On Self-Excitation of a Brushless Electromechanical Amplifier with Adjustable Frequency
Abstract Generated abstract
This paper examines self-excitation in a brushless electromechanical amplifier with adjustable output frequency, focusing on loss of voltage control caused by current through return diodes exceeding the controlled semiconductor-switch current. The authors formulate steady-state circuit equations for the control, load, and rotor circuits, incorporate the effect of a demagnetizing or bias winding on the magnetizing impedance, and derive a condition for operation without self-excitation. Using representative machine parameters, they calculate self-excitation boundaries for no-load operation under different rotor resistance regulation and capacitive compensation schemes. The analysis indicates that stable operation over a wide frequency range, especially with simultaneous stator and rotor capacitive compensation, requires negative feedback, which the authors propose to provide by a bias winding.
Full Text
ON THE SELF-EXCITATION OF A COMMUTATORLESS ELECTRIC MACHINE AMPLIFIER WITH ADJUSTABLE FREQUENCY
Sh. S. Roiz, A. I. Skorospeleshkin
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
A commutatorless electric machine amplifier with adjustable frequency (Fig. 1) is an electromechanical–semiconductor complex consisting of a controlled semiconductor commutator and an electric-machine part.
Fig. 1. Circuit of a commutatorless electric machine amplifier with adjustable frequency.
The CSC in the circuit (Fig. 1) is represented by semiconductor switches \(T_1, T_2 \ldots T_6\) and return diodes \(D_1, D_2 \ldots D_6\) in the control circuit. The electric-machine part of the amplifier consists of an asynchronous generator (windings \(W_2, W_3\)) and the control winding \(W_1\). The rotor of the amplifier is rotated by means of a drive motor.
The CSC, by ensuring a regular sequence of connection and disconnection of the phases of the control winding to a direct-current source, creates in the amplifier a rotating control magnetic flux. The control flux induces an emf in the output winding \(W_2\) and in the rotor winding \(W_3\).
By changing the switching frequency of the phases of the control winding, it is possible to regulate the frequency at the amplifier output practically from several hertz to
maximum frequency, determined by the number of poles of the asynchronous generator \(p_1\) and the rotational speed of the rotor \(n\).
\[ f_{\text{out}}=p_1 n. \tag{1} \]
To ensure generator operation, the rotor of the asynchronous generator must rotate in the same direction as the excitation field, and the rotor speed must be greater.
Operating with negative slip, the generator delivers active power, supplied from the shaft, to the load and to the control winding \(W_1\), owing to which the control power is sharply reduced.
When a commutatorless EMU operates in the rated mode, self-excitation of the amplifier may occur as a result of parametric resonance. In this case the amplifier loses frequency control. Stable operation of the amplifier is maintained at a gain factor of 20–200. Higher values of the gain factor are limited by the accuracy of capacitor tuning, load instability, and temperature factors [1].
At a certain value of slip, load resistance, and active resistance in the rotor circuit, the amplifier may also lose control with respect to the output voltage. This is due to the fact that the current \(I_y''\), flowing through the return diodes, may turn out to be greater than the current \(I_y'\), flowing through the semiconductor switches. The resulting control current in this case may be equal to zero or take negative values, although the output emf \(E_2\) differs significantly from zero. We have chosen to regard such a mode as a self-excitation mode.
In order to eliminate self-excitation, we propose placing a demagnetizing winding \(W_{\text{po}}\) on the back of the amplifier stator and connecting it in series either into the dc circuit of the UPC or into the circuit of the return diodes. The purpose of the return diodes in the UPC circuit is that, when the amplifier operates in no-load mode or under a small load, the resultant magnetic flux induces an emf in the control winding. The alternating current flowing through the control winding is rectified by the return diodes and enters the dc network. The presence of return diodes in the UPC circuit has a favorable effect on the operation of the UPC, reducing the magnitude of commutation overvoltages on the power elements of the UPC.
Let us investigate the boundaries of self-excitation when the amplifier operates over the entire partial range. For this purpose we shall compose a system of equations for the commutatorless EMU in the steady-state mode. In setting up the equations, the following assumptions are adopted:
- Sinusoidal currents flow through the windings \(W_1, W_2, W_3\).
