Energy Relationships in Commutatorless DC and AC Electric Machine Amplifiers
Abstract Generated abstract
This paper derives energy relationships for several brushless electric-machine amplifiers, including variable-frequency, phase-sensitive increased-frequency, fixed-frequency AC, and DC configurations. Using electromagnetic power, slip, torque, winding losses, iron losses, rectifier losses, and drive-motor power balances, it formulates expressions for input power, output power, drive power, efficiency, and amplification factor over different operating ranges and phase-sequence conditions. The resulting energy diagrams and equations show how power is distributed between stages and indicate comparative performance, with the phase-sensitive amplifier exceeding the adjustable-frequency type in gain and efficiency, and the fixed-frequency AC amplifier exceeding the DC version under the stated loss assumptions.
Full Text
ENERGY RELATIONS IN BRUSHLESS DC AND AC ELECTRIC MACHINE AMPLIFIERS
A. I. SKOROSPECHKIN, E. F. OBERGAN, Sh. S. ROIZ
(Recommended by the seminar of the Departments of Electrical Machines and General Electrical Engineering)
One of the principal indices of any electric-machine amplifier is the gain factor and the efficiency. To determine them it is necessary to know the powers of the input, the output, and the drive motor.
We are developing DC and AC amplifiers. The schematic diagram of a brushless variable-frequency amplifier, explaining its operating principle, is shown in Fig. 1.
Fig. 1. Schematic diagram of a variable-frequency amplifier.
Windings 1 and 2 belong to the first stage, windings 3 and 4 to the second. The number of poles of the windings of the first stage is \(2p_1\), and of the second, \(2p_2\). At the input of the first stage a semiconductor commutator (UPK) is installed, necessary for obtaining the rotating field of winding 1.
Rotation of the converter is accomplished by means of an asynchronous motor with the number of poles \(2p_1\).
The analytical dependence \(f_{\text{out}} = f(f_{\text{in}})\) of the output frequency on the input frequency is expressed by the equation
\[ f_{\text{out}} = - \frac{(p_2 \pm p_1)n}{60} \pm \frac{p_1 n_1}{60}. \tag{1} \]
The signs \((\pm)\) in the parentheses determine the order of phase alternation of windings 2 and 3.
The signs \((\pm)\) before the expression \(\dfrac{p_1 n_1}{60}\) determine the direction of rotation of the field of winding 1.
\(n\) and \(n_1\) are the rotational speeds of the driving motor and of the flux of winding 1.
Figure 2 gives the frequency characteristic; here the ratio of the numbers of pole pairs
\[ \frac{P_2}{P_1} \]
is taken as equal to 3.
Fig. 2. Frequency characteristic
\[
\left(\text{dependence } f_{\mathrm{out}} = f(f_{\mathrm{in}})\right)
\]
Dividing the entire frequency range into a number of sections \((ab, bc, cd,\) and \(de)\), as shown in Fig. 2, makes it possible to consider more simply the energy relations of the variable-frequency amplifier when it operates over the entire frequency characteristic.
The output power of the amplifier \(P_4\) is equal to:
\[ P_4 = P_{\psi 4} - P_{\mathrm{Cu}4}, \tag{2} \]
where \(P_{\psi 4}\) is the electromagnetic power of output winding 4 of the amplifier;
\(P_{\mathrm{Cu}4}\) is the copper loss in winding 4.
The electromagnetic power \(P_{\psi 4}\) for any frequency range is, in the general case, determined by the following expression:
\[ P_{\psi 4} = M\omega_4 = M(\omega + \omega_3) = P_{m2} + P_{\psi 3}, \tag{3} \]
where \(P_{\psi 3} = M \cdot \omega_3\) is the electromagnetic power of winding 3;
\[ P_{m2} = M \cdot \omega \]
is the electromagnetic power of winding 4, equal to the mechanical power on the shaft of the 2nd stage;
\(M\) is the electromagnetic torque due to the interaction of flux \(\Phi_3\) and current \(I_4\);
\(\omega\) is the angular speed of the amplifier rotor;
\[ \omega_3 = (\omega \pm \omega_1)\frac{p_1}{p_2} \]
is the angular speed of rotation of flux \(\Phi_3\) relative to the rotor;
\[ \omega_4 = \omega \pm \omega_3 \]
is the angular speed of rotation of flux \(\Phi_3\) relative to the stator.
