Discontinuous Current Modes of Fully Controlled Converters

A. I. Zaitsev, V. N. Mishin

In investigating discontinuous-current modes we shall use the assumptions and notation of the quantities set forth above [1].

Let us consider the various discontinuous-current modes in the following order of decreasing load.

1. The load current decreases to zero during operation of the power valves in the region of angular coordinates

\[ \arcsin \varepsilon > \nu_n < \frac{\pi}{2}. \]

In this mode the valve is switched on upon arrival of the control signal and conducts current due to the self-induction emf of the load inductance. If the magnitude of the energy stored in the electromagnetic field of the inductance is small, the current may fall to zero before the condition for stable operation of the valve is fulfilled: \(\nu_n > \arcsin \varepsilon\). After such natural turn-off of the valve, if the turn-on signal is retained on its control elements, the device will switch on again when the angular coordinate of the line voltage becomes equal to \(\nu_n = \nu_{B2} = \arcsin \varepsilon\). Upon the repeated switching-on of the valve, the load current, and consequently also the valve current, will begin to increase from a zero value.

Thus, in this mode the current of the power valve during the conduction interval consists of two pulses: the first, beginning from some value, gradually decreases to zero, while the second increases from zero to a certain value at which the valve is artificially turned off. It should be noted that the mode under consideration is very characteristic of systems with leading control angles.

The load current is described by the equations [2]

\[ j_{n1} = \cos \Theta \sin(\nu_{B1} - \Theta + \nu) - \varepsilon + [\varepsilon - \cos \Theta \sin(\nu_{B1} - \Theta) + j_{0n1}] e^{-\nu \operatorname{ctg}\Theta}, \tag{1} \]

\[ j_{n2} = \cos \Theta \sin(\nu_{B2} - \Theta + \nu) - \varepsilon + [\varepsilon - \cos \Theta \sin(\nu_{B2} - \Theta)] e^{-\nu \operatorname{ctg}\Theta}, \tag{2} \]

\[ j_n = (\varepsilon + j_{0n}) e^{-\nu \operatorname{ctg}\Theta} - \varepsilon, \tag{3} \]

where: \(\nu_{B1}\) and \(\nu_{B2}\) are the angular coordinates of the first and second switchings-on of the power valves;

\(\nu\) is the angular coordinate measured from the beginning of the corresponding current pulses.

The initial values of the current are equal to [2]

\[ j_{\mathrm{on}}=\cos\Theta\sin(\nu_{B2}-\Theta+\lambda_{2})-\varepsilon+ [\varepsilon-\cos\Theta\sin(\nu_{B2}-\Theta)]e^{-\lambda_{2}\ctg\Theta}, \tag{4} \]

\[ j_{\mathrm{on}1}= \{\cos\Theta\sin(\nu_{B2}-\Theta+\lambda_{2}) +[\varepsilon-\cos\Theta\sin(\nu_{B2}-\Theta)]e^{-\lambda_{2}\ctg\Theta}\} \times \]

\[ \times e^{-\left(\frac{2\pi}{m}-\lambda\right)\ctg\Theta}-\varepsilon, \tag{5} \]

where \(\lambda\) is the prescribed duration of conduction of the power valves;
\(\lambda_{2}\) is the duration of the second conduction of the power valves.

In calculations it is more convenient to use the duration of the second conduction of the power valves as the independent variable. Then the unknown angular coordinate is the duration of the first conduction, which can be found from the condition

\[ (j_{\mathrm{in}})_{\nu=\lambda_{1}}=0. \tag{6} \]

After substituting expression (1) into (6), we obtain an equation determining the unknown quantity

\[ \cos\Theta\sin(\nu_{n1}-\Theta)-\varepsilon+ [\varepsilon-\cos\Theta\sin(\nu_{B1}-\Theta)+j_{\mathrm{on}1}]e^{-\lambda_{1}\ctg\Theta}, \tag{7} \]

where

\[ \nu_{n1}=\nu_{B1}+\lambda_{1}. \]

