Conversion of Direct Current into Single-Phase Variable-Frequency Current

I. G. Kuleev

One of the significant shortcomings of asynchronous motors with a squirrel-cage rotor is the absence of an economically advantageous method of speed regulation. A method for smooth speed regulation of these motors has not yet been found.

Stepwise regulation by increasing the number of windings on the stator leads to a considerable reduction in the power factor and in the motor efficiency, while increasing the overall dimensions. In this connection, the search for an economically advantageous method of frequency conversion for regulating the speed of asynchronous motors with a squirrel-cage rotor is a very important problem. It could successfully compete with the motor-generator method currently used on longitudinal planing machines. It would undoubtedly be more advantageous than multispeed asynchronous motors, which have found some use at present also on machine tools. Recently, high-frequency installations \((300 \div 1000)\) cycles/sec have come into use. At such a frequency the electric motors of saws used in logging operations are supplied. In mines it is also desirable to have, for supplying electric drills, a frequency within the indicated range, since at increased frequency the dimensions of a hand drill become smaller. The examples cited do not exhaust the national-economic needs for frequency converters.

Fig. 1

In the present work, the process of frequency conversion is considered according to the circuit shown in Fig. 1. Here \(ab\) and \(bd\) are chokes connected by their ends, \(r_1\) is the active resistance of the choke, \(L_1\) is its inductance, \(C\) is the capacitance, \(r\) is the load active resistance, and \(U\) is the voltage of the direct-current source. The proposed circuit of a self-excited ...

the inverter has the following advantages over the circuit shown in Fig. 2, which is currently in widespread use. In the circuit of Fig. 2 there is a choke \(L_0\) and a transformer \(T\); in the circuit of Fig. 1 there is only a choke. In cost, the transformer is more expensive than the choke. Under transient conditions of the transformer, the currents due to the load flow in accordance with the time constant of the leakage fields. This time constant is, of course, smaller than the time constant of the choke, for which it is associated

Fig. 2

with the full flux. An increase in the time constant has a very favorable effect on commutation: the larger it is, the slower the rise of the anode current.

When converting direct current into alternating current according to the circuit of Fig. 1 or 2, a sinusoidal voltage at the output will be obtained in the case when the natural oscillation frequency of the circuit and the frequency of the voltage applied to the grid are equal. When the frequency of the grid voltage is greater than the natural oscillation frequency of the circuit, a peaked form of the curve is obtained across the resistance \(r\) (Fig. 3).

Fig. 3

When the frequency of the grid voltage is less than the natural oscillation frequency of the circuit, a triangular form of the output-voltage curve is obtained, similar to that shown in Fig. 4.

When the inductance of the choke is greater than the leakage inductance of the transformer, a smaller capacitance of the oscillatory circuit will be required,

to obtain an industrial frequency, since the natural frequency of the circuit is determined by the formula:

\[ \omega_0=\sqrt{\frac{1}{LC}}. \tag{1} \]

Reducing the capacitance of the circuit will additionally make the installation cheaper. The inductance \(L_0\) is not required in the author’s circuit, since the inductance of the choke \(ab\) is sufficient to obtain, with a small capacitance, an industrial frequency and a sinusoidal voltage at the output.

Fig. 4

In the author’s circuit there are 2 chokes, \(ab\) and \(bd\), between which there is only an electrical connection; this eliminates the magnetic influence of the left-hand part on the operating process of the right-hand part of the circuit. The magnetic independence of the halves of the circuit, on the one hand, simplifies the phenomena occurring in the circuit and, on the other hand, leads to a more sinusoidal form of the voltage curve.

In addition, the circuit of Fig. 1 provides for smooth regulation of the frequency of the output voltage over wide limits.

Calculation of the capacitor capacitance

As is known, ionic controlled valves have the following properties: after the extinction of the arc, the grid of the valve restores its controlling action not instantaneously, but after a certain interval of time determined by the process of deionization of the interelectrode space. This interval of time is usually called the deionization time and is denoted by \(t_0\). The existence of the deionization time makes it necessary not only to extinguish the conducting valve, but also, for a certain interval of time, to prevent its re-ignition until the grid of the valve has restored its blocking action.

Since the anode–cathode voltage of the extinguishing valve is determined by the voltage on the commutating capacitance and because, at the moment of commutation and for some subsequent interval of time, the anode is negative with respect to the cathode, conditions are created that prevent re-ignition. Re-ignition becomes possible only after the voltage on the commutating capacitance passes through zero. By this moment in time the grid must have restored its blocking action.