- The control winding \(W_1\) is regarded as an \(m\)-phase alternating-current winding supplied from an alternating-current source.
- Losses in the steel are not taken into account.
Taking the adopted assumptions into account, we write the following equations for the amplifier circuits:
For the control circuit
\[ \dot U_1'=\dot I_{01}' Z_{01}(H_{\text{п}})+\dot I_1'(r_1'+j x_{\sigma 1}'), \tag{2} \]
where
\[ \dot U_1'=\dot U_1 K_{u1},\quad \dot I_1'=\dot I_1 \frac{1}{K_{i1}},\quad r_1'=r_1 K_i,\quad x_{\sigma 1}'=x_{\sigma 1}K_1, \]
\[ K_{u1}=\frac{W_2 K_{w2}}{W_1 K_{w1}} \quad \text{— voltage reduction coefficient of winding } W_1; \]
\[ K_{i1}=\frac{m_2 W_2 K_{w2}}{m_1 W_1 K_{w1}} \quad \text{— current reduction coefficient of winding } W_1; \]
\[
K_1=\left(\frac{W_2K_{W_2}}{W_1K_{W_1}}\right)^2\frac{m_2}{m_1}
\]
is the coefficient for reducing the resistances of winding \(W_1\).
\(Z_{01}(H_\mu)\) is the resistance of the magnetizing circuit, taking into account the influence of the auxiliary magnetizing winding.
Note: The parameters of windings \(W_1\) and \(W_3\) are reduced to the output winding \(W_2\).
For the load circuit,
\[ 0=\dot I_{01}Z_{01}(H_\mu)+\dot I_2\left[(r_2+R_2)+j(x_{\delta 2}-x_2)\right], \]
where
\[ R_2=\frac{R_{\mathrm{н}}x_{c2}(x_{c2}-x_{\mathrm{н}})-R_{\mathrm{н}}x_{c2}x_{\mathrm{н}}} {R_{\mathrm{н}}^2+(x_{c2}-x_{\mathrm{н}})^2}, \]
\[ x_2=\frac{R_{\mathrm{н}}^2x_{c2}-x_{c2}(x_{c2}-x_{\mathrm{н}})x_{\mathrm{н}}} {R_{\mathrm{н}}^2+(x_{c2}-x_{\mathrm{н}})^2} \]
\[
Z_{\mathrm{н}}=R_{\mathrm{н}}+jx_{\mathrm{н}}
\]
is the total load impedance.
For the rotor circuit,
\[ 0=\dot I_{01}Z_{01}(H_\mu)+\dot I'_3\left[\left(\frac{r'_3}{S}+R_{\mathrm{э}3}\right)+j(x_{\sigma 3}-x_{\mathrm{э}3})\right], \tag{4} \]
where
\[ R_{\mathrm{э}3}=\frac{R_3^{\prime 1\,2}x_{c3}^{\prime 2}} {S^3\left[R_3^{1\,2}+\left(\frac{x'_{c3}}{S}\right)^2\right]}, \]
\[ x_{\mathrm{э}3}=\frac{R_3^{1\,2}x'_{c3}} {S^2\left[R_3^{1\,2}+\left(\frac{x'_{c3}}{S}\right)^2\right]}, \]
\[ \dot I'_3=I_3\frac{1}{K_{i3}},\qquad r'_3=r_3K_3,\qquad x'_{\sigma 3}=x_{\sigma 3}K_3,\qquad x'_{c3}=x_{c3}K_3,\qquad R'_3=R_3K_3, \]
\[
K_{i3}=\frac{m_2W_2K_{W_2}}{m_3W_3K_{W_3}}
\]
is the coefficient for reducing the current of winding \(W_3\);
\[
K_3=\frac{m_2}{m_3}\left(\frac{W_2K_{W_2}}{W_3K_{W_3}}\right)^2
\]
is the coefficient for reducing the resistances of winding \(W_3\);
\[
S=\frac{\omega_y-\omega}{\omega_y}
\]
is the slip of the asynchronous generator,
\[ \dot I_{01}=\dot I'_1+\dot I_2+\dot I'_3. \]
The relation between the CVFC and the control winding \(W_1\) is written in the form
\[ U_1=K_uU_y, \tag{5} \]
where \(K_u\) is the voltage transfer coefficient in the presence of return diodes in the CVFC circuit;
\(U_y\) is the voltage of the direct-current network at the CVFC input,
\[ I_y=I'_y+I''_y, \tag{6} \]
where \(I'_y=K_I I_{1p}\) is the control current flowing through the semiconductor switches, required to create the magnetic flux;
\(I''_y=K_I I_{1a}\) is the control current flowing through the return diodes, caused by the active component of the rotor current \(I_3\);
\(K_I\) is the current transfer coefficient in the presence of return diodes in the CVFC circuit.