If the zero valve is absent or does not operate, then the duration of the second conduction of the power valves is equal to

\[ \lambda_{2}=\frac{2\pi}{m}-(\arcsin\varepsilon-\nu_{B1}). \tag{8} \]

The upper boundary of the mode is the region of continuous currents, calculated from equation [1],

\[ \cos\Theta\sin(\arcsin\varepsilon-\Theta)-\varepsilon+ \]

\[ +\cos\Theta \frac{ e^{\left(\lambda-\frac{2\pi}{m}\right)\ctg\Theta}\cdot \sin(\nu_{B1}-\Theta+\lambda)-\sin(\nu_{B1}-\Theta) }{ \left(1-e^{-\frac{2\pi}{m}\ctg\Theta}\right) e^{(\arcsin\varepsilon-\nu_{B1})\ctg\Theta} }=0, \tag{9} \]

and the lower boundary is found from the condition

\[ (j_{\mathrm{on}1})=0, \tag{10} \]

whence, after substituting (5), we obtain

\[ \{\cos\Theta\sin(\nu_{B2}-\Theta+\lambda_{2})+ \]

\[ +[\varepsilon-\cos\Theta\sin(\nu_{B2}-\Theta)]e^{-\lambda_{2}\ctg\Theta}\} e^{-\left(\frac{2\pi}{m}-\lambda\right)\ctg\Theta}=\varepsilon. \tag{10a} \]

Having determined the boundaries of existence of the mode, we proceed to the calculation of the average and rms values of the system currents and of their spectral composition.

Taking into account that the current of the power valves in a period consists of two pulses, after integration of (1)—(3) we obtain

\[ j_{\mathrm{av.i}}=\frac{1}{2\pi} \left\{ \left[ \sin\frac{\lambda_{1}}{2}\sin\left(\nu_{B1}+\frac{\lambda_{1}}{2}\right) +\sin\frac{\lambda_{2}}{2}\sin\left(\nu_{B2}+\frac{\lambda_{2}}{2}\right) \right]2- \right. \]

\[ \left. -(\varepsilon+j_{\mathrm{on}1})(1-e^{-\lambda_{n}\ctg\Theta})\tg\Theta -\varepsilon\lambda \right\}, \tag{11} \]

\[ j_{\mathrm{av.n}}=\frac{m}{2\pi}\left[(\varepsilon+j_{0\mathrm{n}1})(e^{-\lambda \operatorname{ctg}\Theta}+1)\operatorname{tg}\Theta-\varepsilon\lambda_{\mathrm{n}}\right]. \tag{12} \]

\[ j_{\mathrm{av}}=\frac{m}{\pi}\left[\sin\frac{\lambda_1}{2}\sin\left(\psi_{B1}+\frac{\lambda_1}{2}\right)+ \sin\frac{\lambda_2}{2}\sin\left(\psi_{B2}+\frac{\lambda_2}{2}\right)\right] -\frac{m}{2\pi}(\lambda' + \lambda_{\mathrm{n}})\varepsilon, \tag{13} \]

where

\[ \lambda_{\mathrm{n}}=\frac{2\pi}{m}-\lambda_{\mathrm{n}2}-\psi_{B2}+\psi_{B1} =\frac{2\pi}{m}-\lambda; \]

\[ \lambda'=\lambda_1+\lambda_2. \]

