INVESTIGATION OF FULLY CONTROLLED CONVERTERS

A. I. Zaitsev, V. N. Mishin

Recommended by the scientific seminar of the Electromechanical Faculty

CONTINUOUS CURRENT MODE

Great prospects are opening up for converter technology in connection with the development and application of fully controlled semiconductor valves. Fully controlled valve converters (FCVC) can also be built using transistors, electron tubes, and thyristors with forced commutation. FCVCs allow an installation to operate with voltage regulation at leading control angles, which turns the valve installation from a consumer into a generator of reactive energy, eliminating the main drawback of controlled valve systems [1].

FCVCs can operate in the most advantageous mode, “rectifying” the corresponding portion of the sinusoid of the network voltage, i.e., they make it possible to regulate the power factor of the installation, orienting it toward the optimum value for each specific case. The law of control of the valves of such systems must take into account the reactive-energy requirements of the given network. If there is inductance in the dc circuit, such operation of an FCVC is possible only with a zero valve.

Let us investigate the statics of single-cycle fully controlled converters under the assumption that the installation is supplied from an absolutely rigid network of infinite power, the voltage drops across the valves \((E_v)\) are equal and do not depend on the magnitude of the current, and the load parameters \((L, R, E)\) are constant. The circuit of the case under consideration is shown in Fig. 1, where the fully controlled valve is represented by an automatic switch \(K\), which performs instantaneous switching in the power and zero circuits at the corresponding instants of time.

We shall carry out the investigation of the FCVC in a system of relative units [2].

Fig. 1.

Let us denote:

\(U_m\)—amplitude value of the phase voltage of the network;

\(\omega\)—network frequency;

\(m\)—number of converter phases,

\[ \frac{E+E^{B}}{U_m}=\varepsilon \]

—the relative emf;

\[ \frac{I}{\dfrac{U_m}{R}}=j \]

—the relative current;

\[ \omega \frac{L}{R}=\operatorname{tg}\Theta \]

—the load parameter.

The instantaneous values of the current through the power and neutral valves are described by the following equations, respectively [3].

\[ \left\{ \begin{aligned} j_u &= \cos \Theta \sin(\nu_B-\Theta+\nu)-\varepsilon +\left[\varepsilon-\cos \Theta \sin(\nu_B-\Theta)+j_{ou}\right]e^{-\nu \operatorname{ctg}\Theta},\\ j_n &= (\varepsilon+j_{on})e^{-\nu \operatorname{ctg}\Theta}-\varepsilon, \end{aligned} \right. \tag{1} \]

where: \(\nu_B\)—phase of the network voltage corresponding to the switching-on of the power valve;

\(\nu\)—angular coordinate, measured from the instant of closing of one or another circuit;

\(j_{ou}, j_{on}\)—initial values of the currents of the power and neutral valves, respectively.

The initial values of the currents are found as:

\[ j_{on}=(j_u)_{\nu=\lambda}, \]

\[ j_{ou}=(j_n)_{\nu=\frac{2\pi}{m}-\lambda}, \tag{2} \]

where: \(\lambda\)—duration of conduction of the power valves.

Substituting conditions [2] into the system of equations [1], we obtain

\[ j_{ou}= \frac{\cos \Theta\left[e^{\lambda \operatorname{ctg}\Theta}\cdot \sin(\nu_B-\Theta+\lambda)-\sin(\nu_B-\Theta)\right]} {e^{\frac{2\pi}{m}\operatorname{ctg}\Theta}-1} -\varepsilon, \tag{3} \]

\[ j_{on}= \frac{\cos \Theta\left[\sin(\nu_B-\Theta+\lambda)-e^{-\lambda \operatorname{ctg}\Theta}\cdot \sin(\nu_B-\Theta)\right]} {1-e^{-\frac{2\pi}{m}\operatorname{ctg}\Theta}} -\varepsilon. \tag{4} \]

If the neutral valve is absent or does not participate in the operation, then in the continuous-load-current mode

\[ \lambda=\frac{2\pi}{m}. \]

The conditions for the existence of the continuous-load-current mode will be:

\[ j_{ou}\geq 0,\quad \text{if } \nu_B\geq \arcsin \varepsilon, \tag{5} \]

and

\[ (j_u)\geq 0,\quad \nu=\arcsin \varepsilon-\nu_B,\quad \text{if } \nu_B\leq \arcsin \varepsilon. \tag{6} \]