Thus, if the interval of time, counted from the moment of commutation, during which the voltage on the commutating capacitance reaches zero, is greater than the deionization time, then reliable commutation will be ensured. In Fig. 5 the voltage curve on the commutating capacitance is presented, which explains what has been said.

The capacitance of the capacitor must be such as to compensate the lag of the current caused by the inductance of the circuit and, in addition, it must ensure a current lead of not less than \(t_0\). The capacitance of the converter,

Fig. 5

for which the natural frequency of the circuit is made equal to the intended frequency, proves to be no less than that which is necessary for creating the lead time.

Fig. 6

As applied to the circuit of Fig. 6, we can obtain the following. The conductance of the capacitor is \(b_c=\omega_0 C\), and the active conductance of the circuit is

\[ g=\frac{r}{r^2+\omega_0^2 L^2}=\frac{r}{z^2}. \]

The inductive conductance of the same circuit is

\[ b=\frac{\omega_0 L}{r^2+\omega_0^2 L^2}. \]

The total conductance of branch \(efgk\)

\[ \gamma^{1}=\sqrt{\frac{r^{2}}{z^{4}}+\left(\frac{\omega_{0}Cz^{2}-\omega_{0}L}{z^{2}}\right)^{2}}= \]

\[ =\frac{1}{z^{2}}\sqrt{r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)^{2}}. \]

The active resistance of branch \(efgk\)

\[ r^{1}=\frac{rz^{2}}{\left[r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)^{2}\right]}=\frac{g}{\gamma^{2}}. \]

The inductive resistance of circuit \(efgk\)

\[ x'_{1}=\left(\omega_{0}C-\frac{\omega_{0}L}{z^{2}}\right):\frac{r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)}{z^{4}}= \]

\[ =\frac{(\omega_{0}Cz^{2}-\omega_{0}L)z^{2}}{r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)^{2}}. \]

The total resistance of the oscillatory circuit

\[ z=\sqrt{(r'+r_{1})^{2}+(x'+x_{1})^{2}}. \]

The phase-shift angle of the branch \(\varphi\) is determined from the formula:

\[ \operatorname{tg}\varphi=\frac{x'}{r^{1}}= \frac{\dfrac{(\omega_{0}Cz^{2}-\omega_{0}L)z^{2}}{r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)^{2}}} {\dfrac{rz^{2}}{r^{2}+(\omega_{0}Cz^{2}-\omega_{0}L)^{2}}} = \frac{\omega_{0}Cz^{2}-\omega_{0}L}{r}. \tag{2} \]

The current of the circuit must lead the voltage; therefore

\[ \operatorname{arc\,tg}\frac{\omega_{0}Cz^{2}-\omega_{0}L}{r}=(\varphi+\varphi'), \quad \text{where } \varphi' \text{ is the lead angle.} \]

Consequently, the capacitance of the circuit

\[ C=\frac{\operatorname{tg}(\varphi-\varphi')r+\omega_{0}L}{\omega_{0}z^{2}}. \]

Analytical study of converter operation

1. Active load

With reference to Fig. 1, when the left anode is operating, we can write

\[ i_{1}r_{1}+L_{1}pi_{1}+\frac{i_{c}}{pC}=u=i_{2}r_{1}+L_{1}pi_{2}, \tag{3} \]

where

\[ i_{1}=i_{c}+i;\quad \frac{i_{c}}{pC}=ir, \tag{4} \]

whence

\[ i_{c}=pirC. \tag{5} \]

Substituting the value of $i_c$ into equation (4), we obtain

\[ i_1=pirC+i. \tag{6} \]

Replacing $i_1$ in equation (3) by its value according to equation (6), we shall have:

\[ pirCr_1+ir_1+L_1p^2irC+L_1pi+ir=0 \]

or

\[ p^2L_1rCi+pi(L_1+rCr_1)+i(r_1+r)=u. \]

Dividing out the factor at $p^2$, we shall have:

\[ pi+pi\,\frac{L+rr_1C}{LrC}+i\,\frac{r_1+r}{L_1rC}=\frac{u}{L_1rC} \]

or

\[ p^2i+a_1pi+a_0i=k, \]

where

\[ a_1=\frac{L_1+rr_1C}{L_1rC},\qquad a_0=\frac{r_1+r}{L_1rC};\qquad k=\frac{u}{L_1rC}. \]

