MODULATION CYCLOCONVERTER FOR AUTONOMOUS SOURCES OF STABLE FREQUENCY

V. A. DENISOV, Sh. S. ROIZ

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

In the present article, a single-phase semiconductor direct-coupled frequency converter, supplied by an increased-frequency voltage modulated in amplitude, is considered as a modulation cycloconverter. The named modulation cycloconverter is the main component of a modulated machine-valve autonomous source of stable frequency [1, 2].

The power section of the modulation cycloconverter (Fig. 1) is made according to a bridge circuit and contains four fully controlled bidirectional semiconductor valves. The control circuit is built on logic elements.

Fig. 1. Schematic diagram of a modulation cycloconverter

Fig. 2. Timing diagrams illustrating the operation of the modulation cycloconverter

The operating principle of the circuit is explained by Fig. 2 and is as follows. The pulse shapers \(\Phi_1\) and \(\Phi_2\) convert the modulated input voltage \(U_1\) into rectangular pulses. The pulses \(\Phi_1\) correspond to the positive half-wave of \(U_1\), and the pulses \(\Phi_2\) to the negative half-wave. Next, \(\Phi_1\) and \(\Phi_2\) are fed to a coincidence circuit formed from four AND elements, to which pulses from a flip-flop are also applied (the states of the flip-flop are denoted in the diagram by \(T_1\) and \(T_2\)), controlled by the frequency setter. The control pulses thus obtained are amplified in amplifiers \(Y_1\) and \(Y_2\) and then act on power valves 1, 2, 3, and 4.

For the positive half-wave of the voltage \(U_2\), the positive pulses \(\Phi_1\) control valves 1 and 4, while the negative pulses \(\Phi_2\) control valves 2 and 3. For the negative half-wave of the voltage \(U_2\), the positive pulses \(\Phi_1\) control valves 2 and 3, while the negative pulses \(\Phi_2\) control valves 1 and 4. Such control and the use in the power circuit of fully controlled bidirectional valves ensure precise stabilization of the frequency of the output voltage \(U_2\), bidirectional energy exchange between the source and the load regardless of the nature of the load, and considerably simplify the structure of the logical control circuit.

Let us consider the electromagnetic processes in the cycloconverter ...

when it operates into an active-inductive load, taking into account the following assumptions: 1) the forward resistance of the power valves is equal to zero, and the reverse resistance is infinite; 2) the phenomenon of current overlap by the valves during commutation is absent; 3) the ratio of the carrier frequency to the modulating frequency is equal to a multiple integer. In the analysis and derivation of the equations we shall assume that the supply input voltage is modulated according to a sinusoidal law with double frequency, i.e.,

\[ U_1=\left(1-\frac{a}{2}-\frac{a}{2}\cos 2\omega t\right)\sin k\omega t \quad \text{for } \quad 0\leq \omega t\leq \pi, \]

where \(a\) is the depth of amplitude modulation;
\(\omega\) is the angular output frequency of the cycloconverter;
\(k\) is the multiplicity of the ratio of the carrier and modulating frequencies.

Then, under the accepted assumptions, the voltage at the output of the cycloconverter can be represented as shown in Fig. 3.

Fig. 3. Approximate form of the cycloconverter output voltage

In the interval \(0\leq \omega t\leq \frac{\pi}{k}\), the following equation is valid for the load circuit:

\[ L\frac{di_0}{dt}+Ri_0=\left(1-\frac{a}{2}-\frac{a}{2}\cos 2\omega t\right)\sin k\omega t, \]

where \(R\) and \(L\) are the active and inductive resistances of the load circuit.

After integration and applying the condition \(i_0(t=0)=0\), we obtain

\[ i^0(t)=f(t)-f(0)\exp(-\omega t\operatorname{ctg}\varphi), \tag{1} \]

where

\[ \operatorname{ctg}\varphi=\frac{R}{\omega L}; \]

\[ \begin{aligned} f(t)=& \frac{1-\frac{a}{2}}{\sin\varphi\left(\operatorname{ctg}^2\varphi+k^2\right)} \left(\operatorname{ctg}\varphi\sin k\omega t-k\cos k\omega t\right) \\ &-\frac{a}{4\sin\varphi\left[\operatorname{ctg}^2\varphi+(k+2)^2\right]} \left[\operatorname{ctg}\varphi\sin (k+2)\omega t-(k+2)\cos k\omega t\right] \\ &-\frac{a}{4\sin\varphi\left[\operatorname{ctg}^2\varphi+(k-2)^2\right]} \left[\operatorname{ctg}\varphi\sin (k-2)\omega t-(k-2)\cos k\omega t\right]. \end{aligned} \]

