14.2.1 Phase-Controlled Single-Phase AC Voltage Controller

For a full-wave symmetrical phase control, the SCRs T₁ and T₂ in Fig. 14.1A are gated at α and π+α, respectively, from the zero crossing of the input voltage; by varying α, the power flow to the load is controlled through voltage control in alternate half cycles. As long as one SCR is carrying current, the other SCR remains reverse-biased by the voltage drop across the conducting SCR. The principle of operation in each half cycle is similar to that of the controlled half-wave rectifier, and one can use the same approach for analysis of the circuit.

Operation with R-load. Fig. 14.2 shows the typical voltage and current waveforms for the single-phase bidirectional phase-controlled ac voltage controller of Fig. 14.1A with a resistive load. The output voltage and current waveforms have half-wave symmetry and so no dc component.

If $v_{\mathrm{s}}=\sqrt{2}{V}_{\mathrm{s}}sin\omega t$ is the source voltage, the rms output voltage with T₁ triggered at α can be found from the half-wave symmetry as

$V_{\mathrm{o}}={\left[\frac{1}{\pi }{\displaystyle \underset{\alpha }{\overset{\pi }{\int }}}2V_{\mathrm{s}}^2{sin}^2\omega td\left(\omega t\right)\right]}^{\frac{1}{2}}=V_{\mathrm{s}}{\left[1-\frac{\alpha }{\pi }+\frac{sin2\alpha }{2\pi }\right]}^{\frac{1}{2}}$

Note that V_o can be varied from V_s to 0 by varying α from 0 to π:

$\mathrm{The}\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{value}\mathrm{of}\mathrm{load}\mathrm{current},I_{\mathrm{o}}=\frac{V_{\mathrm{o}}}{R}$

$\mathrm{The}\mathrm{input}\mathrm{power}\mathrm{factor}=\frac{P_{\mathrm{o}}}{VA}=\frac{V_{\mathrm{o}}}{V_{\mathrm{s}}}={\left[1-\frac{\alpha }{\pi }+\frac{sin2\alpha }{2\pi }\right]}^{\frac{1}{2}}$

$\mathrm{The}\mathrm{average}\mathrm{S}\mathrm{C}\mathrm{R}\mathrm{current},I_{\mathrm{A},\mathrm{S}\mathrm{C}\mathrm{R}}=\frac{1}{2\pi R}{\displaystyle \underset{\alpha }{\overset{\pi }{\int }}}\sqrt{2}{V}_{\mathrm{s}}sin\omega t\mathrm{d}\left(\omega t\right)$

Since each SCR carries half the line current, the rms current in each SCR is

$I_{\mathrm{o},\mathrm{S}\mathrm{C}\mathrm{R}}=\frac{I_{\mathrm{o}}}{\sqrt{2}}$

Operation with RL Load. Fig. 14.3 shows the voltage and current waveforms for the controller in Fig. 14.1A with RL load. Due to the inductance, the current carried by the SCR T₁ may not fall to zero at $\omega t=\pi$ when the input voltage goes negative and may continue till $\omega t=\beta$, the extinction angle, as shown. The conduction angle,

$\theta =\beta -\alpha$

of the SCR depends on the firing delay angle α and the load impedance angle ϕ.

The expression for the load current I_o(ωt), when conducting from α to β, can be derived in the same way as that used for a phase-controlled rectifier in a discontinuous mode by solving the relevant Kirchhoff's voltage equation:

$i_{\mathrm{o}}\left(\omega t\right)=\frac{\sqrt{2}V}{Z}\left[sin\left(\omega t-\varphi \right)\right.\left.-sin\left(\alpha -\varphi \right)e^{\left(\alpha -\omega t\right)/tan\varphi }\right],\alpha <\omega t<\beta$

where $Z={\left(R^2+{\omega }^2{L}^2\right)}^{\frac{1}{2}}=\mathrm{load}\mathrm{impedance}$ and $\varphi =\mathrm{load}\mathrm{impedance}\mathrm{angle}={\displaystyle {tan}^{-1}}\left(\omega L/R\right)$.

The angle β, when the current i_o falls to zero, can be determined from the following transcendental equation resulted by using $i_{\mathrm{o}}\left(\omega t=\beta \right)=0$ in Eq. (14.7):

$sin\left(\beta -\varphi \right)=sin\left(\alpha -\varphi \right)-sin\left(\alpha -\varphi \right)e^{\left(\alpha -\beta \right)/tan\varphi }$

From Eqs. (14.6) and (14.8), one can obtain a relationship between θ and α for a given value of ϕ, as shown in Fig. 14.4, which shows that as α is increased, the conduction angle θ decreases, and thus, the rms value of the current decreases.

The rms output voltage

$\begin{array}{c}{V}_{\mathrm{o}}={\left[\frac{1}{\pi }{\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}2V_{\mathrm{s}}^2{sin}^2\omega td\left(\omega t\right)\right]}^{\frac{1}{2}}\\ =\frac{V_{\mathrm{s}}}{\pi }{\left[\beta -\alpha +\frac{sin2\alpha }{2}-\frac{sin2\beta }{2}\right]}^{\frac{1}{2}}\end{array}$

V_o can be evaluated for two possible extreme values of $\varphi =0$ when $\beta =\pi$, and $\varphi =\pi /2$ when $\beta =2\pi -\alpha$, and the envelope of the voltage control characteristics for this controller is shown in Fig. 14.5.

The rms SCR current can be obtained from Eq. (14.7) as follows:

$I_{\mathrm{o},\mathrm{S}\mathrm{C}\mathrm{R}}={\left[\frac{1}{2\pi }{\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}{i}_{\mathrm{o}}^{2}d\left(\omega t\right)\right]}^{\frac{1}{2}}$

$\mathrm{The}\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{load}\mathrm{current},I_o=\sqrt{2}{I}_{\mathrm{o},\mathrm{S}\mathrm{C}\mathrm{R}}$

$\mathrm{The}\mathrm{average}\mathrm{value}\mathrm{of}\mathrm{S}\mathrm{C}\mathrm{R}\mathrm{current},I_{\mathrm{A},\mathrm{S}\mathrm{C}\mathrm{R}}=\frac{1}{2\pi }{\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}{i}_{\mathrm{o}}d\left(\omega t\right)$

Gating Signal Requirements. For the inverse-parallel SCRs as shown in Fig. 14.1A, the gating signals of SCRs must be isolated from one another because there is no common cathode. For R-load, each SCR stops conducting at the end of each half cycle and under this condition, single short pulses may be used for gating as shown in Fig. 14.2. With RL load, however, this single short pulse gating is not suitable as shown in Fig. 14.6. When SCR T₂ is triggered at $\omega t=\pi +\alpha$, SCR T₁ is still conducting due to the load inductance. By the time the SCR T₁ stops conducting at β, the gate pulse for SCR T₂ has already ceased and T₂ will fail to turn on resulting the converter to operate as a single-phase rectifier with conduction of T₁ only. This necessitates the application of a sustained gate pulse in the form of either a continuous signal for the half-cycle period or better a train of pulses (carrier frequency gating) as the former results in increased dissipation in SCR gate circuit, and a large isolating pulse transformer is required.

Operation with $\alpha <\varphi$. If $\alpha =\varphi$, then from Eq. (14.8),

$sin\left(\beta -\varphi \right)=sin\left(\beta -\alpha \right)=0$

and

$\beta -\alpha =\theta =\pi$

As the conduction angle θ cannot exceed π and the load current must pass through zero, the control range of the firing angle is $\varphi \le \alpha \le \pi$. With narrow gating pulses and $\alpha <\varphi$, only one SCR will conduct resulting in a rectifier action as shown. Even with a train of pulses, if $\alpha <\varphi$, the changes in the firing angle will not change the output voltage and current but both the SCRs will conduct for the period π with T₁ becoming on at $\omega t=\pi$ and T₂ at $\omega t+\pi$.

The duration of this dead zone ($\alpha =0$ to ϕ), which varies with the load impedance angle ϕ, is not a desirable feature in closed-loop control schemes. An alternative approach to the phase control with respect to the input voltage zero crossing has been visualized in which the firing angle is defined with respect to the instant when it is the load current, not the input voltage, that reaches zero, this angle being called the hold-off angle (γ) or the control angle (as marked in Fig. 14.3). This method needs sensing the load current, which may otherwise be required in a closed-loop controller for monitoring or control purposes.

Power Factor and Harmonics. As in the case of phase-controlled rectifiers, the important limitations of the phase-controlled ac voltage controllers are the poor power factor and the introduction of harmonics in the source currents. As seen from Eq. (14.3), the input power factor depends on α, and as α increases, the power factor decreases.