Let us express the powers \(P_{\psi 3}\) and \(P_{m2}\) through the electromagnetic power \(P_{\psi 4}\):
\[ P_{\psi 3} = P_{\psi 4} S_4, \tag{4a} \]
\[ P_{m2} = P_{\psi 4}(1 - S_4), \tag{4b} \]
where
\[ S_4=-\frac{\omega_4-\omega}{\omega_4} =\frac{\pm\omega_2}{\omega_4} =\frac{1}{1\pm \dfrac{\omega p_2}{(\omega\pm\omega_1)p_1}} \]
is the slip of the field of winding 4 relative to winding 3.
For the amplifier under consideration, for the ranges \(ab\) and \(bc\), \(S_4<0\); for \(cd\) and \(de\), \(S_4>0\). This follows from the fact that for the ranges \(ab\) and \(bc\), \(\omega>\omega_3\).
The electromagnetic power of winding 2, taking losses into account, for the ranges \(ab\) and \(bc\) is determined by the following expression:
\[ P_{\psi 2}=P_{\psi 4}\cdot S_4+\sum p_3-p_{\mathrm{Cu}2}. \tag{5a} \]
For the ranges \(cd\) and \(de\),
\[ P_{\psi 2}=P_{\psi 4}\cdot S_4+\sum p_3+p_{\mathrm{Cu}2}, \tag{5b} \]
where
\[ \sum p_3=p_{\mathrm{Cu}3}+p_{\mathrm{Fe}3} \]
is the sum of the copper losses of winding 3 and the iron losses caused by the field \(\Phi_3\);
\(p_{\mathrm{Cu}2}\) is the copper loss of winding 2.
The difference between expressions (5a) and (5b) is that, when the amplifier operates in the ranges \(ab\) and \(bc\), winding 2 is a load with respect to winding 3, whereas in the ranges \(cd\) and \(de\) the reverse is true.
The electromagnetic power \(P_{\psi 2}\) of winding 2 is equal to the product of the electromagnetic torque \(M_1\), caused by the interaction of the flux \(\Phi_1\) and the current \(I_2\), and the angular speed of rotation \(\omega_2\) of the flux \(\Phi_1\) relative to winding 2:
\[ P_{\psi 2}=M_1\cdot\omega_2=M_1(\omega_1+\omega)=P_{\psi 1}+P_{m1}, \tag{6} \]
where
\[ P_{\psi 1}=M_1\cdot\omega_1 \]
is the electromagnetic power of winding 1;
\[ P_{m1}=M\cdot\omega \]
is the electromagnetic power of winding 2, equal to the mechanical power on the shaft of the first stage.
Let us express the powers \(P_{\psi 1}\) and \(P_{m1}\) through the electromagnetic power \(P_{\psi 2}\):
\[ P_{\psi 1}=P_{\psi 2}\cdot\frac{1}{S_1}, \tag{7a} \]
\[ P_{m1}=P_{\psi 2}\left(1-\frac{1}{S_1}\right), \tag{7b} \]
where
\[ S_1=\frac{\omega_2}{\omega_1} =\frac{\omega_1\pm\omega}{\omega_1} \]
is the slip of the field \(\Phi_1\) relative to winding 2.
For the ranges \(bc\) and \(cd\), \(S_1<0\); for the ranges \(ab\) and \(de\), \(S_1>0\). This is true for the amplifier under consideration, since \(\omega>\omega_1\).
Using expressions 2, 3, 4, 3a, 5b, 6, 7a, 7b, we construct the energy diagrams of the amplifier for each of the ranges (\(ab\), \(bc\), \(cd\), and \(de\)).
In Fig. 3 (a, b, c, d) the energy diagrams of the amplifier are presented for its operation in the ranges \(ab\), \(bc\), \(cd\), and \(de\).
The energy diagrams clearly show the distribution of powers in the amplifier, the exchange of energy between stages, and the direction of the powers in one stage or another.