Thus, the limiting values of the angular coordinates at which the continuous-current mode in the load still exists, taking into account conditions [5]—[6] and expressions [1] and [3], are found from the equations:

\[ \frac{\cos \Theta \left\{ e^{\lambda_{\mathrm{gr}}\operatorname{ctg}\Theta}\cdot \sin(\gamma_B-\Theta+\lambda_{\mathrm{gr}})-\sin(\gamma_B-\Theta)\right\}} {e^{\frac{2\pi}{m}\operatorname{ctg}\Theta}-1}-\varepsilon=0, \tag{7} \]

if

\[ \gamma_B \geqslant \arcsin \varepsilon \]

and

\[ \varepsilon-\cos\Theta\, \frac{ e^{\left(\lambda_{\mathrm{gr}}-\frac{2\pi}{m}\right)\operatorname{ctg}\Theta} \cdot \sin(\gamma_B-\Theta+\lambda_{\mathrm{gr}})-\sin(\gamma_B-\Theta)} {\left(1-e^{-\frac{2\pi}{m}\operatorname{ctg}\Theta}\right)e^{(\arcsin\varepsilon-\gamma_B)\operatorname{ctg}\Theta}} =0 \tag{8} \]

if

\[ \gamma_B \leqslant \arcsin \varepsilon . \]

If \(\lambda=\dfrac{2\pi}{m}\), using [7] or [8] we find the value of the firing angle \(\gamma_B\) at which the limiting mode of continuous load current exists. In systems with a neutral valve it is more convenient to prescribe the firing angle and to find from [7] or [8] the value of the limiting conduction duration. For converters with an uncontrolled neutral valve, under the valve-control law providing the minimum possible firing angles, and \(\lambda \leqslant \dfrac{2\pi}{m}\), the firing angle is equal to zero.

Having calculated the boundaries of existence of the continuous-current mode, one may proceed to determine the average and rms values of the current in the system elements, as well as the spectral composition of the current flowing through the converter power valves.

By integrating expressions (1), we obtain the average values of the current of the load, power valves, and neutral valves in relative units, respectively,

\[ j_{\mathrm{cp}}=\frac{m}{\pi}\sin\frac{\lambda}{2}\sin\left(\gamma_B+\frac{\lambda}{2}\right)-\varepsilon, \tag{9} \]

\[ j_{\mathrm{cp.i}}= \frac{\cos\Theta}{2\pi} \left[ 2\sin\frac{\lambda}{2}\sin\left(\gamma_B-\Theta+\frac{\lambda}{2}\right) + A\,\operatorname{tg}\Theta\left(1-e^{-\lambda\operatorname{ctg}\Theta}\right) \right] -\frac{\lambda}{2\pi}\varepsilon, \tag{10} \]

\[ j_{\mathrm{cp.p}}= \frac{m}{2\pi}\sin\Theta \left[ 2\sin\frac{\lambda}{2}\cos\left(\gamma_B-\Theta+\frac{\lambda}{2}\right) - A\left(1-e^{-\lambda\operatorname{ctg}\Theta}\right) \right] -\varepsilon\left(1-\frac{m}{2\pi}\lambda\right), \tag{11} \]

where

\[ A= \frac{ e^{\left(\lambda-\frac{2\pi}{m}\right)\operatorname{ctg}\Theta} \cdot \sin(\gamma_B-\Theta+\lambda)-\sin(\gamma_B-\Theta)} {1-e^{-\frac{2\pi}{m}\operatorname{ctg}\Theta}} . \]

For convenience of notation, let us express the rms values of the currents through the corresponding quadratic areas of the current pulses:

\[ j_{\mathrm{d}}=\sqrt{\frac{m}{2\pi}\left(S_{\mathrm{kv.i}}+S_{\mathrm{kv.p}}\right)}, \tag{12} \]

\[ j_{\mathrm{d.i}}=\sqrt{\frac{1}{2\pi}S_{\mathrm{kv.i}}}, \tag{13} \]

\[ j_{\mathrm{d.p}}=\sqrt{\frac{m}{2\pi}S_{\mathrm{kv.p}}}, \tag{14} \]

where the quadratic current areas of the power and zero valves are found as

\[ S_{\mathrm{sq},n}=\int_{0}^{\lambda} j_n^{2}(\nu)\,d\nu . \tag{15} \]

\[ S_{\mathrm{sq},p}=\int_{0}^{\frac{2\pi}{m}-\lambda} j_n^{2}(\nu)\,d\nu . \tag{16} \]