The parameter \(Z_{01}(H_{\mathrm{p}})\) in the equivalent circuit (Fig. 2), constructed from equations (1)—(6), is equal to
\[ Z_{01}(H_{\mathrm{p}})=\frac{E_2}{I_{01}(H_{\mathrm{p}})} \tag{7} \]
In the absence of current in the bias winding,
\[ Z_{01}(H_{\mathrm{p}})=Z_{01}=\frac{E_2}{I_{01}} . \tag{8} \]
We shall express the values of the currents \(I_{01}(H_{\mathrm{p}})\) and \(I_{01}\) in terms of the m.m.f. of the winding \(W_2\),
\[ I_{01}(H_{\mathrm{p}})=\frac{\Sigma F_2(H_{\mathrm{p}})}{0.9\,m_2 W_2 K_{W2}}, \tag{9a} \]
\[ I_{01}=\frac{\Sigma F_2}{0.9\,m_2 W_2 K_{W2}}, \tag{9б} \]
Fig. 2. Equivalent circuit of a commutatorless EMA with adjustable frequency.
where \(\Sigma F_2(H_{\mathrm{p}})\) is the total m.m.f. of the magnetic circuit taking into account the current of the bias winding;
\(\Sigma F_2\) is the total m.m.f. of the magnetic circuit without taking into account the bias winding.
Solving (9a) and (9б) jointly, we obtain
\[ I_{01}(H_{\mathrm{p}})=I_{01}\frac{\Sigma F_2(H_{\mathrm{p}})}{\Sigma F_2}. \tag{10} \]
Taking (10) into account, the parameter \(Z_{01}\) is equal to
\[ Z_{01}(H_{\mathrm{p}})=Z_{01}\frac{\Sigma F_2(H_{\mathrm{p}})}{\Sigma F_2}. \tag{11} \]
The total values of the m.m.f. \(\Sigma F_2(H_{\mathrm{p}})\) and \(\Sigma F_2\), expressed in terms of the m.m.f. of the sections of the magnetic circuit, are equal to
\[ \Sigma F_2(H_{\mathrm{p}})=F_{01}+F_{zp}+F_{zc}+F_{jp}+F'_{jc}, \tag{12a} \]
\[ \Sigma F_2=F_{01}+F_{zp}+F_{zc}+F_{jp}+F_{jc}, \tag{12б} \]
where
\[
F_{01}
\]
is the m.m.f. of the air gap;
\[
F_{zp}\ \text{and}\ F_{zc}
\]
are the m.m.f. of the rotor and stator teeth;
\[
F_{jp}
\]
is the m.m.f. of the rotor yoke;
\[
F_{jc}
\]
is the m.m.f. of the stator yoke without taking into account the bias winding;
$F'_{jc}$ is the m.m.f. of the stator back with allowance for the premagnetizing winding.
Taking (12a) and (12b) into account, the parameter $Z_{01}(H_{\mathrm{п}})$ is equal to
\[ Z_{01}(H_{\mathrm{п}})=Z_{01}K_{\mathrm{п}}, \tag{13} \]
where
\[ K_{\mathrm{п}}=\frac{1}{1+\dfrac{F'_{jc}}{\sum F_2}} \]
is a coefficient allowing for the influence of the premagnetizing winding.