For calculating the input powers \(P_1\) and those of the drive motor \(P_{\mathrm{qv}}\), their expressions have been obtained for each range.
For the range \(ab\),
\[ P_1=P_{\psi 4}\left[ -\frac{\omega_1 p_1}{\omega(p_2-p_1)-\omega_1 p_1} \right] +\left(\sum p_3-p_{\mathrm{Cu}2}\right)\frac{\omega_1}{\omega+\omega_1} +p_{\mathrm{Fe}1}+p_{\mathrm{Cu}1}, \tag{8a} \]
Fig. 3. Energy diagrams of a variable-frequency amplifier
\(p_{\mathrm{Cu}1}\) — losses in the copper of winding 1,
\(p_{\mathrm{Fe}1}\) — losses in the steel caused by rotation of the field \(\Phi_1\),
\[ P_{\mathrm{dv}} = P_{m1} + P_{m2} + p_{\mathrm{Fe}4} + p_{\mathrm{Fe}2} + p_{\mathrm{mx}} = \]
\[ = P_{\psi4}\left[1+\frac{\omega_1 p_1}{\omega (p_2-p_1)-\omega_1 p_1}\right] + \left(\sum p_3 - p_{\mathrm{Cu}2}\right)\frac{\omega}{\omega+\omega_1} + \]
\[ + p_{\mathrm{Fe}4} + p_{\mathrm{Fe}2} + p_{\mathrm{mx}}, \tag{8б} \]
where \(p_{\mathrm{Fe}2}, p_{\mathrm{Fe}4}\) are losses in the steel, covered at the expense of the driving motor.
For the range \(bc\)
\[ P_1 = P_{\psi4}\left[ \frac{\omega_1 p_1}{\omega (p_2-p_1)+\omega_1 p_1} \right] + \left(\sum p_3 - p_{\mathrm{Cu}2}\right) \frac{\omega_1}{\omega_1-\omega} + \]
\[ + p_{\mathrm{Fe}1} + p_{\mathrm{Cu}1}, \tag{9a} \]
\[ P_{\mathrm{dv}}=P_{\psi4}\left[1-\frac{\omega_1p_1}{\omega(p_2-p_1)+\omega_1p_1}\right] +(\sum p_3-p_{\mathrm{Cu}2})\frac{\omega}{\omega-\omega_1} +p_{\mathrm{Fe}4}+p_{\mathrm{Fe}2}+p_{\mathrm{mx}}. \tag{9b} \]
For range cd
\[ P_1=P_{\psi4}\left[-\frac{\omega_1p_1}{\omega(p_2-p_1)+\omega_1p_1}\right] +(\sum p_3+p_{\mathrm{Cu}2})\frac{\omega}{\omega_1-\omega} +\sum P_1, \tag{10a} \]
\[ P_{\mathrm{dv}}=P_{\psi4}\left[1+\frac{\omega_1p_1}{\omega(p_2-p_1)-\omega_1p_1}\right] +(\sum p_3+p_{\mathrm{Cu}2})\frac{\omega}{\omega-\omega_1} +p_{\mathrm{Fe}4}+p_{\mathrm{Fe}2}+p_{\mathrm{mx}}. \tag{10b} \]
For range de
\[ P_1=P_{\psi4}\left[-\frac{\omega_1p_1}{\omega(p_2-p_1)+\omega_1p_1}\right] +(\sum p_3+p_{\mathrm{Cu}2})\frac{\omega_1}{\omega_1+\omega} +\sum p_1, \tag{11a} \]
\[ P_{\mathrm{dv}}=P_{\psi4}\left[1-\frac{\omega_1p_1}{\omega(p_2+p_1)+\omega_1p_1}\right] +(\sum p_3+p_{\mathrm{Cu}2})\frac{\omega}{\omega+\omega_1} +p_{\mathrm{Fe}4}+p_{\mathrm{Fe}2}+p_{\mathrm{mx}}. \tag{11b} \]
The value of the efficiency in the general case for any range is determined by the expression
\[ \eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{dv}}+P_{\mathrm{in}}} =\frac{P_4}{P_{\mathrm{dv}}+P_{\mathrm{in}}} = \]
\[ =\frac{P_{\psi4}-p_{\mathrm{Cu}4}} {P_{\psi4}+(\sum p_3\pm p_{\mathrm{Cu}2})+p_{\mathrm{Fe}2}+p_{\mathrm{Fe}4}+p_{\mathrm{mx}}+p_{\mathrm{Cu}1}+p_{\mathrm{Fe}1}}. \tag{12} \]
The value of the amplification factor in the general case for any range is determined by the following formula
\[ K_y=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}=\frac{P_4}{P_1}= \]
\[ =\frac{P_{\psi4}-p_{\mathrm{Cu}4}} {P_{\psi4}\left[\pm\frac{\omega_1p_1}{\omega(p_2\pm p_1)\pm\omega_1p_1}\right] +(\sum p_3\pm p_{\mathrm{Cu}2})\frac{\omega_1}{\omega_1\pm\omega} +\sum p_1}. \tag{13} \]
Thus, the energy relations obtained by us make it possible to correctly determine the power of the semiconductor commutator (SPC), the power of the drive motor \(P_{\mathrm{dv}}\), while expressions (12) and (13) make it possible to calculate the amplification factor and the efficiency.