We shall express the quadratic area of the current pulses through a power valve as

\[ S_{\mathrm{sq}}=S_{\mathrm{sq.i}1}+S_{\mathrm{sq.i}2}, \]

where \(S_{\mathrm{sq.i}1}\) and \(S_{\mathrm{sq.i}2}\) are found with the aid of the expression

\[ \begin{aligned} S_{\mathrm{sq.i.k}}={}&0.5\cos^2\Theta\left[ \lambda-\sin\lambda\cdot \cos^2\left(\psi_B-\Theta+\frac{\lambda}{2}\right) -\right.\\ &\left.{}-4\varepsilon\cos\Theta\sin\frac{\lambda}{2} \sin\left(\psi_B-\Theta+\frac{\lambda}{2}\right) +\varepsilon^2\lambda +B\sin2\Theta\left[\sin\psi_B-e^{-\lambda\operatorname{ctg}\Theta}\times\right.\right.\\ &\left.\left.{}\times\sin(\psi_B+\lambda)\right]\right] -B(1-e^{-\lambda\operatorname{ctg}\Theta}) \left[2\varepsilon-0.5B(1+e^{-\lambda\operatorname{ctg}\Theta})\right]\operatorname{tg}\Theta, \end{aligned} \tag{14} \]

in which, when calculating the corresponding areas, it is necessary to substitute

1) for \(S_{\mathrm{sq.i}1}\): \(\psi_B=\psi_{B1}\), \(\lambda=\lambda_{\mathrm{i}1}\), \(B=B_1=\varepsilon-\cos\Theta\sin(\psi_{B1}-\Theta)+j_{0\mathrm{i}1}\);

2) for \(S_{\mathrm{sq.i}2}\): \(\psi_B=\psi_{B2}\), \(\lambda=\lambda_{\mathrm{i}2}\), \(B=B_2=\varepsilon-\cos\Theta\sin(\psi_{B2}-\Theta)\).

The quadratic area of the zero-valve current pulse is equal to

\[ S_{\mathrm{sq.n}}=(\varepsilon+j_{0\mathrm{n}})(1-e^{-\lambda_{\mathrm{n}}\operatorname{ctg}\Theta}) \left[0.5(\varepsilon+j_{0\mathrm{n}})(1+e^{-\lambda_{\mathrm{n}}\operatorname{ctg}\Theta}) -2\varepsilon\right]\operatorname{tg}\Theta+\varepsilon^2\lambda_{\mathrm{n}}. \tag{15} \]

The coefficients of the fundamental harmonic of the Fourier series for the current of the power valves have the form

\[ \begin{aligned} a_1={}&\frac{1}{\pi}\left\{ \frac{\cos\Theta}{2}\left[ \lambda'\cos(\psi_{B1}-\Theta) -\sin\lambda_1\cdot\cos(\psi_{\mathrm{n}1}-\Theta) -\sin\lambda_2\cos(\psi_{B2}-\Theta+\lambda) \right]\right.\\ &{}-2\varepsilon\left[ \sin^2\frac{\lambda_1}{2} +\sin\frac{\lambda_2}{2}\cdot\sin\left(\lambda-\frac{\lambda_2}{2}\right) \right] +B_1\sin\Theta\left[\sin\Theta-e^{-\lambda_1\operatorname{ctg}\Theta}\times\right.\\ &\left.\left.{}\times\sin(\Theta+\lambda_1)\right] +B_2\sin\Theta\left[ \sin(\psi_{B2}-\psi_{B1}+\Theta) -e^{-\lambda_2\operatorname{ctg}\Theta}\cdot\sin(\lambda+\Theta) \right]\right\}, \end{aligned} \tag{16} \]

\[ \begin{aligned} b_1={}&\frac{1}{\pi}\left\{ \frac{\cos\Theta}{2}\left[ \lambda'\sin(\psi_{B1}-\Theta) +\sin\lambda_1\cdot\sin(\psi_{\mathrm{n}1}-\Theta) +\right.\right.\\ &\left.{}\quad+\sin\lambda_2\sin(\psi_{B2}+\lambda-\Theta) \right] -\varepsilon\left[ \sin\lambda_1-2\sin\frac{\lambda_2}{2}\cos\left(\lambda-\frac{\lambda_2}{2}\right) \right]+\\ &{}+B_1\sin\Theta\left[ \cos\Theta-e^{-\lambda_1\operatorname{ctg}\Theta}\cdot\cos(\Theta+\lambda_1) \right] +B_2\sin\Theta\left[ \cos(\psi_{B2}-\psi_{B1}+\Theta) -\right.\\ &\left.\left.{}-e^{-\lambda_2\operatorname{ctg}\Theta}\cdot\cos(\lambda+\Theta) \right]\right\}; \end{aligned} \tag{17} \]