The magnetizing force $F'_{jc}$ is determined from the expression
\[ F'_{jc}=H'_{jc}l_{jc}\xi, \tag{14} \]
where $H'_{jc}$ is the rms value of the field intensity in the stator back under simultaneous magnetization by direct and alternating current;
$l_{jc}$ is the length of the magnetic line of force of the stator back;
$\xi$ is a coefficient allowing for the nonuniformity of the flux along the magnetic line of force.
The determination of $H_{jc}$ is carried out by the known method (2), where the instantaneous value of the field intensity in the stator back is equal to
\[ H'_{jc}=\alpha \operatorname{Sh}\beta\left(B_{\mathrm{п}}+B_{\max}\sin\omega t\right), \tag{15} \]
where $\alpha$ and $\beta$ are coefficients of approximation of the steel;
$B_{\mathrm{п}}$ is the constant component of induction;
$B_{\max}$ is the amplitude value of the alternating component of induction.
The rms value of the field intensity $H'_{jc}$ is determined by the formula
\[ H'_{jc}=1.41\sqrt{(1+\varepsilon_{\mathrm{п}})^2\left[I_{2n+1}(jb_{\mathrm{м}})\right]^2+\varepsilon_{\mathrm{п}}^2\left[I_{2n}(jb_{\mathrm{м}})\right]^2}, \tag{16} \]
where
\[ \varepsilon_{\mathrm{п}}=\frac{h_{\mathrm{п}}}{I_c(jb_{\mathrm{м}})},\qquad h_{\mathrm{п}}=\frac{H_{\mathrm{п}}}{\alpha},\qquad b_{\mathrm{м}}=\beta B_{\max}, \]
$H_{\mathrm{п}}$ is the magnetic-field intensity in the stator back due to the current of the premagnetizing winding,
$I_n$ are Bessel functions of order $n$ of a purely imaginary argument $(jb_{\mathrm{м}})$.
The values of the Bessel functions are determined from a table for modified functions [3].
The amplifier operates without self-excitation if the condition is satisfied
\[ I_y=I'_y-I''_y>0 \tag{17a} \]
or
\[ \left|\dot I'_{1\mathrm{p}}\right|-\left|\dot I'_{1\mathrm{a}}\right|>0. \tag{17b} \]
The components of the currents $\dot I'_{1\mathrm{p}}$ and $\dot I'_{1\mathrm{a}}$ are equal to
\[ \dot I'_{1\mathrm{p}}=I_{01}+\dot I'_{3\mathrm{p}}+\dot I_{2\mathrm{p}}, \tag{18a} \]
\[ \dot I'_{1\mathrm{a}}=\dot I_{2\mathrm{a}}+\dot I'_{3\mathrm{a}}, \tag{18b} \]
where $\dot I'_{3\mathrm{p}}$ and $\dot I'_{3\mathrm{a}}$ are the reactive and active components of the rotor current $\dot I'_3$;
$\dot I_{2\mathrm{p}}$ and $\dot I_{2\mathrm{a}}$ are the reactive and active components of the load current $\dot I_2$.
Substituting the values of the currents $\dot I'_{1\mathrm{p}}$ and $\dot I'_{1\mathrm{a}}$, expressed through the parameters
amplifier, in (17), we obtain
\[ \left| \frac{1}{x_{01}(H_{\mathrm p})} + \frac{x_{\sigma 2}-x_2}{(r_2+R_2)^2+(x_{\sigma 2}-x_2)^2} + \frac{x'_{\sigma 3}-x_{\mathrm e3}} {\left(\dfrac{r'_3}{S}+H_{\mathrm e3}\right)^2+(x'_{\sigma 3}-x_{\mathrm e3})^2} \right| + \left| \frac{r_2+R_2}{(r_2+R_2)^2+(x_{\sigma 2}-x_2)^2} + \frac{\dfrac{r'_3}{S}+R_{\mathrm e3}} {\left(\dfrac{r'_3}{S}+R_{\mathrm e3}\right)^2+(x'_{\sigma 3}-x_{\mathrm e3})^2} \right| \ge 0 . \tag{19} \]
Equation (19) is the condition for the absence of self-excitation of the amplifier under any load.
The most dangerous operating condition from the standpoint of self-excitation is the no-load condition.
Fig. 3.
