On the Definition of Various Types of Power in Electric Circuits

G. E. Pukhov

1. Introduction

In electric circuits under steady-state sinusoidal conditions, three types of power are distinguished: active power \(P\), reactive power \(Q\), and apparent power \(S\). The quantities \(P\) and \(Q\) are found as algebraic sums of the powers on the individual elements of the circuit, that is, by the formulas

\[ P=\sum_{k=1}^{n} P_k=\sum_{k=1}^{n} U_k I_k \cos\varphi_k , \tag{1.1} \]

\[ Q=\sum_{k=1}^{n} Q_k=\sum_{k=1}^{n} U_k I_k \sin\varphi_k . \tag{1.2} \]

The concept of the apparent power \(S\) of an electric multi-terminal network has not yet been clearly established. Still more obscure is the question of apparent and reactive powers under nonsinusoidal conditions.

Let a section be separated from some circuit, connected with the remaining part by means of \(n\) conductors. From our point of view, it is most natural to define the apparent power of such a section as that maximum active power which can be obtained from the remaining part of the circuit for given rms (effective) values of the currents in the connecting conductors and of the voltages applied to the multi-terminal network.

Of course, the above formulation leaves open the question of choosing the reference point for the voltages of the multi-terminal network. The apparent powers of different circuits can be compared only in the case when the reference points of the voltages are chosen in exactly the same way. This very important circumstance should be given special attention. It should be borne in mind that the magnitude of the apparent power by itself, without an indication of the voltage system that determines it, in no way characterizes either the receiving or the supplying part of the electric circuit.

Denoting, for brevity, the voltages \(U_{10}, U_{20}, \ldots, U_{n0}\) (\(U_{k0}\) is the voltage between the initial point and the \(k\)-th conductor) by \(U_1, U_2, \ldots, U_n\), for the definition of apparent power we shall evidently obtain the expression

\[ S=\sum_{k=1}^{n} U_k I_k . \tag{1.3} \]

It is natural to suppose that apparent and active powers are related by the dependence

\[ S^2=P^2+D^2 . \tag{1.4} \]

The physical meaning of the power \(D\), which takes into account the difference between the quantities \(P\) and \(S\), will depend on the type of circuit and its operating mode. We shall call the power \(D\) the accompanying power. For a two-terminal network, \(D=Q\) in the steady-state sinusoidal regime, and \(D=\sqrt{Q^2+T_{\mathrm{и}}^2}\) in the periodic nonsinusoidal regime, where \(T_{\mathrm{и}}\) is the distortion power.

The purpose of the present work is to establish the factors influencing the magnitude of the accompanying power and to clarify the relations between this power and other known types of power.

The exposition will proceed in the following order. First we shall briefly consider the mathematical concepts and operations used below. Then we shall obtain expressions for the effective values of systems of periodic currents and voltages and consider ways of representing them symbolically; and, finally, we shall derive the corresponding expressions for various types of power, first for the sinusoidal regime and then for the periodic nonsinusoidal regime.

2. Some mathematical concepts and operations

Let us call an \(n\)-dimensional vector \(\bar A\) a system of \(n\) numbers arranged in a definite order, and write it as follows:

\[ \bar A= \begin{pmatrix} A_1\\ A_2\\ \cdot\\ A_n \end{pmatrix}, \tag{2.1} \]

where \(A_k\) are the components of the vector. If these components are complex numbers \(\dot A_1,\dot A_2,\ldots,\dot A_n\), then we shall denote the vector by \(\dot{\bar A}\). When \(\dot A_1,\dot A_2,\ldots,\dot A_n\) are replaced by the conjugate complex quantities \(\hat A_1,\hat A_2,\ldots,\hat A_n\), the corresponding vector takes the form \(\hat{\bar A}\).

We define the length (absolute value) of a vector as the square root of the sum of the squares of the absolute values of its components, that is,

\[ |\bar A|=\sqrt{|A_1|^2+|A_2|^2+\cdots+|A_n|^2}. \tag{2.2} \]

Note that the lengths of the vectors \(\dot{\bar A}\) and \(\hat{\bar A}\) are the same, that is, \(|\dot{\bar A}|=|\hat{\bar A}|\). We shall also introduce definitions of the scalar and vector products of two multidimensional vectors.