The harmonic distortion increases and the quality of the input current decreases with increase of firing angle. The variations of low-order harmonics with the firing angle as computed by Fourier analysis of the voltage waveform of Fig. 14.2 (with R-load) are shown in Fig. 14.7. Only odd harmonics exist in the input current because of half-wave symmetry.

14.2.2 Single-Phase AC-AC Voltage Controller with On/Off Control

Integral Cycle Control. As an alternative to the phase control, the method of integral cycle control or burst firing is used for systems with large time constants, for example, heating loads. Here, the switch is turned on for a time t_n with n integral cycles and turned off for a time t_m with m integral cycles (Fig. 14.8). As the SCRs or triacs used here are turned on at the zero crossing of the input voltage and turn off occurs at zero current, supply harmonics and radio-frequency interference are very low.

However, subharmonic frequency components may be generated, which are undesirable because they may set up subharmonic resonance in the power supply system, cause lamp flicker, and may interfere with the natural frequencies of motor loads causing shaft oscillations.

For sinusoidal input voltage, $v=\sqrt{2}{V}_{\mathrm{s}}sin\omega t$, the rms output voltage

$V_{\mathrm{o}}=V_{\mathrm{s}}\sqrt{k}\mathrm{where}k=n/\left(n+m\right)=\mathrm{duty}\mathrm{cycle}\mathrm{and}{V}_{\mathrm{s}}=\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{phase}\mathrm{voltage}$

$\mathrm{The}\mathrm{power}\mathrm{factor}=\sqrt{k}$

which is poorer for lower values of the duty cycle k.

PWM AC Chopper. As in the case of controlled rectifier, the performance of ac voltage controllers can be improved in terms of harmonics, quality of output current, and input power factor by PWM control in PWM ac choppers; the circuit configuration of one such single-phase unit is shown in Fig. 14.9. Here, fully controlled switches S₁ and S₂ connected in antiparallel are turned on and off many times during the positive and negative half cycles of the input voltage, respectively. S₁′ and S₂′ provide the freewheeling paths for the load current when S₁ and S₂ are off. An input capacitor filter may be provided to attenuate the high switching frequency currents drawn from the supply and also to improve the input power factor. Fig. 14.10 shows the typical output voltage and load current waveform for a single-phase PWM ac chopper. It can be shown that the control characteristics of an ac chopper depend on the modulation index M, which theoretically varies from 0 to 1.

Three-phase PWM choppers consist of three single-phase choppers connected in either delta or four-wire (star) configuration.

14.3 Three-Phase AC-AC Voltage Controllers

14.3.1 Phase-Controlled Three-Phase AC Voltage Controllers

Various Configurations. Several possible circuit configurations for three-phase phase-controlled ac regulators with star- or delta-connected loads are shown in Fig. 14.11A–H.

The configurations in (A) and (B) can be realized by three single-phase ac regulators operating independently of each other, and they are easy to analyze. In (A), the SCRs are to be rated to carry line currents and withstand phase voltages, whereas in (B) they should be capable to carry phase currents and withstand the line voltage. In (B), the line currents are free from triplen harmonics while these are present in the closed delta. The power factor in (B) is slightly higher. The firing angle control range for both these circuits is 0–180 degrees for R-load.

The circuits in (C) and (D) are three-phase three-wire circuits and are complicated to analyze. In both these circuits, at least two SCRs, one in each phase, must be gated simultaneously to get the controller started by establishing a current path between the supply lines. This necessitates two firing pulses spaced at 60 degrees apart per cycle for firing each SCR. The operation modes are defined by the number of SCRs conducting in these modes. The firing control range is 0–150 degrees. The triplen harmonics are absent in both these configurations.

Another configuration is shown in (E) when the controllers are connected in delta and the load is connected between the supply and the converter. Here, current can flow between two lines even if one SCR is conducting so each SCR requires one firing pulse per cycle. The voltage and current ratings of SCRs are nearly the same as that of the circuit (B). It is also possible to reduce the number of devices to three SCRs in delta as shown in (F), connecting one source terminal directly to one load circuit terminal. Each SCR is provided with gate pulses in each cycle spaced at 120 degrees apart. In both (E) and (F), each end of each phase must be accessible. The number of devices in (F) is less, but their current ratings must be higher.

As in the case of single-phase phase-controlled voltage regulator, the total regulator cost can be reduced by replacing six SCRs by three SCRs and three diodes, resulting in three-phase half-wave controlled unidirectional ac regulators as shown in (G) and (H) for star- and delta-connected loads. The main drawback of these circuits is the large harmonic content in the output voltage—particularly, the second harmonic because of the asymmetry. However, the dc components are absent in the line. The maximum firing angle in the half-wave controlled regulator is 210 degrees.

14.3.2 Fully Controlled Three-Phase Three-Wire AC Voltage Controller

Star-connected Load with Isolated Neutral. The analysis of operation of the full-wave controller with isolated neutral as shown in Fig. 14.11C is, as mentioned, quite complicated in comparison with that of a single-phase controller, particularly for an RL or motor load. As a simple example, the operation of this controller with a simple star-connected R-load is considered here. The six SCRs are turned on in the sequence 1-2-3-4-5-6 at 60 degrees intervals, and the gate signals are sustained throughout the possible conduction angle.

The output phase voltage waveforms for α=30, 75, and 120 degrees for a balanced three-phase R-load are shown in Fig. 14.12. At any interval, either three SCRs or two SCRs or no SCRs may be on, and the instantaneous output voltages to the load are either a line-to-neutral voltage (three SCRs on) or one-half of the line-to-line voltage (two SCRs on) or zero (no SCR on).

Depending on the firing angle α, there may be three operating modes:

Mode I (also known as Mode 2/3). 0 degrees≤α≤60 degrees; There are periods when three SCRs are conducting, one in each phase for either direction, and there are periods when just two SCRs conduct.

For example, with α=30 degrees in Fig. 14.12A, assume that at $\omega t=0$, SCRs T₅ and T₆ are conducting, and the current through the R-load in a-phase is zero making $v_{\mathrm{an}}=0$. At $\omega t=30\mathrm{degrees}$, T₁ receives a gate pulse and starts conducting; T₅ and T₆ remain on, and $v_{\mathrm{an}}=v_{\mathrm{AN}}$. The current in T₅ reaches zero at 60 degrees, turning T₅ off. With T₁ and T₆ staying on, $v_{\mathrm{an}}=\frac{1}{2}{v}_{\mathrm{AB}}$. At 90 degrees, T₂ is turned on, the three SCRs T₁, T₂, and T₆ are then conducting, and $v_{\mathrm{an}}=v_{\mathrm{AN}}$. At 120 degrees, T₆ turns off, leaving T₁ and T₂ on, so $v_{\mathrm{an}}=\frac{1}{2}{v}_{\mathrm{AC}}$. Thus, with the progress of firing in sequence till α=60 degrees, the number of SCRs conducting at a particular instant alternates between two and three.

Mode II (also known as Mode 2/2). 60 degrees≤α≤90 degrees; Two SCRs, one in each phase, always conduct.

For α=75 degrees as shown in Fig. 14.12B, just prior to α=75 degrees, SCRs T₅ and T₆ were conducting, and $v_{\mathrm{an}}=0$. At 75 degrees, T₁ is turned on, and T₆ continues to conduct, whereas T₅ turns off as v_CN is negative. $v_{\mathrm{an}}=\frac{1}{2}{v}_{\mathrm{AB}}$. When T₂ is turned on at 135 degrees, T₆ is turned off, and $v_{\mathrm{an}}=\frac{1}{2}{v}_{\mathrm{AC}}$. The next SCR to turn on is T₃, which turns off T₁, and $v_{\mathrm{an}}=0$. One SCR is always turned off when another is turned on in this range of α, and the output voltage is either one-half line-to-line voltage or zero.

Mode III (also known as Mode 0/2). 90 degrees≤α≤150 degrees, when none or two SCRs conduct.

For α=120 degrees, Fig. 14.12C, earlier, no SCRs were on, and $v_{\mathrm{an}}=0$. At α=120 degrees, SCR T₁ is given a gate signal, whereas T₆ has a gate signal already applied. Since v_AB is positive, T₁ and T₆ are forward-biased, and they begin to conduct, and $v_{\mathrm{an}}=\frac{1}{2}{v}_{\mathrm{AB}}$. Both T₁ and T₆ turn off when v_AB becomes negative. When a gate signal is given to T₂, it turns on, and T₁ turns on again.

For α>150 degrees, there is no period when two SCRs are conducting and the output voltage is zero at α=150 degrees. Thus, the range of the firing angle control is 0 degrees≤α≤150 degrees.