In the interval \(\dfrac{\pi}{k} \leq \omega t \leq 2\dfrac{\pi}{k}\), the expression for the current takes the form

\[ i(t)=-f(t)+2f\left(\frac{\pi}{k\omega}\right)\exp\left[\left(\frac{\pi}{k}-\omega t\right)\operatorname{ctg}\varphi\right]-f(0)\exp(-\omega t\operatorname{ctg}\varphi). \tag{2} \]

Similarly, we find the current equation in the interval \(m\), when

\[ m\frac{\pi}{k}\leq \omega t \leq (m+1)\frac{\pi}{k}; \]

\[ i_m(t)=(-1)^m f(t)+2\sum_{n=1}^{m}(-1)^{n+1}f\left(\frac{n\pi}{k\omega}\right)\exp \times \]

\[ \times \left[\left(\frac{n\pi}{k}-\omega t\right)\operatorname{ctg}\varphi\right]-f(0)\exp(-\omega t\operatorname{ctg}\varphi). \tag{3} \]

Equation (3) makes it possible to determine the value of the current in the load at any point of the interval \(0\)—\(\pi\).

To determine the current in the intervals of the second half of the period \(\pi\)—\(2\pi\), one may use (3), if the initial value of the current is taken into account,

\[ i_{\text{init}}=i_{m=k-1}\left(t=\frac{\pi}{\omega}\right), \]

\[ i_m(t)=(-1)^m f(t)+2\sum_{n=1}^{m}(-1)^{n+1}f\left(\frac{n\pi}{k\omega}\right)\exp \times \]

\[ \times \left[\left(\frac{n\pi}{k}-\omega t\right)\operatorname{ctg}\varphi\right]-[i_{\text{init}}+f(0)]\exp(-\omega t\operatorname{ctg}\varphi), \tag{4} \]

where

\[ i_{\text{init}}=(-1)^{k-1}f\left(\frac{\pi}{\omega}\right)+2\sum_{n=1}^{k-1}(-1)^{n+1}f\times \]

\[ \times \left(\frac{n\pi}{k\omega}\right)\exp\left[\left(\frac{n\pi}{k}-\pi\right)\operatorname{ctg}\varphi\right]-f(0)\exp(-\pi\operatorname{ctg}\varphi). \]

Since the value of the initial current \(i_{\text{init}}\) increases only insignificantly in the subsequent half-periods, equation (4) is also valid for the established steady-state mode.

Calculations of the current curves carried out on a digital computer using equation (4) made it possible to find the optimal value of the depth of amplitude modulation (\(a=0.6\)), at which the current waveform in the load is practically sinusoidal. This current curve is shown in Fig. 5, a in comparison with the load-current curve in the absence of amplitude modulation (\(a=0\)).

To verify the theoretical results under the assumptions made, an experiment was carried out on a prototype setup.

Here, four valve bridges built on D245A diodes and loaded by power triodes (KT-805A) were used as fully controlled bidirectional valves. The logic circuit is built on elements of the “Logika T” series and contains two T-403 amplifiers, four T-107 elements, a T-102 trigger, and two T-202 relay elements. The latter are connected to the input voltage through a transformer with stabilitrons installed on the secondary windings (Fig. 4).

A comparison of the experimental oscillograms (Fig. 5, b) with the calculated curves (Fig. 5, a) indicates their satisfactory agreement.

Conclusions

1. The investigations carried out fully confirmed the operability of the proposed circuit of a modulation cycloconverter.

2. The obtained equations of the electromagnetic processes make it possible

Fig. 4. Circuit diagrams of nodes of the modulation cycloconverter:

a — module of a fully controlled bidirectional valve,
b — rectangular control-pulse former

Fig. 5. Load-current curves for \(k=10\) and \(\cos \varphi=0.8\):
a — calculated curves, b — experimental oscillograms

construct and evaluate the load-current curve for any value of the amplitude-modulation depth.

REFERENCES

1. M. P. Popov, I. I. Lukin. On the problem of obtaining alternating current of constant frequency with a varying rotation speed of the generator drive. Izvestiya vuzov, “Energetika,” 1970, No. 1.

2. V. A. Denisov, Sh. S. Roiz. Single-phase autonomous source of a specified stable frequency. Abstracts of the final scientific-and-technical conference, IEI, Ivanovo, 1973.