Let us investigate the operating regions of the amplifier in the no-load condition in which self-excitation with respect to the amplitude of the output voltage is absent. The calculation of the regions was carried out for the following parameter values: \(r_2=7.8\ \Omega\), \(L_{\sigma 2}=21.4\ \mathrm{mH}\), \(r_3=1.5\ \Omega\),
\[ L_{\sigma 3}=31.8\ \mathrm{mH},\quad L_{01}=120\ \mathrm{mH},\quad C_2=6\ \mu\mathrm{F},\quad C_3=30\ \mu\mathrm{F}. \]
Mode 1. No-load operation. There is no additional resistance in the rotor circuit \((R'_3=0)\).
The condition for the absence of self-excitation of the amplifier in this mode, taking (13) into account, takes the form
\[ K_{\mathrm p} < \frac{1}{ x_{01} \left\{ \left| \dfrac{1}{x_{c2}-x_{\sigma 2}} + \dfrac{x'_{\sigma 3}}{\left(\dfrac{r'_3}{S}\right)^2+x_{\sigma 3}^{\prime 2}} \right| + \left| \dfrac{\dfrac{r'_3}{S}}{\left(\dfrac{r'_3}{S}\right)^2+x_{\sigma 3}^{\prime 2}} \right| \right\} }. \tag{20} \]
When capacitive compensation \(x_{c2}\le x_{01}+x_{\sigma 2}\) is implemented at the upper frequency of the frequency range, the self-excitation region is bounded by curve 1 (Fig. 3). The region in which self-excitation is absent lies below the curve.
When capacitive compensation \(x_{c2}\ge x_{01}+x_{\sigma 2}\) is implemented over the entire frequency range, the self-excitation region is bounded by curve 2 (Fig. 3).
Analysis of this mode shows that the region of stable operation is preserved over almost the entire frequency range of operation of the amplifier.
Mode 2. No-load mode. The additional resistance in the rotor circuit is regulated according to a specified law.
The law of regulation of \(R'_3\) may be written in the form
\[ R'_3 = R_{\mathrm p}\beta - r'_3, \tag{21} \]
where
\[ R_{\mathrm p} = \frac{r'_3}{\beta_{\max}}, \]
\[ \beta_{\max} = \frac{\omega_{y\max} - \omega}{\omega_{y\max}} \quad \text{is the relative slip.} \]
Such a law ensures constancy of the rotor current \(I'_{3a}\).
When capacitive compensation \(x_{c2} \le x_{01} + x_{\sigma2}\) is implemented at the upper frequency of the frequency range, the region of self-excitation is bounded by curve 3 (Fig. 3).
When capacitive compensation
\[ \frac{x'_{c3}}{S^2} \le x_{01} + x'_{\sigma3} \]
is implemented at the lower frequency of the frequency range, the region of self-excitation is bounded by curve 4 (Fig. 3).
When simultaneous capacitive compensation is implemented on the stator and rotor sides,
\[ x_{c2} + \frac{x'_{c3}}{S^2} \le x_{01} + x_{\sigma2} + x'_{\sigma3}, \]
the region of self-excitation is bounded by curve 5 (Fig. 3).
Analysis of this mode shows that, with simultaneous capacitive compensation on the stator and rotor sides, the amplifier operates without self-excitation over a wide frequency range only when a bias winding is present.
As a result of the investigation carried out, the following conclusions may be drawn:
-
On the basis of the equations of the commutatorless EMA, a condition has been obtained for the absence of self-excitation of the amplifier during operation in no-load mode. For specific amplifier parameters, the boundaries of self-excitation have been calculated.
-
Analysis of amplifier operation in no-load mode has shown that elimination of self-excitation with respect to the amplitude of the output voltage is possible only in the presence of negative feedback. It is proposed that this feedback in the amplifier be provided by the bias winding.
REFERENCES
-
R. I. Zelichenok. Electromachine asynchronous power amplifier, Elektrotekhnika, 1966, No. 12.
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A. I. Bertinov. Aircraft Electric Generators, Oborongiz, Moscow, 1959.
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E. Jahnke, F. Emde, F. Lösch. Special Functions. Nauka, Moscow, 1968.