For star-connected R-load, assuming the instantaneous phase voltages as

$\begin{array}{l}{v}_{\mathrm{AN}}=\sqrt{2}{V}_{\mathrm{s}}sin\omega t\\ v_{\mathrm{B}\mathrm{N}}=\sqrt{2}{V}_{\mathrm{s}}sin\omega t-120\mathrm{degrees}\\ v_{\mathrm{C}\mathrm{N}}=\sqrt{2}{V}_{\mathrm{s}}sin\omega t-240\mathrm{degrees}\end{array}$

the expressions for the rms output phase voltage V_o can be derived for the three modes as follows:

$0\mathrm{degrees}\le \alpha \le 60\mathrm{degrees}{V}_{\mathrm{o}}=V_{\mathrm{s}}{\left[1-\frac{3\alpha }{2\pi }+\frac{3}{4\pi }sin2\alpha \right]}^{\frac{1}{2}}$

$60\mathrm{degrees}\le \alpha \le 90\mathrm{degrees}{V}_{\mathrm{o}}=V_{\mathrm{s}}{\left[\frac{1}{2}+\frac{3}{4\pi }sin2\alpha +sin\left(2\alpha +60\mathrm{degrees}\right)\right]}^{\frac{1}{2}}$

$90\mathrm{degrees}\le \alpha \le 150\mathrm{degrees}{V}_{\mathrm{o}}=V_{\mathrm{s}}{\left[\frac{5}{4}-\frac{3\alpha }{2\pi }+\frac{3}{4\pi }sin\left(2\alpha +60\mathrm{degrees}\right)\right]}^{\frac{1}{2}}$

For star-connected pure L-load, the effective control starts at α>90 degrees, and the expressions for two ranges of α are as follows:

$90\mathrm{degrees}\le \alpha \le 120\mathrm{degrees}{V}_{\mathrm{o}}=V_{\mathrm{s}}{\left[\frac{5}{2}-\frac{3\alpha }{\pi }+\frac{3}{2\pi }sin2\alpha \right]}^{\frac{1}{2}}$

$120\mathrm{degrees}\le \alpha \le 150\mathrm{degrees}{V}_{\mathrm{o}}=V_{\mathrm{s}}{\left[\frac{5}{2}-\frac{3\alpha }{\pi }+\frac{3}{2\pi }sin\left(2\alpha +60\mathrm{degrees}\right)\right]}^{\frac{1}{2}}$

The control characteristics for these two limiting cases (ϕ=0 for R-load and ϕ=90 degrees for L-load) are shown in Fig. 14.13. Here also, like the single-phase case, the dead zone may be avoided by controlling the voltage with respect to the control angle or hold-off angle (γ) from the zero crossing of current in place of the firing angle α.

RL Load. The analysis of the three-phase voltage controller with star-connected RL load with isolated neutral is quite complicated, since the SCRs do not cease to conduct at voltage zero, and the extinction angle β is to be known by solving the transcendental equation for the case. In this case, the mode II operation disappears [1], and the operation shift from mode I to mode III depends on the so-called critical angle α_crit [2,3], which can be evaluated from a numerical solution of the relevant transcendental equations. Computer simulation by either PSPICE program [4,5] or switching-variable approach coupled with an iterative procedure [6] is a practical means of obtaining the output voltage waveform in this case. Fig. 14.14 shows typical simulation results using the later approach [6] for a three-phase voltage controller-fed RL load for α=60, 90, and 105 degrees, which agree with the corresponding practical oscillograms given in [7].

Delta-Connected R-Load. The configuration is shown in Fig. 14.11B. The voltage across an R-load is the corresponding line-to-line voltage when one SCR in that phase is on. Fig. 14.15 shows the line and phase currents for α=130 and 90 degrees with an R-load. The firing angle α is measured from the zero crossing of the line-to-line voltage, and the SCRs are turned on in the sequence as they are numbered. As in the single-phase case, the range of firing angle is $0\le \alpha \le 180$ degrees. The line currents can be obtained from the phase currents as follows:

$\begin{array}{l}{i}_{\mathrm{a}}=i_{\mathrm{ab}}-i_{\mathrm{c}\mathrm{a}}\hfill \\ i_{\mathrm{b}}=i_{\mathrm{b}\mathrm{c}}-i_{\mathrm{ab}}\hfill \\ i_{\mathrm{c}}=i_{\mathrm{c}\mathrm{a}}-i_{\mathrm{b}\mathrm{c}}\hfill \end{array}$

The line currents depend on the firing angle and may be discontinuous as shown. Due to the delta connection, the triplen harmonic currents flow around the closed delta and do not appear in the line. The rms value of the line current varies between the range

$\sqrt{2}{I}_{\mathrm{\Delta }}\le I_{\mathrm{L},\mathrm{r}\mathrm{m}\mathrm{s}}\le \sqrt{3}{I}_{\mathrm{\Delta }.\mathrm{r}\mathrm{m}\mathrm{s}}$

as the conduction angle varies from very small (large α) to 180 degrees ($\alpha =0$).

14.4.1 Single-Phase to Single-Phase Cycloconverter

Though rarely used, the operation of a single-phase to single-phase cycloconverter is useful to demonstrate the basic principle involved. Fig. 14.16A shows the power circuit of a single-phase bridge type of cycloconverter, which has the same arrangement as that of a single-phase dual converter. The firing angles of the individual two-pulse two-quadrant bridge converters are continuously modulated here, so that each ideally produces the same fundamental ac voltage at its output terminals as marked in the simplified equivalent circuit in Fig. 14.16B. Because of the unidirectional current-carrying property of the individual converters, it is inherent that the positive half cycle of the current is carried by the P-converter and the negative half cycle of the current by the N-converter regardless of the phase of the current with respect to the voltage. This means that for a reactive load, each converter operates in both rectifying and inverting region during the period of the associated half cycle of the low-frequency output current.

Operation with R-Load. Fig. 14.17 shows the input and output voltage waveforms with a pure R-load for a $50-16\frac{2}{3}-\mathrm{Hz}$ cycloconverter. The P- and N-converters operate for alternate T_o/2 periods. The output frequency (1/T_o) can be varied by varying T_o and the voltage magnitude by varying the firing angle α of the SCRs. As shown in the figure, three cycles of the ac input wave are combined to produce one cycle of the output frequency to reduce the supply frequency to one-third across the load.

If α_P is the firing angle of the P-converter, the firing angle of the N-converter α_N is $\pi -{\alpha }_{\mathrm{P}}$, and the average voltage of the P-converter is equal and opposite to that of the N-converter.

The inspection of Fig. 14.17 shows that the waveform with α remaining fixed in each half cycle generates a square wave having a large low-order harmonic content. A near approximation to sine wave can be synthesized by a phase modulation of the firing angles as shown in Fig. 14.18 for a 50–10 Hz cycloconverter. The harmonics in the load voltage waveform are less compared with earlier waveform. The supply current, however, contains a subharmonic at the output frequency for this case as shown.

Operation with RL Load. The cycloconverter is capable of supplying loads of any power factor. Fig. 14.19 shows the idealized output voltage and current waveforms for a lagging power factor load where both the converters are operating as rectifier and inverter at the intervals marked. The load current lags the output voltage, and the load current direction determines which converter is conducting. Each converter continues to conduct after its output voltage changes polarity, and during this period, the converter acts as an inverter, and the power is returned to the ac source. Inverter operation continues till the other converter starts to conduct. By controlling the frequency of oscillation and the depth of modulation of the firing angles of the converters (as shown later), it is possible to control the frequency and the amplitude of the output voltage.

The load current with RL load may be continuous or discontinuous depending on the load phase angle, ϕ. At light load inductance or for $\varphi \le \alpha \le \pi$, there may be discontinuous load current with short zero-voltage periods. The current wave may contain even harmonics and subharmonic components. Further, as in the case of dual converter, although the mean output voltage of the two converters is equal and opposite, the instantaneous values may be unequal and a circulating current can flow within the converters. This circulating current can be limited by having a center-tapped reactor connected between the converters or can be completely eliminated by logical control similar to the dual converter case when the gate pulses to the converter remaining idle are suppressed, when the other converter is active. In practice, a zero-current interval of short duration is needed, in addition, between the operation of the P- and N-converters to ensure that the supply lines of the two converters are not short-circuited. With circulating current-free operation, the control scheme becomes complicated if the load current is discontinuous.

In the case of the circulating current scheme, the converters are kept in virtually continuous conduction over the whole range, and the control circuit is simple. To obtain reasonably good sinusoidal voltage waveform using the line-commutated two-quadrant converters and eliminate the possibility of the short circuit of the supply voltages, the output frequency of the cycloconverter is limited to a much lower value of the supply frequency. The output voltage waveform and the output frequency range can be improved further by using converters of higher pulse numbers.

14.4.3 Cycloconverter Control Scheme

Various possible control schemes, analog and digital, for deriving trigger signals and for controlling the basic cycloconverter have been developed over the years.