The image of this equation will be:

\[ p^2i(p)-pi_0-pi'_0+a_1pi(p)-a_1pi_0+a_0i(p)=k, \]

whence

\[ i(p)=\frac{k+p^2i_0+pi_0+a_1pi_0}{p^2+a_1p+a_0}. \]

The characteristic equation

\[ p^2+a_1p+a_0=0 \]

has two roots

\[ p_1=-b+j\omega_0\quad \text{and}\quad p_2=-b-j\omega_0, \]

where

\[ b=\frac{a_1}{2}=\frac{L_1+r_1rC}{2L_1rC}=\frac{1}{2rC}+\frac{r_1}{2L_1}, \]

\[ \omega_0=\sqrt{a_0-\frac{a_1^2}{4}} =\sqrt{\frac{r_1+r}{L_1rC}-\frac{1}{4}\left(\frac{L_1+r_1rC}{L_1rC}\right)^2}. \tag{7} \]

The sought current $i$ is determined by a linear differential equation of the second order. Its solution will be the following expression:

\[ i=A_0+A_1e^{p_1t}+A_2e^{p_2t}. \tag{8} \]

We shall obtain the value of the coefficient $A_0$ if, in equation (8), we single out the constant quantity, for which it is sufficient to set $P$ equal to zero,

\[ A_0=\frac{k}{a_0}=\frac{uL_1rC}{L_1rC(r_1+r)}=\frac{u}{r_1+r}. \]

The coefficients $A_1$ and $A_2$ must satisfy the condition that, at the instants $t_0=0$ and $t=T$, the current $i$ and its derivative with respect to time must respectively have equal and opposite values.

\[ A)\quad A_0+A_1+A_2=-(A_0+A_1e^{p_1T}+A_2e^{p_2T}), \]

\[ B)\quad A_1p_1+A_2p_2=-(A_1p_1e^{p_1T}+A_2p_2e^{p_2T}). \]

Multiplying equation \(A\) by \(P_2\) and subtracting equation \(B\) from it, we obtain

\[ 2A_0p_2-A_1(p_1-p_2)=A_1e^{p_1T}(p_1-p_2) \]

or

\[ 2A_0p_2=A_1(p_1-p_2)(1+e^{p_1T}), \]

whence

\[ \begin{aligned} A_1 &=\frac{2A_0p_2}{(p_1-p_2)(1+e^{p_1T})} =-\frac{2A_0(b+j\omega_0)} {(-b+j\omega_0+b+j\omega_0)(1+e^{p_1T})} = \\ &=-\frac{2A_0(b+j\omega_0)}{2j\omega_0(1+e^{p_1T})} = \\ &=-\frac{A_0(b+j\omega_0)} {j\omega_0[1+e^{-bT}(\cos\omega_0T+j\sin\omega_0T)]} = \\ &=-\frac{A_0\sqrt{b^2+\omega_0^2}\,e^{\psi}} {j\omega_0\sqrt{1+2e^{-bT}\cos\omega_0T+e^{-2bT}\cos^2\omega_0T+e^{-2bT}\sin^2\omega_0T}\,e^{\varphi}} = \\ &=-\frac{A_0\sqrt{b^2+\omega_0^2}\,e^{\psi}} {j\omega_0\sqrt{1+2e^{-bT}\cdot\cos\omega_0T+e^{-2bT}}} = \\ &=-\frac{A_0\sqrt{b^2+\omega_0^2}\,e^{j(\psi-\varphi)}} {j\omega_0\sqrt{1+2e^{-bT}\cdot\cos\omega_0T+e^{-2bT}}}, \end{aligned} \]

where

\[ \operatorname{tg}\psi=\frac{\omega_0}{b}; \qquad \operatorname{tg}\varphi=\frac{e^{-bT}\cdot\sin\omega_0T} {1+e^{-bT}\cdot\cos\omega_0T}. \]

Similarly we find

\[ A_2= \frac{A_0\sqrt{b^2+\omega_0^2}\cdot e^{-j(\psi-\varphi)}} {j\omega_0\sqrt{1+2e^{-bT}\cos\omega_0T+e^{-2bT}}}. \]

Substituting the values \(A_0\), \(A_1\), and \(A_2\) into equation (8), we obtain

\[ i(t)=\frac{u}{r_1+r} \left[ 1-\frac{2\sqrt{b^2+\omega_0^2}\,e^{-bt}\cdot\sin(\omega_0t+\psi-\varphi)} {\omega_0\sqrt{1+2e^{-bT}\cos\omega_0T+e^{-2bT}}} \right]. \tag{9} \]

The voltage at the terminals of the resistance will be

\[ u(t)=i(t)r-u_c(t). \tag{10} \]

2. No-load operation

The circuit for this case of operation is shown in Fig. 7.