and the coefficients of the higher harmonics are calculated as

\[ \begin{aligned} a_k={}&\frac{\cos\Theta}{\pi(1-k)} \left\{ \cos\left[\psi_{B1}-\Theta+\frac{\lambda_1}{2}(1-k)\right]\cdot \sin\frac{\lambda_1}{2}(1-k)+\right.\\ &\left.{}+\cos\left[\psi_{B2}-\Theta+\frac{\lambda_2}{2}(1+k)-k\lambda\right]\cdot \sin\frac{\lambda_2}{2}(1-k) \right\}- \end{aligned} \]

\[ \begin{aligned} &-\frac{\cos\Theta}{\pi(1+k)} \Bigl\{ \cos\bigl[\nu_{B1}-\Theta+\frac{\lambda_1}{2}(1+k)\bigr]\sin\frac{\lambda_1}{2}(1+k)+{}\\ &\qquad{}+\cos\bigl[\nu_{B2}-\Theta+\frac{\lambda_1}{2}(1-k)+k\lambda\bigr]\sin\frac{\lambda_2}{2}(1+k) \Bigr\}+{}\\ &{}+\frac{B_2}{\pi(k^2+\operatorname{ctg}^2\Theta)} \Bigl\{ \bigl[k-e^{-\lambda_1\operatorname{ctg}\Theta}\bigl(\operatorname{ctg}\Theta\sin k\lambda_1 +k\cos k\lambda\bigr)\bigr]\frac{B_1}{B_2}+{}\\ &\qquad{}+\operatorname{ctg}\Theta\bigl[\sin k(\nu_{B2}-\nu_{B1}) -e^{-\lambda_2\operatorname{ctg}\Theta}\sin k\lambda\bigr] +k\bigl[\cos k(\nu_{B2}-\nu_{B1})\\ &\qquad{}-e^{-\lambda_2\operatorname{ctg}\Theta}\cos k\lambda\bigr] \Bigr\} \frac{2\varepsilon}{\pi k} \biggl[\sin^2 k\frac{\lambda_1}{2} +\sin k\Bigl(\lambda-\frac{\lambda_2}{2}\Bigr)\sin k\frac{\lambda_2}{2}\biggr], \tag{18} \end{aligned} \]

\[ \begin{aligned} b_k={}&\frac{1}{\pi}\biggl\{ \frac{\cos\Theta}{1-k^2} \bigl[\cos(\nu_{B1}-\Theta)-\cos k\lambda_1\cdot\cos(\nu_{n1}-\Theta)\\ &\qquad{}-k\sin k\lambda_1\cdot\sin(\nu_{n1}-\Theta) +\cos(\nu_{B2}-\Theta)\cdot\cos k(\nu_{B2}-\nu_{B1})\\ &\qquad{}-\cos(\nu_{B2}-\Theta+\lambda_2)\cdot\cos k\lambda +k\sin(\nu_{B2}-\Theta)\sin(\nu_{B2}-\nu_{B1})\\ &\qquad{}-k\sin(\nu_{B2}-\Theta+\lambda_2)\sin k\lambda\bigr] -\frac{\varepsilon}{k}\biggl[ \sin k\lambda_1+2\cos k\Bigl(\lambda-\frac{\lambda_2}{2}\Bigr)\sin k\frac{\lambda_2}{2} \biggr]+{}\\ &\qquad{}+\frac{\operatorname{ctg}\Theta}{k^2+\operatorname{ctg}^2\Theta} \bigl[ B_1+B_1e^{-\lambda_1\operatorname{ctg}\Theta} (k\cdot\tg\Theta\cdot\sin k\lambda_1-\cos k\lambda_1)\\ &\qquad{}+B_2ke^{-\lambda_2\operatorname{ctg}\Theta}\times \tg\Theta\cdot\sin k\Bigl(\lambda-\frac{\lambda_2}{2}\Bigr) -B_2e^{-\lambda_2\operatorname{ctg}\Theta}\cdot\cos k\lambda\\ &\qquad{}-B_2k\tg\Theta\sin(\nu_{B2}-\nu_{B1})k +B_2\cos k(\nu_{B2}-\nu_{B1}) \bigr] \biggr\}. \tag{19} \end{aligned} \]