Out of the several possible signal combinations, it has been shown [8] that a sinusoidal reference signal ($e_{\mathrm{r}}=E_{\mathrm{r}}sin{\omega }_{\mathrm{o}}t$) at desired output frequency f_o and a cosine modulating signal ($e_{\mathrm{m}}=E_{\mathrm{m}}cos{\omega }_{\mathrm{i}}t$) at input frequency f_i is the best combination possible for comparison to derive the trigger signals for the SCRs (Fig. 14.26 [9]), which produces the output waveform with the lowest total harmonic distortion. The modulating voltages can be obtained as the phase-shifted voltages (B-phase for A-phase SCRs, C-phase voltage for B-phase SCRs, and so on) as explained in Fig. 14.27, where at the intersection point “a”

$\begin{array}{c}{E}_{\mathrm{m}}sin\left({\omega }_{\mathrm{i}}t-120\mathrm{degrees}\right)=-E_{\mathrm{r}}sin\left({\omega }_{\mathrm{o}}t-\varphi \right)\\ \mathrm{or}cos\left({\omega }_{\mathrm{i}}t-30\mathrm{degrees}\right)=\left(E_{\mathrm{r}}/E_{\mathrm{m}}\right)sin\left({\omega }_{\mathrm{o}}t-\varphi \right)\end{array}$

From Fig. 14.27, the firing delay for A-phase SCR, $\alpha =\left({\omega }_{\mathrm{i}}t-30\mathrm{degrees}\right)$

$\mathrm{So},cos\alpha =\left(E_{\mathrm{r}}/E_{\mathrm{m}}\right)sin\left({\omega }_{\mathrm{o}}t-\varphi \right)$

The cycloconverter output voltage for continuous-current operation,

$V_{\mathrm{o}}=V_{\mathrm{do}}cos\alpha =V_{\mathrm{do}}\left(E_{\mathrm{r}}/E_{\mathrm{m}}\right)sin\left({\omega }_{\mathrm{o}}t-\varphi \right)$

in which the equation shows that the amplitude, frequency, and phase of the output voltage can be controlled by controlling corresponding parameters of the reference voltage, thus making the transfer characteristic of the cycloconverter linear. The derivation of the two complimentary voltage waveforms for the P-group or N-group converter “banks” in this way is illustrated in Fig. 14.28. The final cycloconverter output wave shape is composed of alternate half-cycle segments of the complementary P-converter and the N-converter output voltage waveforms that coincide with the positive and negative current half cycles, respectively.

Control Circuit Block Diagram. Fig. 14.29 [10] shows a simplified block diagram of the control circuit for a circulating current-free cycloconverter implemented with ICs in the early seventies in the Power Electronics Laboratory at IIT Kharagpur in India. The same circuit is also applicable to a circulating current cycloconverter with the omission of the converter group selection and blanking circuit.

The synchronizing circuit produces the modulating voltages ($e_{\mathrm{a}}=-K{v}_{\mathrm{b}}$, $e_{\mathrm{b}}=-K{v}_{\mathrm{c}}$, $e_{\mathrm{c}}=-K{v}_{\mathrm{a}}$), synchronized with the mains through step-down transformers and proper filter circuits.

The reference source produces variable-magnitude variable-frequency three-phase voltages (e_ra, e_rb, e_rc) for comparison with the modulation voltages. Various ways, analog or digital, have been developed to implement this reference source as in the case of the PWM inverter. In one of the early analog schemes, Fig. 14.30 [10], for a three-pulse cycloconverter, a variable-frequency UJT relaxation oscillator of frequency 6f_d triggers a ring counter to produce a three-phase square-wave output of frequency f_d that is used to modulate a single-phase fixed frequency (f_c) variable amplitude sinusoidal voltage in a three-phase full-wave transistor chopper. The three-phase output contains ($f_{\mathrm{c}}-f_{\mathrm{d}}$), ($f_{\mathrm{c}}+f_{\mathrm{d}}$), ($3f_{\mathrm{d}}+f_{\mathrm{c}}$), etc. frequency components from where the “wanted” frequency component ($f_{\mathrm{c}}-f_{\mathrm{d}}$) is filtered out for each phase using a low-pass filter. For example, with $f_{\mathrm{c}}=500\mathrm{Hz}$ and frequency of the relaxation oscillator varying between 2820 and 3180 Hz, a three-phase 0–30 Hz reference output can be obtained with the facility for phase sequence reversal.

The logic and trigger circuit for each phase involves comparators for comparison of the reference and modulating voltages and inverters acting as buffer stages. The outputs of the comparators are used to clock the flip-flops or latches whose outputs in turn feed the SCR gates through AND gates and pulse amplifying and isolation circuits; the second input to the AND gates is from the converter group selection and blanking circuit.

In the converter group selection and blanking circuit, the zero crossing of the current at the end of each half cycle is detected and is used to regulate the control signals to either P-group or N-group converters depending on whether the current goes to zero from negative to positive or positive to negative, respectively. However, in practice, the current being discontinuous passes through multiple zero crossings while changing direction, which may lead to undesirable switching of the converters. So, in addition to the current signal, the reference voltage signal is also used for the group selection, and a threshold band is introduced in the current signal detection to avoid inadvertent switching of the converters. Further, a delay circuit provides a blanking period of appropriate duration between the converter groups switching to avoid line-to-line short circuits [10]. In some schemes, the delays are not introduced when a small circulating current is allowed during crossover instants limited by reactors of limited size, and this scheme operates in the so-called “dual mode”—circulating current and circulating current-free mode for minor and major portions of the output cycle, respectively. A different approach to the converter group selection, based on the closed-loop control of the output voltage where a bias voltage is introduced between the voltage transfer characteristics of the converters to reduce circulating current, is discussed in [8].

Improved Control Schemes. With the development of microprocessors and PC-based systems, digital software control has taken over many tasks in modern cycloconverters, particularly in replacing the low-level reference waveform generation and analog signal comparison units. The reference waveforms can be easily generated in the computer, stored in the EPROMs, and accessed under the control of a stored program and microprocessor clock oscillator. The analog signal voltages can be converted to digital signals by using analog-to-digital converters (ADCs). The waveform comparison can then be made with the comparison features of the microprocessor system. The addition of time delays and intergroup blanking can also be achieved with digital techniques and computer software. A modification of the cosine firing control using communication principles, such as regular sampling in preference to the natural sampling (discussed so far) of the reference waveform, yielding a stepped sine wave before comparison with the cosine wave [11] has been shown to reduce the presence of subharmonics (discussed later) in the circulating current cycloconverter and facilitate microprocessor-based implementation, as in the case of PWM inverter.

For a six-pulse noncirculating current cycloconverter-fed synchronous motor drive with a vector control scheme and a flux observer, a PC-based hybrid control scheme (a combination of analog and digital control) has been reported in [12]. Here, the functions such as comparison, group selection, blanking between the groups and triggering signal generation, and filtering and phase conversion are left to the analog controller and digital controller that take care of more serious tasks such as voltage decoupling for current regulation; flux estimation using observer; speed, flux and field current regulators using PI-controllers; and position and speed calculation leading to an improvement of sampling time and design accuracy.

14.4.4 Cycloconverter Harmonics and Input Current Waveform

The exact wave shape of the output voltage of the cycloconverter depends on (1) the pulse number of the converter, (2) the ratio of the output to input frequency (f_o/f_i), (3) the relative level of the output voltage, (4) the load displacement angle, (5) the circulating current or circulating current-free operation, and (6) the method of control of the firing instants. The harmonic spectrum of a cycloconverter output voltage is different and more complex than that of a phase-controlled converter. It has been revealed [8] that because of the continuous “to-and-fro” phase modulation of the converter firing angles, the harmonic distortion components (known as necessary distortion terms) have frequencies that are sums and differences between multiples of output and input supply frequencies.

Circulating Current-Free Operation. A derived general expression for the output voltage of a cycloconverter with circulating current-free operation [8] shows the following spectrum of harmonic frequencies for the 3-pulse, 6-pulse, and 12-pulse cycloconverters employing cosine modulation technique:

$\begin{array}{c}3-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}=\left|3\left(2k-1\right)f_{\mathrm{i}}\pm 2n{f}_{\mathrm{o}}\right|\mathrm{and}\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|\\ 6-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}=\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|\\ 12-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}=\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|\end{array}$

where k is any integer from 1 to infinity and n is any integer from 0 to infinity.