The phenomena in the circuit are described by the equation

\[ i_1r_1+L_1pi_1+\frac{i_1}{pC}=u. \]

The transform of this equation is:

\[ r_1i(p)-L_1pi_0+L_1pi_1(p)+\frac{i(p)}{Cp}+u_0=u, \]

whence

\[ i_c(p)=i_1(p)=\frac{L_1 p i_0-u-u_0+u}{r_1+L_1p+\dfrac{1}{Cp}} = \]

\[ =\frac{p(L_1 p i_0-u_0+u)}{L_1(p^2+a_1p+a_0)}, \]

Fig. 7

where

\[ a_1=\frac{r_1}{L_1}; \qquad a_0=\frac{1}{L_1C}. \]

The voltage at the capacitor terminals is

\[ u_c(p)=\frac{i_1(p)}{pC}= \frac{L_1 p i_0-u_0+u}{L_1C\left(p^2+\dfrac{r_1}{L_1}p+\dfrac{1}{L_1C}\right)}. \tag{11} \]

The characteristic equation has two roots:

\[ p_1=-b+j\omega_0 \quad \text{and} \quad p_2=-b-j\omega_0, \]

where

\[ b=\frac{a_1}{2}=\frac{r_1}{2L_1} \qquad \text{and} \qquad \omega_0=\sqrt{\frac{1}{L_1C}-\frac{r_1^2}{4L_1^2}}. \]

The voltage at the capacitor terminals is determined by a linear differential equation of the second order.

Its solution is

\[ u_c=A_0+A_1\cdot e^{p_1t}+A_2e^{p_2t}. \tag{12} \]

We obtain the value of the coefficient \(A_0\) if, in expression (12), we set \(p\) equal to zero:

\[ A_0=u-u_0. \]

The coefficients \(A_1\) and \(A_2\) must satisfy the condition that at the instants \(t=0\) and \(t=T\) the voltage and its time derivative must respectively have equal and opposite values:

\[ \begin{aligned} \text{a)}\quad &A_0+A_1+A_2=-(A_0+A_1e^{p_1T}+A_2e^{p_2T}),\\ \text{b)}\quad &A_1p_1+A_2p_2=-(A_1p_1e^{p_1T}+A_2p_2e^{p_2T}). \end{aligned} \tag{13} \]

Multiplying equation (a) by \(P_2\) and subtracting equation (b) from it, we obtain:

\[ A_1=-\frac{A_0\sqrt{b^2+\omega_0^2}\,e^{(\psi-\varphi)}}{-j\omega_0\sqrt{1+2e^{-bT}\cdot\cos\omega_0T+e^{-2bT}}}. \]

Here

\[ \tg\psi=\frac{\omega_0}{b};\qquad \tg\varphi=\frac{e^{-bT}\cdot\sin\omega_0T}{1+e^{-bT}\cdot\cos\omega_0T}. \]

Similarly we find

\[ A_2=\frac{A_0\sqrt{b^2+\omega_0^2}\,e^{(\psi-\varphi)}}{j\omega_0\sqrt{1+2e^{-bT}\cdot\cos\omega_0T+e^{-2bT}}}. \]

Substituting the values of \(A_0\), \(A_1\), and \(A_2\) into equation (12), we obtain

\[ u_c(t)=(u-u_0)\cdot \left[ 1- \frac{2\sqrt{b^2+\omega_0^2}\,e^{-bt}\cdot\sin(\omega_0t+\psi-\varphi)} {\omega_0\sqrt{1+2e^{-bT}\cdot\cos\omega T+e^{-2bT}}} \right]. \tag{14} \]

3. Inductive Load

Since the duration of the processes in the inverter is within the limits of only one half-period, the attenuation of the current caused by the active resistances of the circuit \(r\) and \(r_1\), when they are small, may be neglected.

