On the Question of Electromagnetic Power
Abstract Generated abstract
This paper addresses inconsistent definitions and formulas for electromagnetic power in the teaching and analysis of electrical machines, especially across synchronous and asynchronous machines. Using vector diagrams and energy relations, it distinguishes between the power converted into or from the electromagnetic energy flux and the power actually transmitted through the air gap, accounting for stator copper and iron losses. The authors propose a unified expression for this quantity for electrical machines and argue that, if electromagnetic power is reserved for air-gap power, the flux-related quantity should instead be termed internal power.
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ON THE QUESTION OF ELECTROMAGNETIC POWER
G. A. SIPAILOV and E. V. KONONENKO
(Presented by the scientific seminar of the Electromechanical Faculty)
As has already been noted [4], a substantial shortcoming of the method of presenting a course on electrical machines is that machines of different types are considered in isolation. As a result, one and the same concept is sometimes given different content when different types of machines are considered. Such an inconsistency occurs above all in the question of electromagnetic power.
Electromagnetic power is usually defined as the power transmitted by the rotating magnetic field from the stator to the rotor, or conversely [1; 2; 3]; however, the analytical expressions for electromagnetic power do not always correspond to this definition.
For example, for a synchronous motor the electromagnetic power \(P_M\) is equal to [2]:
\[ P_M = P_1 - p_{M1}, \tag{1} \]
and for an asynchronous motor [2]:
\[ P_M = P_1 - p_{M1} - p_{c1}, \tag{2} \]
where \(P_1\) is the electrical power supplied to the motor,
\(p_{M1}\) is the loss in the stator copper,
\(p_{c1}\) is the loss in the stator iron.
From the energy diagram [1], the electromagnetic power of a synchronous motor is equal to:
\[ P_M = P_1 - p_{M1} - p_{c1} - p'_{\delta}, \tag{3} \]
where \(p'_{\delta}\) denotes additional losses depending on the leakage fields of the fundamental frequency, etc. The same picture is observed in generators as well.
As a result, the student does not form a clear idea of the essence of electromagnetic power: according to some equations (2, 3) it is the power transmitted from the stator to the rotor; according to another equation (1) it is the power equal to that part of the electrical power supplied to the motor which is converted into the power of the electromagnetic energy flux and which differs from the power transmitted through the air gap by the magnitude of the stator iron losses.
In order to obtain a unified conception of the question of electromagnetic power for all types of electrical machines when analyzing energy relations, vector diagrams may be used.
For example, for an asynchronous motor, whose vector diagram is presented in Fig. 1, the power supplied to the stator winding from the network is equal to:
\[ P_1 = m_1 U_1 I_1 \cos \varphi_1. \tag{4} \]
From the vector diagram (Fig. 1)
\[ U_1 \cos \varphi_1 = E_1 \cos \psi_1 + I_1 r_1, \]
then
\[ P_1 = m_1 E_1 I_1 \cos \psi_1 + m_1 I_1^2 r_1 . \tag{5} \]
As is known, the electrical power supplied to the motor is partly expended in covering the copper losses of the stator \(p_{M1}=m_1 I_1^2 r_1\), while the remaining part is converted into the power of the electromagnetic energy flux \(P_{\mathrm{e}}\).
From equation (5),
\[ P_{\mathrm{e}} = m_1 E_1 I_1 \cos \psi_1 . \tag{6} \]
Part of the power of the rotating flux \(P_{\mathrm{e}}\) is expended in covering the iron losses of the stator \(p_{c1}\), and the remaining part of the power \(P_{12}\) is transmitted electromagnetically through the air gap,
\[ P_{\mathrm{e}} = P_{12} + p_{c1}. \tag{7} \]
The magnitude of the iron losses can be expressed analytically through the active component of the current of the ideal no-load operation of an asynchronous motor.