It may be observed that for certain ratios of f_o/f_i, the order of harmonics may be less or equal to the desired output frequency. All such harmonics are known as subharmonics, since they are not higher multiples of the input frequency. These subharmonics may have considerable amplitudes (e.g., with a 50 Hz input frequency and 35 Hz output frequency, a subharmonic of frequency $3\times 50-4\times 35=10\mathrm{Hz}$ is produced whose magnitude is 12.5% of the 35 Hz component [11]) and are difficult to filter and so objectionable. Their spectrum increase with the increase of the ratio f_o/f_i and so limits its value at which a tolerable waveform can be generated.

Circulating-Current Operation. For circulating current operation with continuous current, the harmonic spectrum in the output voltage is the same as that of the circulating current-free operation except that each harmonic family now terminates at a definite term, rather than having infinite number of components. They are

$\begin{array}{c}3-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}\left|3\left(2k-1\right)f_{\mathrm{i}}\pm 2n{f}_{\mathrm{o}}\right|,n\le 3\left(2k-1\right)+1\\ \mathrm{and}\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|,\left(2n+1\right)\le \left(6k+1\right)\\ 6-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|,\left(2n+1\right)\le \left(6k+1\right)\\ 12-\mathrm{pulse}:f_{\mathrm{o}\mathrm{H}}\left|6k{f}_{\mathrm{i}}\pm \left(2n+1\right)f_{\mathrm{o}}\right|,\left(2n+1\right)\le \left(12k+1\right)\end{array}$

The amplitude of each harmonic component is a function of the output voltage ratio for the circulating current cycloconverter and the output voltage ratio and the load displacement angle for the circulating current-free mode.

From the point of view of maximum useful attainable output-to-input frequency ratio (f_i/f_o) with the minimum amplitude of objectionable harmonic components, a guideline is available in [8] for it as 0.33, 0.5, and 0.75 for 3-, 6-, and 12-pulse cycloconverter, respectively. However, with the modification of the cosine-wave modulation timings like regular sampling [11] in the case of circulating current cycloconverters only and using a subharmonic detection and feedback control concept [13,14] for both circulating and circulating current-free cases, the subharmonics can be suppressed, and useful frequency range for the naturally commutated cycloconverters can be increased.

Other Harmonic Distortion Terms. Besides the harmonics as mentioned, other harmonic distortion terms consisting of frequencies of integral multiples of desired output frequency appear if the transfer characteristic between the output and reference voltages is not linear. These are called unnecessary distortion terms that are absent when the output frequencies are much less than the input frequency. Further, some practical distortion terms may appear due to some practical nonlinearities and imperfections in the control circuits of the cycloconverter, particularly at relatively lower levels of output voltage.

Input Current Waveform. Although the load current, particularly for higher pulse cycloconverters can be assumed to be sinusoidal, the input current is more complex being made of pulses. Assuming the cycloconverter to be an ideal switching circuit without losses, it can be shown from the instantaneous power balance equation that in cycloconverter supplying a single-phase load, the input current has harmonic components of frequencies $\left(f_{\mathrm{I}}\pm 2f_{\mathrm{o}}\right)$ called characteristic harmonic frequencies that are independent of pulse number, and they result in an oscillatory power transmittal to the ac supply system. In the case of cycloconverter feeding a balanced three-phase load, the net instantaneous power is the sum of the three oscillating instantaneous powers when the resultant power is constant, and the net harmonic component is much reduced compared with that of a single-phase load case. In general, the total rms value of the input current waveform consists of three components: in-phase, quadrature, and the harmonic. The in-phase component depends on the active power output, while the quadrature component depends on the net average of the oscillatory firing angle and is always lagging.

14.4.5 Cycloconverter Input Displacement/Power Factor

The input supply performance of a cycloconverter such as displacement factor or fundamental power factor, input power factor, and the input current distortion factor is defined similar to those of the phase-controlled converter. The harmonic factor for the case of a cycloconverter is relatively complex as the harmonic frequencies are not simple multiples of the input frequency but are sums and differences between multiples of output and input frequencies.

Irrespective of the nature of the load, leading, lagging, or unity power factor, the cycloconverter requires reactive power decided by the average firing angle. At low-output voltage, the average phase displacement between the input current and the voltage is large, and the cycloconverter has a low input displacement and power factor. Besides load displacement factor and output voltage ratio, another component of the reactive current arises due to the modulation of the firing angle in the fabrication process of the output voltage [8]. In a phase-controlled converter supplying dc load, the maximum displacement factor is unity for maximum dc output voltage. However, in the case of the cycloconverter, the maximum input displacement factor is 0.843 with unity power factor load [8,15]. The displacement factor decreases with reduction in the output voltage ratio. The distortion factor of the input current is given by (I₁/I), which is always less than 1, and the resultant power factor (= distortion factor × displacement factor) is thus much lower (around 0.76 maximum) than the displacement factor, and this is a serious disadvantage of the naturally commutated cycloconverter (NCC).

14.4.6 Effect of Source Impedance

The source inductance introduces commutation overlap and affects the external characteristics of a cycloconverter similar to the case of a phase-controlled converter with the dc output. It introduces delay in transfer of current from one SCR to another and results in a voltage loss at the output and a modified harmonic distortion. At the input, the source impedance causes “rounding off” of the steep edges of the input current waveforms, resulting in reduction in the amplitudes of higher order harmonic terms and a decrease in the input displacement factor.

14.4.7 Simulation Analysis of Cycloconverter Performance

The nonlinearity and discrete time nature of practical cycloconverter systems, particularly for discontinuous-current conditions, make an exact analysis quite complex, and a valuable design and analytic tool is a digital computer simulation of the system. Two general methods of computer simulation of the cycloconverter waveforms for RL and induction motor loads with circulating current and circulating current-free operation have been suggested in [16] where one of the methods that is very fast and convenient is the crossover point method that gives the crossover points (intersections of the modulating and reference waveforms) and the conducting phase numbers for both P- and N-converters from which the output waveforms for a particular load can be digitally computed at any interval of time for a practical cycloconverter.

14.4.8 Power Quality Issues

Degradation of power quality (PQ) in a cycloconverter-fed system due to subharmonics/interharmonics in the input and the output has been a subject of recent studies [14,17]. In [17], the study includes the impact of cycloconverter control strategies on the total harmonic distortion (THD), distribution transformers, and communication lines, while in [14], the PQ indices are suitably defined and the effect on THD, input/output displacement factor, and input/output power factor for a cycloconverter-fed synchronous motor drive is studied together with a development of a simple feedback method of reduction of subharmonics/low-frequency interharmonics for the improvement of the power quality. The implementation of this scheme, detailed in [14], requires a simple modification of the control circuit of the cycloconverter in contrast to the expensive power level active filters otherwise required for suppression of such harmonics [18].

14.4.9 Forced Commutated Cycloconverter

The naturally commutated cycloconverter (NCC) with SCRs as devices, so far discussed, is sometimes referred to as a restricted frequency changer; in view of the allowance on the output voltage quality ratings, the maximum output voltage frequency is restricted ($f_{\mathrm{o}}\ll f_{\mathrm{i}}$), as mentioned earlier. With devices replaced by fully controlled switches like forced commutated SCRs, power transistors, IGBTs, and GTOs, a forced commutated cycloconverter can be built where the desired output frequency is given by $f_{\mathrm{o}}=\left|f_{\mathrm{s}}-f_{\mathrm{i}}\right|$, where f_s is the switching frequency, which may be larger or smaller than the f_i. In the case when $f_{\mathrm{o}}\ge f_{\mathrm{i}}$, the converter is called unrestricted frequency changer (UFC) and when $f_{\mathrm{o}}\le f_{\mathrm{i}}$, it is called a slow switching frequency changer (SSFC). The early FCC structures have been comprehensively treated in [15]. It has been shown that in contrast with the NCC, where the input displacement factor (IDF) is always lagging, in UFC, it is leading when the load displacement factor is lagging and vice versa, and in SSFC, it is identical to that of the load. Further, with proper control in an FCC, the input displacement factor can be made unity displacement factor frequency changer (UDFFC) with concurrent composite voltage waveform or controllable displacement factor frequency changer (CDFFC), where P-converter and N-converter voltage segments can be shifted relative to the output current wave for control of IDF continuously from lagging via unity to leading.

In addition to allowing bilateral power flow, UFCs offer an unlimited output frequency range, offer good input voltage utilization, do not generate input current and output voltage subharmonics, and require only nine bidirectional switches (Fig. 14.31) for a three-phase to three-phase conversion. The main disadvantage of the structures treated in [15] is that they generate large unwanted low-order input current and output voltage harmonics that are difficult to filter out, particularly for low-output voltage conditions. This problem has largely been solved with an introduction of an imaginative PWM voltage control scheme in [19], which is the basis of the newly designated converter called the matrix converter (also known as PWM cycloconverter) that operates as a generalized solid-state transformer with significant improvement in voltage and input current waveforms resulting in sine-wave input and sine-wave output as discussed in the next section.

