Cr}{v}_{c_q}+\frac{-\left({\rho }_{i_{dd}}/\sqrt{3}\right)-{\rho }_{i_{qd}}}{2C}{i}_{o_d}\\ +\frac{-\left({\rho }_{i_{dq}}/\sqrt{3}\right)-{\rho }_{i_{qq}}}{2C}{i}_{o_q}+\frac{1}{2\sqrt{3}Cr}{v}_{i{g}_d}+\frac{1}{2Cr}{v}_{i{g}_q}\end{array}\right.$

As a consequence, neglecting ripples, all the input variables become time invariant, allowing a better understanding of the sliding-mode controller design, and the choice of the most adequate state-space vector. Using this state-space model, the input i_igd and i_igq currents are

$\left\{\begin{array}{l}{i}_{i{g}_d}=i_{l_d}+\frac{l}{r}\left(\frac{d{i}_{l_d}}{dt}-{\omega }_i{i}_{lq}\right)\\ i_{i{g}_q}=i_{l_q}+\frac{l}{r}\left(\frac{d{i}_{l_q}}{dt}+{\omega }_i{i}_{l_d}\right)\end{array}\right.\iff \left\{\begin{array}{l}{i}_{i{g}_d}=i_{l_d}-\frac{1}{2r}{v}_{c_d}-\frac{1}{2\sqrt{3}r}{v}_{c_q}+\frac{1}{r}{v}_{i{g}_d}\\ i_{i{g}_q}=i_{l_q}+\frac{1}{2\sqrt{3}r}{v}_{c_d}-\frac{1}{2r}{v}_{c_q}+\frac{1}{r}{v}_{i{g}_q}\end{array}\right.$

The input power factor controller should consider the input-output power constraint (Eq. (35.185)) (the converter losses and ripples are neglected), obtained as a function of the input and output voltages and currents (the input voltage v_igq is equal to zero in the chosen dq rotating frame). The choice of one output voltage vector automatically defines the instantaneous value of the input i_igd(t) current:

$v_{i{g}_d}{i}_{i{g}_d}\approx \frac{1}{3}\left(\frac{\sqrt{3}}{2}{v}_{o{l}_d}+\frac{1}{2}{v}_{o{l}_q}\right)i_{o_d}+\frac{1}{3}\left(-\frac{1}{2}{v}_{o{l}_d}+\frac{\sqrt{3}}{2}{v}_{o{l}_q}\right)i_{o_q}$

Therefore, only the sliding surface associated to the i_igq current is needed, expressed as a function of the system state variables and based on the state-space model determined in Eq. (35.183):

$\begin{array}{l}\frac{\mathrm{d}{i}_{i{g}_q}}{\mathrm{d}t}=-{\omega }_i{i}_{l_d}-\frac{1}{3Cr}{i}_{l_q}+\left(-\frac{1}{6\sqrt{3}C{r}^2}+\frac{{\omega }_i}{2r}+\frac{1}{2\sqrt{3}l}\right)v_{c_d}\\ +\left(\frac{{\omega }_i}{2\sqrt{3}r}+\frac{1}{6C{r}^2}-\frac{1}{2l}\right)v_{c_q}+\frac{1}{3Cr}\left({\rho }_{i_{qd}}{i}_{o_d}+{\rho }_{i_{qq}}{i}_{o_q}\right)\\ +\left(-\frac{1}{3C{r}^2}+\frac{1}{l}\right)v_{i{g}_q}+\frac{1}{r}\frac{\mathrm{d}{v}_{i{g}_q}}{dt}\end{array}$

As the derivative of the input i_igq(t) current depends directly on the control variables ${\rho }_{i_{qd}}$ and ${\rho }_{i_{qq}}$, the sliding function $S_{i_q}\left(e_{i_q},t\right)$ will depend only on the input current error $e_{i_q}=i_{i{g}_{\mathit{qref}}}-i_{i{g}_q}$:

$S_{i_q}\left(e_{i_q},t\right)=k_{i_q}\left(i_{i{g}_{\mathit{qref}}}-i_{i{g}_q}\right)$

As the sliding-mode stability is guaranteed if $S_{i_q}\left(e_{i_q},t\right){\stackrel{•}{S}}_{i_q}\left(e_{i_q},t\right)<0$, the criterion to choose the state-space vectors is [23]