By the scalar product of \(n\)-dimensional vectors \(\bar A\) and \(\bar B\) we shall understand a number equal to the algebraic sum of their corresponding (first, second, etc.) components, that is,

\[ \bar A\cdot\bar B=A_1B_1+A_2B_2+\cdots+A_nB_n=\sum_{k=1}^{n} A_kB_k. \tag{2.3} \]

Scalar products \(\dot{\bar A}\cdot\hat{\bar B}\), \(\hat{\bar A}\cdot\bar B\), etc., are defined analogously. For example,

\[ \dot{\bar A}\cdot\hat{\bar B}=\dot A_1\hat B_1+\dot A_2\hat B_2+\cdots+\dot A_n\hat B_n =\sum_{k=1}^{n}\dot A_k\hat B_k. \tag{2.4} \]

The vector product of \(n\)-dimensional vectors \(\bar A\) and \(\bar B\) will be called the \(\dfrac{n(n-1)}{2}\)-dimensional vector defined by the expression:

\[ \bar A \times \bar B = \left| \begin{array}{c} A_1\\ A_2\\ \cdot\\ A_n \end{array} \right| \times \left| \begin{array}{c} B_1\\ B_2\\ \cdot\\ B_n \end{array} \right| = \left| \begin{array}{c} A_1B_2-A_2B_1\\ A_1B_3-A_3B_1\\ \cdots\\ A_1B_n-A_nB_1\\ A_2B_3-A_3B_2\\ \cdots\\ A_2B_n-A_nB_2\\ \cdots\\ A_{n-1}B_n-A_nB_{n-1} \end{array} \right| \tag{2.5} \]

For example, the vector product of two four-dimensional vectors

\[ \bar A = \left| \begin{array}{c} A_1\\ A_2\\ A_3\\ A_4 \end{array} \right| \quad \text{and} \quad \bar B = \left| \begin{array}{c} B_1\\ B_2\\ B_3\\ B_4 \end{array} \right| \tag{2.5} \]

represents a \(\dfrac{4(4-1)}{2}=6\)-dimensional vector

\[ \bar A \times \bar B = \left| \begin{array}{c} A_1B_2-A_2B_1\\ A_1B_3-A_3B_1\\ A_1B_4-A_4B_1\\ A_2B_3-A_3B_2\\ A_2B_4-A_4B_2\\ A_3B_4-A_4B_3 \end{array} \right| \tag{2.6} \]

The length of the vector \(\bar B=\bar A \times \bar B\) is found in the usual way

\[ |\bar B| = \sqrt{|B_1|^2+|B_2|^2+\cdots+ \left|B_{\frac{n(n-1)}{2}}\right|^2}, \tag{2.7} \]

where \(B_k\) are the components of the \(\dfrac{n(n-1)}{2}\)-dimensional vector \(\bar B\).

By direct substitution one could prove that, between the products of the lengths of vectors and the absolute values of their vector and scalar products, there exist relations expressed by the dependence

\[ |\bar A \cdot \bar B|^2+|\bar A \times \bar B|^2 = |\bar A|^2|\bar B|^2 . \tag{2.8} \]

If the components of the vectors are real numbers, then this formula simplifies to

\[ |\bar A \cdot \bar B|^2+|\bar A \times \bar B|^2 = |\bar A|^2|\bar B|^2 . \tag{2.9} \]

When the components of vectors are, in turn, certain, let us say, \(s\)-dimensional vectors, then the scalar product is found in the usual way\(^{1)}\)

\[ \bar{\bar A}\cdot \bar{\bar B} = \sum_{k=1}^{n}\bar A_k\cdot \bar B_k = \sum_{k=1}^{n}\sum_{l=1}^{s} A_{kl}B_{kl}, \tag{2.10} \]

\(^{1)}\) The double bar placed above a vector indicates that its components are also vectors.