14.5.1 Operation and Control of Matrix Converter

The converter in Fig. 14.31 connects any input phase (A, B, and C) to any output phase (a, b, and c) at any instant. When connected, the voltages v_an, v_bn, and v_cn at the output terminals are related to the input voltages V_Ao, V_Bo, and V_Co as

$\left[\begin{array}{c}\hfill v_{\mathrm{an}}\hfill \\ \hfill v_{\mathrm{b}\mathrm{n}}\hfill \\ \hfill v_{\mathrm{c}\mathrm{n}}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill S_{\mathrm{Aa}}\hfill & \hfill S_{\mathrm{Ba}}\hfill & \hfill S_{\mathrm{C}\mathrm{a}}\hfill \\ \hfill S_{\mathrm{Ab}}\hfill & \hfill S_{\mathrm{B}\mathrm{b}}\hfill & \hfill S_{\mathrm{C}\mathrm{b}}\hfill \\ \hfill S_{\mathrm{A}\mathrm{c}}\hfill & \hfill S_{\mathrm{B}\mathrm{c}}\hfill & \hfill S_{\mathrm{C}\mathrm{c}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_{\mathrm{A}\mathrm{o}}\hfill \\ \hfill v_{\mathrm{Bo}}\hfill \\ \hfill v_{\mathrm{C}\mathrm{o}}\hfill \end{array}\right]$

where S_Aa through S_Cc are the switching variables of the corresponding switches shown in Fig. 14.31. For a balanced linear star-connected load at the output terminals, the input phase currents are related to the output phase currents by

$\left[\begin{array}{c}\hfill i_{\mathrm{A}}\hfill \\ \hfill i_{\mathrm{B}}\hfill \\ \hfill i_{\mathrm{C}}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill S_{\mathrm{Aa}}\hfill & \hfill S_{\mathrm{Ab}}\hfill & \hfill S_{\mathrm{A}\mathrm{c}}\hfill \\ \hfill S_{\mathrm{Ba}}\hfill & \hfill S_{\mathrm{B}\mathrm{b}}\hfill & \hfill S_{\mathrm{B}\mathrm{c}}\hfill \\ \hfill S_{\mathrm{C}\mathrm{a}}\hfill & \hfill S_{\mathrm{C}\mathrm{b}}\hfill & \hfill S_{\mathrm{C}\mathrm{c}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_{\mathrm{a}}\hfill \\ \hfill i_{\mathrm{b}}\hfill \\ \hfill i_{\mathrm{c}}\hfill \end{array}\right]$

Note that the matrix of the switching variables in Eq. (14.33) is a transpose of the respective matrix in Eq. (14.32). The matrix converter should be controlled using a specific and appropriately timed sequence of the values of the switching variables, which will result in balanced output voltages having the desired frequency and amplitude, while the input currents are balanced and in phase (for unity IDF) or at an arbitrary angle (for controllable IDF) with respect to the input voltages. As the matrix converter, in theory, can operate at any frequency and at the output or input, including zero, it can be employed as a three-phase ac-dc converter, dc/three-phase ac converter, or even a buck/boost dc chopper and thus as a universal power converter.

The control methods adopted so far for the matrix converter are quite complex and are subjects of continuing research [21–39]. Out of several methods proposed for independent control of the output voltages and input currents, two methods are of wide use and will be briefly reviewed here: (1) the Venturini method based on a mathematical approach of transfer function analysis and (2) the space vector modulation (SVM) approach (as has been standardized now in the case of PWM control of the dc link inverter).

Venturini Method. Given a set of three-phase input voltages with constant amplitude V_i and frequency $f_{\mathrm{i}}={\omega }_{\mathrm{i}}/2\pi$, this method calculates a switching function involving the duty cycles of each of the nine bidirectional switches and generates the three-phase output voltages by sequential piecewise sampling of the input waveforms. These output voltages follow a predetermined set of reference or target voltage waveforms and with a three-phase load connected, a set of input currents I_i and angular frequency ω_i should be in phase for unity IDF or at a specific angle for controlled IDF.

A transfer function approach is used in [36] to achieve the abovementioned features by relating the input and output voltages and the output and input currents as

$\left[\begin{array}{c}\hfill V_{\mathrm{o}1}\left(t\right)\hfill \\ \hfill V_{\mathrm{o}2}\left(t\right)\hfill \\ \hfill V_{\mathrm{o}3}\left(t\right)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill m_{11}\left(t\right)\hfill & \hfill m_{12}\left(t\right)\hfill & \hfill m_{13}\left(t\right)\hfill \\ \hfill m_{21}\left(t\right)\hfill & \hfill m_{22}\left(t\right)\hfill & \hfill m_{23}\left(t\right)\hfill \\ \hfill m_{31}\left(t\right)\hfill & \hfill m_{32}\left(t\right)\hfill & \hfill m_{33}\left(t\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill V_{\mathrm{i}1}\left(t\right)\hfill \\ \hfill V_{\mathrm{i}2}\left(t\right)\hfill \\ \hfill V_{\mathrm{i}3}\left(t\right)\hfill \end{array}\right]$

$\left[\begin{array}{c}\hfill I_{\mathrm{i}1}\left(t\right)\hfill \\ \hfill I_{\mathrm{i}2}\left(t\right)\hfill \\ \hfill I_{\mathrm{i}3}\left(t\right)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill m_{11}\left(t\right)\hfill & \hfill m_{21}\left(t\right)\hfill & \hfill m_{31}\left(t\right)\hfill \\ \hfill m_{12}\left(t\right)\hfill & \hfill m_{22}\left(t\right)\hfill & \hfill m_{32}\left(t\right)\hfill \\ \hfill m_{13}\left(t\right)\hfill & \hfill m_{23}\left(t\right)\hfill & \hfill m_{33}\left(t\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill I_{\mathrm{o}1}\left(t\right)\hfill \\ \hfill I_{\mathrm{o}2}\left(t\right)\hfill \\ \hfill I_{\mathrm{o}3}\left(t\right)\hfill \end{array}\right]$

where the elements of the modulation matrix, m_ij(t), ($i,j=1,2,3$) represent the duty cycles of a switch connecting output phase i to input phase j within a sample switching interval. The elements of m_ij(t) are limited by the constraints discussed above and are mathematically written as

$0\le m_{ij}\left(t\right)\le 1\mathrm{and}{\displaystyle \sum _{j=1}^3}{m}_{ij}\left(t\right)=1\left(i=1,2,3\right)$

The set of three-phase target or reference voltages to achieve the maximum voltage transfer ratio for unity IDF is

$\left[\begin{array}{c}\hfill v_{o1}\hfill \\ \hfill v_{o2}\hfill \\ \hfill v_{o3}\hfill \end{array}\right]=V_{om}\left[\begin{array}{c}\hfill cos{\omega }_{o}t\hfill \\ \hfill cos\left({\omega }_{o}t-120\right)\hfill \\ \hfill cos\left({\omega }_{o}t-240\right)\hfill \end{array}\right]+\frac{V_{im}}{4}\left[\begin{array}{c}\hfill cos\left(3{\omega }_{i}t\right)\hfill \\ \hfill cos\left(3{\omega }_{i}t\right)\hfill \\ \hfill cos\left(3{\omega }_{i}t\right)\hfill \end{array}\right]-\frac{V_{om}}{6}\left[\begin{array}{c}\hfill cos\left(3{\omega }_{o}t\right)\hfill \\ \hfill cos\left(3{\omega }_{o}t\right)\hfill \\ \hfill cos\left(3{\omega }_{o}t\right)\hfill \end{array}\right]$

where V_om and V_im are the magnitudes of output and input fundamental voltages of angular frequencies ω_o and ω_i, respectively. With $V_{\mathrm{o}\mathrm{m}}\le 0.866V_{\mathrm{i}\mathrm{m}}$, a general formula for the duty cycles m_ij(t) is derived in [36]. For unity IDF condition, a simplified formula is

$\begin{array}{c}{m}_{ij}=\frac{1}{3}\left\{1+2qcos({\omega }_{1}t-2\left(j-1\right)60\mathrm{degrees}\right.)\\ \times \left[,cos\left({\omega }_{\mathrm{o}}t-2\left(i-1,\right)60\mathrm{degrees}\right)+\frac{1}{2\sqrt{3}}cos\left(,3{\omega }_{\mathrm{i}}t\right)-\frac{1}{6}cos\left(,3{\omega }_{\mathrm{o}}t\right),\right]\\ -\frac{2q}{3\sqrt{3}}\left[cos\left(4{\omega }_{\mathrm{i}}t-2\left(j-1\right)60\mathrm{degrees}\right)\right.\\ \left.\left.-cos\left(2{\omega }_{\mathrm{i}}t-2\left(1-j\right)60\mathrm{degrees}\right)\right]\right\}\end{array}$

where $i,j=1,2,3$ and $q=V_{\mathrm{o}\mathrm{m}}/V_{\mathrm{i}\mathrm{m}}$.