Substituting expressions (1) into (15) and (16) and integrating, after simplification we obtain

\[ \begin{aligned} S_{\mathrm{sq},n}={}&0.5\cos^{2}\Theta \left[\lambda-\sin\lambda\cos 2\left(\nu_B-\Theta+\frac{\lambda}{2}\right)\right] +\varepsilon^{2}\lambda \\ &-4\varepsilon\cos\Theta\sin\frac{\lambda}{2}\, \sin\left(\nu_B-\Theta+\frac{\lambda}{2}\right) +0.25A^{2}\left(1-e^{-2\lambda\operatorname{ctg}\Theta}\right)\sin 2\Theta \\ &+2A\sin\Theta\left\{\cos^{2}\Theta \left[\sin\nu_B-e^{-\lambda\operatorname{ctg}\Theta}\sin(\nu_B+\lambda)\right] -\left(1-e^{-\lambda\operatorname{ctg}\Theta}\right)\varepsilon\right\}, \end{aligned} \]

\[ \begin{aligned} S_{\mathrm{sq},p}={}&0.5\operatorname{tg}\Theta\,(\varepsilon+j_{op}) \left[1-e^{-\left(\frac{2\pi}{m}-\lambda\right)\operatorname{ctg}\Theta}\right] \\ &\times\left\{\left[1+e^{-\left(\frac{2\pi}{m}-\lambda\right)\operatorname{ctg}\Theta}\right](\varepsilon+j_{op})-4\varepsilon\right\} +\left(\frac{2\pi}{m}-\lambda\right)\varepsilon^{2}. \end{aligned} \tag{18} \]

The current of the power valves of the converter may be represented in the form of the following Fourier series [2]

\[ j_n=j_{\mathrm{avg},n}+\sum_{k=1}^{\infty} j_{kn}\sin(k\nu+\psi_k), \tag{19} \]

where

\[ j_{kn}=\sqrt{a_k^{2}+b_k^{2}} \tag{20} \]

is the amplitude of the \(k\)-th harmonic of the current pulse;

\[ \psi_k=\operatorname{acr\,tg}\frac{b_k}{a_k}. \tag{21} \]

is the phase of the \(k\)-th harmonic, measured from the beginning of the pulse. The phase of the \(k\)-th harmonic relative to the AC mains voltage is found as

\[ \varphi_k=\nu_B-\operatorname{arc\,tg}\frac{b_k}{a_k}. \tag{22} \]

The coefficients of the Fourier series are determined by means of the integrals

\[ a_k=\frac{1}{\pi}\int_{0}^{\lambda} j_n(\nu)\sin k\nu\,d\nu, \tag{23} \]

\[ b_k=\frac{1}{\pi}\int_{0}^{\lambda} j_n(\nu)\cos k\nu\,d\nu, \tag{24} \]

whence, after transformations, we obtain the coefficients of the first harmonic \((k=1)\)

\[ b_1=\frac{\cos\Theta}{2\pi}\left\{\lambda\sin(\nu_B-\Theta)+\sin\lambda\,\sin(\nu_B-\Theta+\lambda)+\right. \]

\[ +2A\left[\cos\Theta-e^{-\lambda\operatorname{ctg}\Theta}\cdot\cos(\Theta+\lambda)\right]\sin\Theta\}-\frac{\varepsilon}{\pi}\sin\lambda, \tag{25} \]

\[ a_1=\frac{\cos\Theta}{2\pi}\left\{\lambda\cos(\nu_B-\Theta)-\sin\lambda\cos(\nu_B-\Theta+\lambda)+\right. \]

\[ \left.+2A\left[\sin\Theta-e^{-\operatorname{ctg}\Theta}\cdot\sin(\Theta+\lambda)\right]\sin\Theta\right\} -\frac{2\varepsilon}{\pi}\sin^2\frac{\lambda}{2} \tag{26} \]

and the coefficients of the higher harmonics \((k>1)\)