Another alternating-current amplifier is the increased-frequency amplifier (phase-sensitive). The schematic diagram explaining the principle of its operation is presented in Fig. 4.
To construct the energy diagrams we shall use equations 2—11b.
The powers \(P_{\psi3}, P_{m2}, P_{\psi2}, P_{m1}, P_{\psi1}\) are determined by the following formulas:
\[ P_{\psi3}=P_{\psi4}\cdot S_4 =P_{\psi4}\cdot\frac{1}{1\pm\frac{p_2}{p_1}}, \tag{14a} \]
Fig. 4. Schematic circuit diagram of a phase-sensitive AC amplifier.
\[ P_{m2}=P_{\psi4}(1-S_4)=P_{\psi4}\left(1-\frac{1}{1\pm \dfrac{p_2}{p_1}}\right), \tag{14b} \]
\[ P_{\psi2}=P_{\psi4}\cdot S_4+\left(\sum p_3 \pm p_{\mathrm{Cu}2}\right), \tag{15a} \]
\[ P_{\psi1}=0, \tag{15b} \]
\[ P_{m1}=P_{\psi2}, \tag{15c} \]
\[ \sum p_3=p_{\mathrm{Cu}3}. \tag{16} \]
From expression (15c) it follows that the power of the synchronous generator (the first cascade) is equal to the mechanical power on the shaft of the first cascade. The power of winding 1 is equal to the copper losses, \(P_1=p_{\mathrm{Cu}1}\).
The power of the driving motor is determined by the following formula:
\[ P_{\mathrm{dr}}=P_{m2}+P_{m1}+p_{\mathrm{Fe}2}+p_{\mathrm{Fe}4} =P_{\psi4}+p_{\mathrm{Cu}3}\pm p_{\mathrm{Cu}2}+p_{\mathrm{Fe}2}+p_{\mathrm{Fe}4}+p_{\mathrm{mx}}. \tag{17} \]
Using equations (14a--17), we construct the energy diagrams.
The energy diagram shown in Fig. 5a applies to the case in which windings 2 and 3 have a direct phase sequence.
The energy diagram shown in Fig. 5b applies to the reverse phase sequence.
The gain coefficient and efficiency are calculated from the following expressions:
\[ K_y=\frac{P_{\psi4}-p_{\mathrm{Cu}4}}{p_{\mathrm{Cu}1}}, \tag{18} \]
\[ \eta=\frac{P_{\psi4}-p_{\mathrm{Cu}4}} {P_{\psi4}+p_{\mathrm{Cu}3}\pm p_{\mathrm{Cu}2}+p_{\mathrm{Fe}2}+p_{\mathrm{Fe}4}}. \tag{19} \]
Comparing expressions (12) and (13) for the adjustable-frequency amplifier and (18), (19) for the phase-sensitive amplifier, one may conclude:
The gain coefficient and efficiency of the phase-sensitive amplifier are higher than those of the adjustable-frequency amplifier.