The method developed as above is based on a direct transfer function (DTF) approach using a single modulation matrix for the matrix converter, employing the switching combinations of all the three groups in Table 14.1. Another approach called indirect transfer function (ITF) approach [30,31] considers the matrix converter as a combination of PWM voltage source rectifier-PWM voltage source inverter (VSR-VSI) and employs the already well-established VSR and VSI PWM techniques for MC control utilizing the switching combinations of group II and group III only of Table 14.1. The drawback of this approach is that the IDF is limited to unity, and the method also generates higher and fractional harmonic components in the input and the output waveforms. This method can also be found in [40].

A simplified direct transfer function approach at unity IDF can be developed. Eq. (14.34) and the constraint equations can be rewritten as follows:

${\underbrace{\left[\begin{array}{c}\hfill v_{o1}\hfill \\ \hfill v_{o1}\hfill \\ \hfill v_{o1}\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]}Y}_{6\times 1}={\underbrace{\left[\begin{array}{ccccccccc}\hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill & \hfill v_{\mathit{in}1}\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]}A}_{6\times 9}{\underbrace{\left[\begin{array}{c}\hfill m_{11}\hfill \\ \hfill m_{12}\hfill \\ \hfill m_{13}\hfill \\ \hfill m_{21}\hfill \\ \hfill m_{22}\hfill \\ \hfill m_{23}\hfill \\ \hfill m_{31}\hfill \\ \hfill m_{32}\hfill \\ \hfill m_{33}\hfill \end{array}\right]}X}_{9\times 1}$

X is the modulation vector, and the elements can be derived as

$X=A^{\mathrm{T}}{\left(A{A}^{\mathrm{T}}\right)}^{-1}Y$

AA^T when solved will have terms ${\displaystyle {\sum }_{i=1}^3{v}_{{\mathrm{in}}_i}^2}$ and ${\displaystyle {\sum }_{i=1}^3{v}_{{\mathrm{in}}_i}}$, so, let ${\displaystyle {\sum }_{i=1}^3{v}_{{\mathrm{in}}_i}^2=L}$ and ${\displaystyle {\sum }_{i=1}^3{v}_{{\mathrm{in}}_i}}=M$. Each term of X, when (14.39) is solved, is of similar pattern and can be shown as

$m_{ji}=\frac{1}{\left(3L-M^2\right)}\left[v_{{\mathrm{o}}_j}\left(3v_{{\mathrm{in}}_i}-M\right)-M{v}_{{\mathrm{in}}_i}+L\right]$

where M_ji is the time-based index corresponding to the switch S_ji that connects j-th output to the i-th input. The time duration for which this switch is on, is calculated by multiplying this index with sampling time. Further solving by considering m balanced input phases, and finally neglecting the terms with denominators with high value, solution for m×n matrix converter is written as

$m_{ji}=\frac{1}{3}\left(1+\frac{2v_{{\mathrm{in}}_i}{v}_{{\mathrm{o}}_j}}{V_{\mathrm{in}}^2}\right)$

A detailed simulation with this modulation scheme can be seen in [41].

SVM Method. The space vector modulation is a well-documented inverter PWM control technique that yields high-voltage gain and less harmonic distortion compared with the other modulation techniques. Here, the three-phase input currents and output voltages are represented as space vectors, and SVM is simultaneously applied to the output voltage and input current space vectors, while the matrix converter is modeled as a rectifying and inverting stage by the indirect modulation method (Fig. 14.33). Applications of SVM algorithm to control of matrix converters have appeared extensively in the literature [33–38,42–47] and shown to have inherent capability to achieve full control of the instantaneous output voltage vector and the instantaneous current displacement angle even under supply voltage disturbances. The algorithm is based on the concept that the MC output line voltages for each switching combination can be represented as a voltage space vector defined by

$v_{\mathrm{o}}=\frac{2}{3}\left[v_{\mathrm{ab}}+v_{\mathrm{b}\mathrm{c}}exp\left(j120\mathrm{degrees}\right)+v_{\mathrm{c}\mathrm{a}}exp\left(-j120\mathrm{degrees}\right)\right]$

Out of the three groups in Table 14.1, only the switching combinations of group II and group III are employed for the SVM method. Group II consists of switching state voltage vectors having constant angular positions and are called active or stationary vectors. Each subgroup of group II determines the position of the resulting output voltage space vector and the six-state space voltage vectors form a six-sextant hexagon used to synthesize the desired output voltage vector. Group III comprises the zero vectors positioned at the center of the output voltage hexagon, and these are suitably combined with the active vectors for the output voltage synthesis.

The modulation method involves selection of the vectors and their on-time computation. At each sampling period T_s, the algorithm selects four active vectors related to any possible combinations of output voltage and input current sectors in addition to the zero vector to construct a desired reference voltage. The amplitude and the phase angle of the reference voltage vector are calculated, and the desired phase angle of the input current vector is determined in advance. For computation of the on-time periods of the chosen vectors, these are combined into two sets leading to two new vectors adjacent to the reference voltage vector in the sextant and having the same direction as the reference voltage vector. Applying the standard SVM theory, the general formulas derived for the vector on-times that satisfy, at the same time, the reference output voltage and input current displacement angle in [36] are

$\begin{array}{l}{t}_1=\frac{2q{T}_{\mathrm{s}}}{\sqrt{3}cos{\varphi }_{\mathrm{i}}}sin\left(60\mathrm{degrees}-{\theta }_{\mathrm{o}}\right)\cdot sin\left(60\mathrm{degrees}-{\theta }_{\mathrm{i}}\right)\\ t_2=\frac{2q{T}_{\mathrm{s}}}{\sqrt{3}cos{\varphi }_{\mathrm{i}}}sin\left(60\mathrm{degrees}-{\theta }_{\mathrm{o}}\right)\cdot sin{\theta }_{\mathrm{i}}\\ t_3=\frac{2q{T}_{\mathrm{s}}}{\sqrt{3}cos{\varphi }_{\mathrm{i}}}sin{\theta }_{\mathrm{o}}\cdot sin\left(60\mathrm{degrees}-{\theta }_{\mathrm{i}}\right)\\ t_4=\frac{2q{T}_{\mathrm{s}}}{\sqrt{3}cos{\varphi }_{\mathrm{i}}}sin{\theta }_{\mathrm{o}}\cdot sin{\theta }_{\mathrm{i}}\end{array}$

where q is voltage transfer ratio, ϕ_i is the input displacement angle chosen to achieve the desired input power factor (with ${\varphi }_{\mathrm{i}}=0$, a maximum value of $q=0.866$ is obtained), and θ_o and θ_i are the phase displacement angles of the output voltage and input current vectors, respectively, whose values are limited within the 0–60degrees range. The on-time of the zero vector is

$t_{\mathrm{o}}=T_{\mathrm{s}}-{\displaystyle \sum _{i=1}^4}{t}_{\mathrm{i}}$

The integral value of the reference vector is calculated over one sample time interval as the sum of the products of the two adjacent vectors and their on-time ratios, and the process is repeated at every sample instant.

Control Implementation and Comparison of the Two Methods. Both the methods need a digital signal processor (DSP)-based system for their implementation. In one scheme [36] for the Venturini method, the programmable timers, as available, are used to time out the PWM gating signals. The processor calculates the six-switch duty cycles in each sampling interval, converts them to integer counts, and stores them in the memory for the next sampling period. In the SVM method, an erasable programmable read-only memory (EPROM) is used to store the selected sets of active and zero vectors, and the DSP calculates the on-times of the vectors. Then, with an identical procedure as in the other method, the timers are loaded with the vector on-times to generate PWM waveforms through suitable output ports. The total computation time of the DSP for the SVM method has been found to be much less than that of the Venturini method. Comparison of the two schemes shows that while in the SVM method the switching losses are lower, the Venturini method shows better performance in terms of input current and output voltage harmonics.

A direct control method as used in conjunction with the voltage source converters has been developed recently and implemented with a 10 kVA matrix converter [48]. Moreover, the voltage transfer ratio can be increased by going into overmodulation. Enhanced methods employing SVM are discussed in [49,50] in which VTR of 0.92 is achieved.