$\begin{array}{l}{S}_{i_q}\left(e_{i_q},t\right)>0\Rightarrow {\stackrel{.}{S}}_{i_q}\left(e_{i_q},t\right)<0\Rightarrow \frac{\mathrm{d}{i}_{i{g}_q}}{\mathrm{d}t}>\frac{\mathrm{d}{i}_{i{g}_{\mathit{qref}}}}{\mathrm{d}t}\Rightarrow i_{i{g}_q}\uparrow \\ S_{i_q}\left(e_{i_q},t\right)<0\Rightarrow {\stackrel{.}{S}}_{i_q}\left(e_{i_q},t\right)>0\Rightarrow \frac{\mathrm{d}{i}_{i{g}_q}}{\mathrm{d}t}<\frac{\mathrm{d}{i}_{i{g}_{\mathit{qref}}}}{\mathrm{d}t}\Rightarrow i_{i{g}_q}\downarrow \end{array}$

Also, to choose the adequate input current vector, it is necessary (a) to know the location of the output currents, as the input currents depend on the output currents location (Table 35.6), and (b) to know the dq frame location. As in the chosen frame (synchronous with the v_ig1 input voltage), the dq-axis location depends on the v_ig1 input voltage location; the sign of the input current vector i_igq component can be determined knowing the location of the input voltages and the location of the output currents (Fig. 35.66).

Considering the previous example, at sector V_i1 (Fig. 35.65), for an error of C_α=1 and C_β=1, vector −3, +1, or 1 g might be used to control the output voltage. When compared, at sector I_o1 (Fig. 35.66B), these three vectors have positive i_d components and, as a result, will have a similar effect on the input i_igd current. However, they have a different effect on the i_igq current: Vector −3 has a positive i_iq component, vector +1 has a negative i_iq component, and vector 1 g has a nearly zero i_iq component. As a result, if the output voltage errors are C_α=1 and C_β=1, at sectors V_i1 and I_o1, vector −3 should be chosen if the input current error is quantized as C_iq=1 (Fig. 35.66B), vector +1 should be chosen if the input current error is quantized as C_iq=−1, and if the input current error is C_iq=0, vector 1 g or −3 might be used.

When the output voltage errors are quantized as zero C_αβ=0, the null vectors of group III should be used only if the input current error is C_iq=0. Otherwise (being $C_{i_q}\ne 0$), the lowest amplitude voltage vectors ({+2, −8, +5, −2, +8, −5} at sector V_i1 of Fig. 35.65B) that were not used to control the output voltages might be chosen to control the input i_igq current as these vectors may have a strong influence on the input i_igq current component (Fig. 35.66B).

To choose one of these six vectors, only the vectors located as near as possible to the output voltages sector (Fig. 35.67) are chosen (to minimize the output voltage ripple), and a five-level comparator is enough. As a result, there will be 9×5=45 error combinations to select 27 space vectors. Therefore, the same vector may have to be used for more than one error combination.

With this reasoning, it is possible to obtain Table 35.7 for sector V_i1, I_o1, and V_o1 and generalize it for all the other sectors.

The experimental results shown in (Fig. 35.68) were obtained with a low-power prototype (1 kW), with two three-level comparators and one five-level comparator. The transistors IGBT were switched at frequencies near 10 kHz.

The results show the response to a step on the output voltage reference (Fig. 35.68A) and on the input reference current (Fig. 35.68B), for a three-phase output load (R=7 Ω and L=15 mH), with k_αβ=100 and k_iq=2. It can be seen that the matrix converter may operate with a near unity input power factor (Fig. 35.68A – f_o=20 Hz) or with lead/lag power factor (Fig. 35.68B), guaranteeing very low ripple on the output currents, a good tracking capability and fast transient response times.

Also, some simulation results were obtained for the output current-controlled matrix converter. The results (Fig. 35.69A) show the response to a step on the output current reference at t=0.22 s, a step of the output current frequency at t=0.28 s, and a step at the input $i_{i{g}_q}$ current at t=0.34 s, for a three-phase output load (R=3 Ω and L=30 mH), with k_ioαβ=10 and k_iq=2. Again, these results show that matrix converter may operate with lead/lag power factor, guaranteeing very low ripple on the output currents, a good tracking capability and fast transient response times for both input and output currents.