1. The load current decreases to zero during the pause.

The minimum possible firing angles of the power valves are determined by the emf values in the dc circuit and by the voltage drop across the valves:

\[ \nu_B\geq \arcsin\varepsilon . \]

The load current during operation of the power and neutral valves is described by equations (2)—(4), in which we put \(\nu_{B2}=\nu_B,\ \lambda_2=\lambda\).

From above, this mode bounds the region of continuous load currents if \(\nu_B\geq\arcsin\varepsilon\); otherwise, it is the above-considered mode of discontinuous currents. Thus, in one case the upper limiting values should be found from (11), and in the other from [1]

\[ \frac{\cos\Theta\{e^{\lambda\operatorname{ctg}\Theta}\sin(\nu_B-\Theta+\lambda)-\sin(\nu_B-\Theta)\}} {e^{\frac{2\pi}{m}\operatorname{ctg}\Theta}-1} -\varepsilon=0, \tag{20} \]

if \(\nu_B\geq\arcsin\varepsilon\).

The lower boundary of the mode under study is found from the condition

\[ j_{\mathrm{op}}=0, \tag{21} \]

or, substituting (4), we obtain the equation

\[ \cos\Theta\sin(\nu_B-\Theta+\lambda)-\varepsilon+ [\varepsilon-\cos\Theta\sin(\nu_B-\Theta)]e^{-\lambda\operatorname{ctg}\Theta}=0. \tag{22} \]

As follows from (21), from below the mode of currents decreasing to zero during the pause is bounded by the region of discontinuous current with natural turn-off of the valves.

The duration of operation of the neutral valve is found using the condition

\[ (j_n)_{\nu=\lambda_n}=0. \tag{23} \]

Substituting expression (4) into (23), we obtain the equation for determining \(\lambda_n\)

\[ \lambda_{\text{p}}=\operatorname{tg}\Theta\{\ln [\cos \Theta \sin (\nu_{\text{B}}-\Theta+\lambda)e^{-\lambda \operatorname{ctg}\Theta}+\varepsilon- \]
\[ -\cos \Theta \sin(\nu_{\text{B}}-\Theta)]-\ln \varepsilon\}-\lambda . \tag{24} \]

The average values of the currents are found as

\[ j_{\text{av i}}=\frac{1}{2\pi}\left\{2\cos\Theta \sin\frac{\lambda}{2}\left(\nu_{\text{B}}-\Theta+\frac{\lambda}{2}\right)-\varepsilon\lambda+\right. \]
\[ \left.+[\varepsilon-\cos\Theta \sin(\nu_{\text{B}}-\Theta)](1-e^{-\lambda \operatorname{ctg}\Theta})\operatorname{tg}\Theta\right\}, \tag{25} \]

\[ j_{\text{av p}}=\frac{m}{2\pi}[(\varepsilon+j_{0\text{p}})\operatorname{tg}\Theta(1-e^{-\lambda \operatorname{ctg}\Theta})-\varepsilon\lambda_{\text{p}}], \tag{26} \]

\[ j_{\text{av}}=\frac{m}{\pi}\sin\frac{\lambda}{2}\sin\left(\nu_{\text{B}}+\frac{\lambda}{2}\right)-\frac{m}{2\pi}(\lambda+\lambda_{\text{p}})\varepsilon . \tag{27} \]

The quadratic areas of the current pulses of the power and neutral valves should be found from expressions (14) and (15), substituting into the latter \(\lambda_{\text{p}}\) from (24).