14.5.2 Commutation and Protection Issues in a Matrix Converter

As the matrix converter has no dc link energy storage, any disturbance in the input supply voltage will affect the output voltage immediately, and a proper protection mechanism has to be incorporated, particularly against overvoltage from the supply and overcurrent in the load side. As mentioned, two types of bidirectional switch configurations have hitherto been used: one, the transistor (now IGBT) embedded in a diode bridge and, the other, the two IGBTs in antiparallel with reverse voltage blocking diodes (shown in Fig. 14.31B and C). In the latter configuration, each diode and IGBT combination operates in two quadrants only, which eliminates the circulating currents otherwise built up in the diode bridge configuration that can be limited by only bulky commutation inductors in the lines.

Commutation. The MC does not contain freewheeling diodes that usually achieve safe commutation in the case of other converters. To maintain the continuity of the output current as each switch turns off, the next switch in sequence must be immediately turned on. In practice, with bidirectional switches, a momentary short circuit may develop between the input phases when the switches cross over, and one solution for the configurations shown in Fig. 14.31B, C, and E is to use a semisoft current commutation using a multistepped switching procedure to ensure safe commutation [51–53]. This method requires independent control of each two-quadrant switches, sensing the direction of the load current and introducing a delay during the change of switching states. The switching rule for proper commutation from S₁ to S₂ of the arrangement shown in Fig. 14.34 for $i_{\mathrm{L}}>0$ with the two-quadrant switches for four-stepped commutation [51,53] is (a) turn off S_1B, (b) turn on S_2A, (c) turn off S_1A, and (d) turn on S_2B.

Analogously, for $i_{\mathrm{L}}<0$, the switching rule will be (a) turn off S_1A, (b) turn on S_2B, (c) turn off S_1B, and (d) turn on S_2A. These steps for both the cases are shown diagrammatically in Fig. 14.35.

Typically, these commutation strategies are now implemented using programmable logic devices such as field-programmable gate array (FPGA) and programmable logic device (PLD) [53,54]. Also, it is worth mentioning that the configuration shown in Fig 14.31D has to stick to two-step commutation that despite of having a considerable advantage of employing less number of switches will produce more spikes in the output voltages.

A robust voltage commutation scheme without sacrificing the line side current waveform quality and with minimal information requirement has been reported recently [55]. Another way to achieve better commutation is by Z-sourcing the input of the matrix converter [56].

Protection Strategies. A clamp capacitor (typically 2 μF for a 3 kW permanent magnet (PM) motor) connected through two three-phase full-bridge diode rectifiers involving additional 12 diodes as shown in Fig. 14.31A (a new configuration has been reported in [57] where the number of additional diodes is reduced to six using the antiparallel switch diodes at the input and output lines of the MC) serves as a voltage clamp for possible voltage spikes under normal and fault conditions. A new passive protection strategy involving suppressor diodes and varistors for excellent overvoltage protection is discussed recently in [58], which allows the removal of the large and expensive diode clamp. A snubberless solution for overvoltage protection is presented in [59].

Input Filter. A three-phase single-stage LC filter at the input consisting of three capacitors in star and three inductors in the line is used to adequately attenuate the higher order harmonics and render sinusoidal input current. Typical values of L and C based on a 415 V converter with a maximum line current of 6.5 A and a switching frequency of 20 kHz are 3 mH and 1.5 μF only [60]. The filter may cause minor phase shift in the input displacement angle that needs correction. Another filter designs can be seen in [28,29]. Ref. [61] lately has employed the input filter capacitors as an energy exchange devices to enhance the ride-through capability of the matrix converter. Ref. [62] discusses the designing of an EMI filter for matrix converter applications.

Fig. 14.36 shows typical experimental waveforms of output line voltage and line current of an MC. The output line current is mostly sinusoidal except a small ripple, when the switching frequency is around 1 kHz only.

14.5.3 Multilevel Matrix converter

Erickson and Al-Naseem [63] introduced a multilevel matrix converter with four-quadrant dc link H-bridge switching cells as shown in Fig 14.37 in 2001. These are suitable for medium- or high-voltage ac-to-ac power conversion. The use of four transistors in the switch cell of Fig. 14.37A allows the average current to be doubled, relative to the conventional matrix converter whose four-quadrant switches are realized using two transistors and two diodes. With dc capacitor, the switch cell is capable of producing instantaneous voltages $+\mathrm{V}$, 0, $-\mathrm{V}$. These converters can both increase and decrease the voltage amplitude with arbitrary power factors. With series connection of switch cells in each branch of the matrix, multilevel switching can be attained with device voltages locally clamped to dc capacitor voltages. Several types of multilevel matrix converter topologies have been proposed over the years, including a novel capacitor-clamped one and with a space vector modulation [64,65]. In [65], the converter utilizes flying capacitors to balance voltage distribution of series-connected bidirectional switches and provide middle-voltage levels.

Recently, with cascaded combination of multiple matrix converter (MC) modules, principles of high-power multimodular matrix converters (MMMCs) built from three-input two-output matrix converter ($3\times 2$ MC) have been presented in [66] with modulation schemes to synthesize sinusoidal waveforms on both sides of the converters. Recent work on modulation strategies on MMMCs can be seen in [67–69]. The circuit shown in Fig. 14.38 is known as MMMC-I. Another structure is discussed in these papers in which nine modules of 3×2 MCs are employed. This is an extension of the first topology and attains a multilevel structure for higher power output and better line side harmonic performance. These topologies feature modular design and bidirectional power flow and enable the use of low-voltage power devices. They may serve as potential candidates for high-power applications where regenerative capability and power density are of importance. However, the drawbacks of them are large number of devices and the use of the input transformer.

14.5.4 Indirect Matrix Converters

A conventional matrix converter (CMC) as shown in Fig. 14.31A requires nine bidirectional switches, usually constructed using 18 IGBTs, to switch the three-input phases between the three-output phases. The indirect matrix converters (IMCs), as seen in Fig. 14.40A also have 18 controllable switches. The difference lies in the control of such a converter. This configuration is easier to implement as the control strategies of controlled rectifiers, and inverters can be directly implemented. This topology was first introduced in [70] and later discussed in [71]. It employs 18 controllable switches out 12 are used in rectification and 6 in inversion process. The sparse matrix converter (SMC) (Fig. 14.39) is derived from the IMC that is split into an input stage and output stage without any capacitance in the dc link. The input stage is arranged so that only nine switches are required, while the output stage has a standard six-switch configuration. Thus, the SMC has a total of 15 switches compared with 18 for a CMC and IMC. The functional equivalence, the controllability, and the operating range of the SMC are the same as CMC despite the reduced number of unipolar turn-off power switches [72]. Further reduction of the number of IGBTs is possible with ultra sparse matrix converter (USMC) [73], Fig. 14.40 having 12 IGBTs, with restricted operation to a unidirectional power flow and controllability of the phase displacement of input voltage and current fundamental to $\pm \pi /6$ [73]. Very recently, a space vector modulation scheme for an indirect three-level sparse matrix converter (I3 SMC), which is a combination of a three-level neutral-clamped voltage source inverter and an indirect matrix converter, has been discussed in [74]. A 100 kHz SiC sparse matrix converter using 1300 V and 4 A SiC JFET cascade devices has been reported in [75], which is suitable for aircraft applications where a low-volume/low-weight converter is desired. Ref. [76] also introduced a topology with 12 switches employing unidirectional switches in the rectification process. Real-time implementation of an IMC employing SVM algorithms can be seen in [77].

14.5.5 Single-Phase Matrix Converters

The first single-phase matrix converter was studied by Zuckerberger in [78]. In this paper, converter operation was shown to step down the voltage and step up the frequency. Later in [79], step-up/down frequency operation with safe commutation strategy was developed. The limitation of above schemes lied in boosting of output voltages beyond input voltage. This was overcome by introducing single-phase Z-source matrix converter by Nguyen et al. shown in Fig. 14.41, and step-up/step-down of both voltage and frequency was achieved [80]. A family of high-frequency isolated single-phase Z-source matrix converter with safe commutation strategy is discussed in [81]. Another single-phase matrix converter can be seen in [82–86]. An ac Z-source converter based on gamma structure with safe commutation is discussed in [87]. A novel T-type single-phase ac-ac converter is proposed in [88]. A single-phase ac-ac modular multilevel converter is discussed for railway traction purpose in [89].

14.6.1 High Frequency Integral-Pulse Cycloconverter

The input to these converters may be sinusoidal or quasi square wave, and it is possible to use integral half-cycle pulse-width modulation (IPM) principle to synthesize the output voltage waves. The advantage is that the devices can be switched at zero voltage reducing the switching loss. This converter can only work at output frequency, which are multiples of the input frequency [90].

14.6.2 High-Frequency Phase-Controlled Cycloconverter

Here, phase control principle as explained earlier is used to synthesize the output voltage. A sawtooth carrier wave is compared with the sine modulating wave to generate the firing instants of the switches. The phase control provides switching at zero current reducing switching loss, but this scheme has more complex control circuit compared with the previous one. However, it can work at any frequency [91].