With reference to the circuit of Fig. 6, for \(r=0\) and \(r_1=0\) we may write:

\[ i_1=i_c+i, \tag{15} \]

\[ L_1pi_1+Lpi=0, \tag{16} \]

\[ \frac{i_c}{pC}=Lpi, \]

whence

\[ i=\frac{i_c}{p^2LC}. \]

Substituting the value of \(i\) into equation (15), we obtain:

\[ i_1=i_c+\frac{i_c}{p^2LC}. \]

Replacing \(i_1\) in equation (16) by its value, we shall have:

\[ L_1\frac{pi_c}{p^2LC}+L_1pi_c+L\frac{pi_c}{p^2LC}=u \]

or

\[ \frac{L_1+L}{pLC}\,i_c+L_1pi_c=u. \]

The transform of the equation is

\[ L_1pi_c(p)-L_1pi_0+\frac{i_c(p)}{p\frac{LC}{L_1+L}} =-u_0+u=u-u_0, \]

whence

\[ i_c(p)= \frac{p(u-u_0)+L_1pi_0} {L_1\left(p^2+\frac{L_1+L}{L_1LC}\right)}, \]

Inverse transformation

\[ u_{(t)}=\frac{k}{Z}\cdot \frac{p(p_k)}{Q'(p_k)} . \tag{17} \]

Here

\[ p(p_k)=(u-u_0+L_1pi_0)e^{-pt} \]

\[ Q=L_1\left(p^2+\frac{L_1+L}{L_1LC}\right);\qquad Q'=2pL_1 . \]

The function \(Q\) has 2 poles

\[ p_{1-2}=\pm j\sqrt{\frac{L_1+L}{L_1LC}}=\pm j\omega_0 . \]

Substituting the values of the roots, we obtain:

\[ p(p_{k_1})=(u-u_0+j\omega_0L_1i_0)e^{-j\omega_0t}, \]

\[ p(p_{k_2})=(u-u_0-j\omega_0L_1i_0)e^{j\omega_0t}, \]

\[ Q'(p_1)=2j\omega_0L_1\qquad Q'(p_2)=-2j\omega_0L . \]

Substituting the values found into equation (17), we shall have:

\[ \begin{aligned} i_c(t) &=\frac{(u-u_0+j\omega_0L_1i_0)e^{j\omega_0t}}{2j\omega_0L_1} +\frac{(u-u_0-j\omega_0L_1i_0)e^{-j\omega_0t}}{-2j\omega_0L_1} =\\ &=\left(\frac{u-u_0}{2j\omega_0L_1}+\frac{i_0}{2}\right)e^{j\omega_0t} -\left(\frac{u-u_0}{2j\omega_0L}+\frac{i_0}{2}\right)e^{-j\omega_0t} =\\ &=i_0\cos \omega_0t-\frac{u-u_0}{\omega_0L_1}\sin \omega_0t . \end{aligned} \]

Experimental part

In the electrical-machines laboratory of the TPI, an installation was assembled consisting of two reactors. The active resistance of the reactor was \(r_1=7.8\) ohms,

Fig. 8

the inductive reactance \(x_1=100\) ohms; \(x_1\) was determined by dividing the normal voltage by the magnetizing-current strength; the capacitance of the capacitor was \(C=40\,\mu F\).

Oscillographic records of the voltage curves at the capacitor terminals were made for 3 different load cases. The curves are given in Figs. 8–10; in Fig. 8 the oscillogram of the capacitor voltage is for no-load operation.

Fig. 9

In Fig. 9, for an active load \(r = 1000\ \Omega\); in Fig. 10, for a mixed load for which \(r = 2\ \Omega,\ x = 170\ \Omega\).

Fig. 10

Conclusions

According to the author’s circuit, a sinusoidal output voltage of the converter can be obtained if the parameters of the circuit are selected appropriately. The capacitance of the capacitor in the author’s circuit is smaller than the capacitance in the circuits of other authors. The processes occurring in the author’s converter are described by second-degree equations, except for the case when the load is mixed. In describing the processes in the author’s converter, all factors are taken into account. In a converter with a transformer, the magnetizing current and the influence of leakage fields are not taken into account when describing the processes. The author’s attempt to take into account the influence of active resistances on the operation of the converter under mixed load led to third-degree equations. Owing to their unwieldiness, they have no practical significance.