\[ a_k=\frac{\cos\Theta}{\pi(1-k^2)} \left[k\sin(\nu_B-\Theta+\lambda)\cos k\lambda-\sin k\lambda\cos(\nu_B-\Theta+\lambda)-\right. \]

\[ \left.-k\sin(\nu_B-\Theta)\right] +\frac{A\cos\Theta}{\pi(k^2+\operatorname{ctg}^2\Theta)} \left[k-e^{-\lambda\operatorname{ctg}\Theta}(\operatorname{ctg}\Theta\sin k\lambda+k\cos k\lambda)\right]- \]

\[ -\frac{\varepsilon}{k\pi}(1-\cos k\lambda), \tag{27} \]

\[ b_k=\frac{\cos\Theta}{\pi(1-k^2)} \left[\cos(\nu_B-\Theta)-\cos k\lambda\cos(\nu_B-\Theta+\lambda)-\right. \]

\[ \left.-k\sin k\lambda\sin(\nu_B-\Theta+\lambda)\right] +\frac{A\cos\Theta\operatorname{ctg}\Theta}{\pi(k^2+\operatorname{ctg}^2\Theta)} \left[1+e^{-\lambda\operatorname{ctg}\Theta}(k\operatorname{tg}\Theta\sin k\lambda-\right. \]

\[ \left.-\cos k\lambda)\right] -\frac{\varepsilon}{k\pi}\sin k\lambda. \tag{28} \]

In calculating circuits with valves it is often necessary to know the amplitude values of the current pulses.

The maximum value of the current of the zero valve is equal to its initial value
\(j_{n\max}=(j_n)_{\nu=0}=j_{\text{op}}\).

We find the angular coordinate corresponding to the maximum current in the power valves from the condition

\[ \frac{dj_n}{d\nu}=0, \]

whence, after substituting (1) and simplifying, we obtain the transcendental equation

\[ \sin\Theta\cos(\nu_B-\Theta+\nu_{\max})\,e^{\nu_{\max}\operatorname{ctg}\Theta} =\varepsilon-\cos\Theta\sin(\nu_B-\Theta)+j_{\text{op}}. \tag{29} \]

Having computed \(\nu_{\max}\) by means of (29) and substituted it into (1), one can find the amplitude value of the current of the power valves. Note that equation (29) is valid only when \(\nu_{\max}<\lambda\); otherwise the current maximum should be found as

\[ j_{n.\max}=(j_n)_{\nu=\lambda}=j_{\text{op}}. \tag{30} \]

When \(\nu_B+\lambda\le\frac{\pi}{2}\), the maximum value of the current pulse always corresponds to condition (30).

The power factor of a fully controlled converter in a transformerless circuit can be calculated as

\[ K_M=\nu\cos\varphi_{(1)}, \tag{31} \]

where

\[ \nu=\frac{j_{(1)}}{j_{d.n}}= \frac{\sqrt{a_1^2+b_1^2}}{\sqrt{2}\,j_{d.n}} \]

is the distortion factor; and the displacement factor \((\cos\varphi)\) is determined by means of (22).

Using the formulas obtained, it is also possible to calculate ordinary converters with lagging control angles. In this case, in systems with a zero valve, it must be borne in mind that,

\[ \lambda = \pi - \nu_{\mathrm{B}}, \quad \text{if } \nu_{\mathrm{B}} \geq \pi \frac{m-2}{m}, \]

or

\[ \lambda = \frac{2\pi}{m}, \quad \text{if } \nu_{\mathrm{B}} \leq \pi \frac{m-2}{m}. \]

On the basis of the formulas obtained, it is possible to select the valves of the converter, to select the motor according to heating conditions (if the fully controlled valve converter is used in an electric-drive system), and to estimate the energy indicators of the installation. The use of a system of relative units makes it possible to calculate the corresponding universal tables and diagrams for specified laws of control of the converter valves, which are convenient for use in engineering practice.

References

1. I. L. Kaganov. Electronic and Ion Converters, Part III, GEI, 1956.
2. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Valve Systems. Publishing House of the Academy of Sciences of the USSR, 1963.
3. A. I. Zaitsev, V. N. Mishin. On the calculation of certain commutators using fully controlled elements. Izvestiya TPI, Vol. 153, 1965.