Another AC amplifier is the fixed-frequency amplifier. The schematic circuit explaining the principle of operation is shown in Fig. 6.
The values of the powers \(P_4\), \(P_{\psi3}\), \(P_{\psi2}\), \(p_{m1}\), \(p_{m2}\), and \(P_1\) are determined from the following expressions:
Fig. 5. Energy diagrams of a phase-sensitive alternating-current amplifier.
Fig. 6. Schematic circuit diagram of a fixed-frequency alternating-current amplifier and a direct-current amplifier.
\[ P_4 = P_{\psi 4} - p_{\mathrm{Cu}4}, \tag{20} \]
\[ P_{m2} = P_{\psi 4} + p_{\mathrm{Fe}4}, \tag{21} \]
\[ P_{\psi 3}=0,\qquad P_3=p_{\mathrm{Cu}3}, \tag{22} \]
\[ P_2 = p_{\mathrm{Cu}2}+p_{\mathrm{Cu}3}+p_{\mathrm{ventilya}1}, \tag{23} \]
\[ P_{m1}=P_2+p_{\mathrm{Fe}2} =p_{\mathrm{Cu}2}+p_{\mathrm{Cu}3}+p_{\mathrm{ventilya}1}+p_{\mathrm{Ee}2}+p_{\mathrm{Fe}1}, \tag{24} \]
\[ P_1=p_{\mathrm{Cu}1}. \tag{25} \]
The power of the driving motor is calculated by the formula
\[ P_{\mathrm{dv}}=P_{\psi 4}+p_{\mathrm{Cu}2}+p_{\mathrm{Cu}3}+p_{\mathrm{ventilya}1}+p_{\mathrm{Ee}2}+p_{\mathrm{mx}}. \tag{26} \]
The efficiency of the amplifier is equal to
\[ \eta= \frac{P_{\psi 4}-p_{\mathrm{Cu}4}} {P_{\psi 4}+p_{\mathrm{Cu}2}+p_{\mathrm{Cu}3}+p_{\mathrm{ventilya}1}+p_{\mathrm{Fe}2}+p_{\mathrm{mx}}+p_{\mathrm{Cu}1}+p_{\mathrm{Fe}4}}. \tag{27} \]
The energy diagram of the amplifier is shown in Fig. 7.
Fig. 7.
The amplification factor is determined from the expression
\[ K_y=\frac{P_{\psi 4}-p_{\mathrm{Cu}4}}{p_{\mathrm{Cu}1}} . \tag{28} \]
A DC amplifier is an amplifier whose circuit is shown in Fig. 6. At the output of the DC amplifier a rectifier block is installed (dashed line).
The energy diagram of the DC amplifier is shown in Fig. 7 (the losses \(P_{\mathrm{vent}\,2}\) are indicated by a dashed line).
The value of the amplification factor and the efficiency are determined from the formulas
\[ K_y=\frac{P_{\psi 4}-p_{\mathrm{Cu}4}-p_{\mathrm{vent},2}}{p_{\mathrm{Cu}1}}, \tag{29} \]
\[ \eta= \frac{P_{\psi 4}-p_{\mathrm{Cu}4}-p_{\mathrm{vent},2}} {P_{\psi 4}+p_{\mathrm{Cu}2}+p_{\mathrm{Cu}3}+p_{\mathrm{vent},1}+p_{\mathrm{Fe}2}+p_{\mathrm{Fe}4}+p_{\mathrm{MX}}}. \tag{30} \]
The values of \(K_y\) and \(\eta\) for the fixed-frequency amplifier are higher than for the DC amplifier. This is evident from expressions (29) and (30).
Thus, the analytical expressions obtained for the powers, \(\eta\), and \(K_y\) make it possible to calculate these quantities, while the energy diagrams presented clearly show the distribution of powers under various operating conditions of the brushless electric-machine amplifiers considered.
REFERENCES
-
V. S. Novokshenov. Studies of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.
-
P. A. Kalyan. “Energy relations in a contactless AC machine operating with variable rotational speed,” Proceedings of Higher Educational Institutions, Energetika, No. 11, 1965.