For the coefficients of the Fourier series the following equations hold:

\[ a_1=\frac{1}{\pi}\left\{\frac{\cos\Theta}{2}[\lambda\cos(\nu_{\text{B}}-\Theta)-\sin\lambda\cos(\nu_{\text{B}}-\Theta+\lambda)]-2\varepsilon\sin^2\frac{\lambda}{2}+\right. \]
\[ \left.+[\varepsilon-\cos\Theta\sin(\nu_{\text{B}}-\Theta)][\sin\Theta-e^{-\lambda\operatorname{ctg}\Theta}\sin(\Theta+\lambda)]\sin\Theta\right\}, \tag{28} \]

\[ b_1=\frac{1}{\pi}\left\{\frac{\cos\Theta}{2}[\lambda\sin(\nu_{\text{B}}-\Theta)+\sin\lambda\sin(\nu_{\text{B}}-\Theta+\lambda)]-\varepsilon\sin\lambda+\right. \]
\[ \left.+[\varepsilon-\cos\Theta\sin(\nu_{\text{B}}-\Theta)][\cos\Theta-e^{-\lambda\operatorname{ctg}\Theta}\cos(\Theta+\lambda)]\sin\Theta\right\}, \tag{29} \]

\[ a_k=\frac{1}{\pi}\left\{\frac{\cos\Theta}{1-k^2}[k\sin(\nu_{\text{B}}-\Theta+\lambda)\cos k\lambda-\sin k\lambda\cos(\nu_{\text{B}}-\Theta+\lambda)-\right. \]
\[ \left.-k\sin(\nu_{\text{B}}-\Theta)]-\frac{\varepsilon}{k}(1-\cos k\lambda)+\right. \]
\[ \left.+\frac{\varepsilon-\cos\Theta\sin(\nu_{\text{B}}-\Theta)}{k^2+\operatorname{ctg}^2\Theta}[k-e^{-\lambda\operatorname{ctg}\Theta}(\operatorname{ctg}\Theta\sin k\lambda+k\cos k\lambda)]\right\}, \tag{30} \]

\[ b_k=\frac{1}{\pi}\left\{\frac{\cos\Theta}{1-k^2}[\cos(\nu_{\text{B}}-\Theta)-\cos k\lambda\cos(\nu_{\text{B}}-\Theta+\lambda)-\right. \]
\[ \left.-k\sin k\lambda\cdot\sin(\nu_{\text{B}}-\Theta+\lambda)]-\frac{\varepsilon}{k}\sin k\lambda+\right. \]
\[ \left.+\frac{\varepsilon-\cos\Theta\sin(\nu_{\text{B}}-\Theta)}{k^2+\operatorname{ctg}^2\Theta}[1+e^{-\lambda\operatorname{ctg}\Theta}(k\operatorname{tg}\Theta\cdot\sin k\lambda-\cos k\lambda)]\operatorname{ctg}\Theta\right\}. \tag{31} \]

When the load is decreased, the mode of discontinuous currents, which fall to zero during the pause, passes into a mode of currents that decrease to zero while flowing through the power valves of the converter. Thus, in the latter discontinuous-current mode the valves are not artificially turned off, and the converter can operate only with lagging control angles. This mode has been investigated in detail in the works of A. A. Bulgakov [3] and is not considered here.

REFERENCES

1. A. I. Zaitsev, V. N. Mishin. Investigation of fully controlled converters. Article in the present collection.

2. A. I. Zaitsev, V. N. Mishin. On the calculation of certain mutators based on fully controlled elements. Izvestiya TPI, vol. 153, 1965.

3. A. A. Bulgakov. Fundamentals of the Dynamics of Valve Systems. Publishing House of the USSR Academy of Sciences, 1963.