From the vector diagram (Fig. 1),
\[ I_1 \cos \psi_1 = I_0 \sin \alpha + I'_2 \cos \psi_2, \]
then
\[ P_{\mathrm{e}} = m_1 E_1 I'_2 \cos \psi_2 + m_1 E_1 I_0 \sin \alpha, \tag{7a} \]
where
\[ p_{c1} = m_1 E_1 I_0 \sin \alpha, \]
\[ P_{12} = m_1 E_1 I'_2 \cos \psi_2 . \tag{8} \]
The power \(P_{12}\) transmitted through the air gap, after subtracting the copper losses \(p_{M2}\) and iron losses \(p_{c2}\) of the rotor, is converted into the total mechanical power of the motor, which is less than the useful power at the motor shaft by the amount of the mechanical and additional losses.
In accordance with this, Fig. 2 presents the energy diagram of an asynchronous motor.
In analogous fashion, the energy relations for an asynchronous generator (Fig. 2b) may be obtained, where \(P_2\) is the mechanical power supplied to the generator, and \(P_1\) is the electrical power delivered.
Thus, the relations between the power \(P_{\mathrm{e}}\) and the electrical power \(P_1\) in asynchronous machines will be:
\[ \begin{aligned} &\text{for motors} \qquad P_{\mathrm{e}} = P_1 - p_{M1},\\ &\text{for generators} \quad P_{\mathrm{e}} = P_1 + p_{M1}. \end{aligned} \tag{9} \]
Relations (9) form the basis for determining the electromagnetic power of synchronous machines [2].
Indeed, for a synchronous generator,
\[ P_{\mathrm{e}} = P_1 + p_{M1} = m_1 U_1 I_1 \cos \psi + m_1 I_1^2 r_1 = m_1 I_1 (U_1 \cos \psi + I_1 r_1), \]
from the vector diagram (Fig. 3),
\[ U_1 \cos \psi + I_1 r_1 = E_1 \cos \psi_1 \]
and
\[ P_{\mathrm{e}} = m_1 E_1 I_1 \cos \psi_1 . \tag{10} \]
Since for a non-salient-pole machine, neglecting saturation, the equality
\[ E_1 \cos \psi_1 = E_0 \cos \psi \]
is valid, then
\[ P_{\mathrm{e}} = m_1 E_0 I_1 \cos \psi . \tag{10a} \]
Fig. 1.
Fig. 3.
Fig. 2.
a) b)
Fig. 4.
a) b)
Analogously, for a synchronous motor,
\[ P_{\psi}=P_{1}-p_{m1}-m_{1}U_{1}I_{1}\cos\varphi-m_{1}I_{1}^{2}r_{1} = m_{1}E_{1}I_{1}\cos\psi_{1}. \tag{10б} \]
Energy diagrams for synchronous machines are presented in Fig. 4 (a—for the generator, b—for the motor).
Conclusions. 1. If \(P_{\psi}\) is regarded as electromagnetic power, then the following definition may be given for it.
Electromagnetic power \(P_{\psi}\) is that part of the electrical power which is converted into the power of the flow of electromagnetic energy (for motors), or that part of the power of the flow of electromagnetic energy which is converted into electrical power (for generators). This power differs from the power transmitted through the air gap by the amount of losses in the stator steel; moreover, for motors
\[ \begin{aligned} P_{\psi}&=P_{12}+P_{c1},\\ \text{for generators}\qquad P_{\psi}&=P_{12}-P_{c1}. \end{aligned} \tag{11} \]
For all electrical machines, including direct-current machines, the general formula for determining \(P_{\psi}\) has the form:
\[ P_{\psi}=m_{1}E_{1}I_{1}\cos\left(\widehat{E_{1}I_{1}}\right). \tag{12} \]
- If the designation “electromagnetic power,” convenient and customary for asynchronous machines, is retained for the power transmitted through the air gap \(P_{12}\), then the powers \(P_{\psi}\) in both asynchronous and synchronous machines should be given another name, for example, internal power.
REFERENCES
- Kostenko M. P. Electrical Machines, General Part. GEI, 1944.
- Piotrovskii L. M. Electrical Machines. GEI, 1949.
- Petrov G. N. Electrical Machines, Part I. GEI, 1940; Part II, GEI, 1947.
- Stolov L. I. On certain questions of the methodology of presenting the course “Electrical Machines.” Elektrichestvo, 11, 1952.