where

\[ \overline{A}= \begin{vmatrix} \overline{A}_1\\ \overline{A}_2\\ \vdots\\ \overline{A}_n \end{vmatrix}, \quad \overline{B}= \begin{vmatrix} \overline{B}_1\\ \overline{B}_2\\ \vdots\\ \overline{B}_n \end{vmatrix}, \quad \overline{A}_k= \begin{vmatrix} A_{k1}\\ A_{k2}\\ \vdots\\ A_{ks} \end{vmatrix}, \quad \overline{B}_k= \begin{vmatrix} B_{k1}\\ B_{k2}\\ \vdots\\ B_{ks} \end{vmatrix} \tag{2.11} \]

The vector product of the vectors \(\overline{A}\) and \(\overline{B}\) is determined by expression (2.6) after first rewriting them in the form:

\[ \overline{A}= \begin{vmatrix} \overline{A}_1\\ \overline{A}_2\\ \vdots\\ \overline{A}_n \end{vmatrix} = \begin{vmatrix} A_{11}\\ A_{12}\\ \vdots\\ A_{1s}\\ A_{21}\\ \vdots\\ A_{2s}\\ \vdots\\ A_{n1}\\ \vdots\\ A_{ns} \end{vmatrix}, \quad \overline{B}= \begin{vmatrix} \overline{B}_1\\ \overline{B}_2\\ \vdots\\ \overline{B}_n \end{vmatrix} = \begin{vmatrix} B_{11}\\ B_{12}\\ \vdots\\ B_{1s}\\ B_{21}\\ \vdots\\ B_{2s}\\ \vdots\\ B_{n1}\\ \vdots\\ B_{ns} \end{vmatrix} \tag{2.12} \]

Thus,

\[ \overline{A}\times\overline{B}= \begin{vmatrix} A_{11}B_{12}-A_{12}B_{11}\\ A_{11}B_{13}-A_{13}B_{11}\\ \cdots\cdots\cdots\cdots\\ A_{n,s-1}B_{ns}-A_{ns}B_{n,s-1} \end{vmatrix} \tag{2.13} \]

The product \(\overline{A}\times\overline{B}\) is, obviously, an \(\dfrac{ns(ns-1)}{2}\)-dimensional vector. The length of this vector is found from the formula

\[ |\overline{B}|= \sqrt{|B_1|^2+|B_2|^2+\cdots+ \left|B_{\frac{ns(ns-1)}{2}}\right|^2}, \tag{2.14} \]

where \(B_k\) is taken from expression (2.13).

3. Effective values of systems of periodic currents and voltages

In order to facilitate the subsequent exposition, we shall use the concepts of the effective (root-mean-square) values of systems of periodic currents and voltages:

\[ U=\sqrt{\sum_{k=1}^{n}\frac{1}{T}\int_{0}^{T}u_k^2\,dt} = \sqrt{\sum_{k=1}^{n}U_k^2}, \tag{3.1} \]

\[ I=\sqrt{\sum_{k=1}^{n}\frac{1}{T}\int_{0}^{T}i_k^2\,dt} = \sqrt{\sum_{k=1}^{n}I_k^2}, \tag{3.2} \]

where \(u=u_k(t)\) and \(i=i_k(t)\) are periodic voltages and currents with period equal to \(T\),

\(U_k\) and \(I_k\) are the effective values of \(u_k(t)\) and \(i(t)\).

The effective quantities \(U\) and \(I\), obviously, may be interpreted as the lengths of the vectors

\[ \dot U = \begin{vmatrix} U_1\\ U_2\\ \vdots\\ U_n \end{vmatrix} \quad \text{and} \quad \dot I = \begin{vmatrix} I_1\\ I_2\\ \vdots\\ I_n \end{vmatrix}, \tag{3.3} \]

which in what follows we shall call the vectors of effective values of systems of voltages and currents.

4. Symbolic representation of systems of periodic voltages and currents

Periodic voltages \(u_k(t)\) and currents \(i_k(t)\), after expansion in a Fourier series, can always be represented in the form:

\[ u_k(t)=\sum_{l=0}^{s}\sqrt{2}U_{kl}\sin(l\omega t+\alpha_{kl}), \tag{4.1} \]

\[ i_k(t)=\sum_{l=1}^{s}\sqrt{2}I_{kl}\sin(l\omega t+\beta_{kl}), \tag{4.2} \]

where \(s\) is the number of harmonics. Further, from the complexes of the effective values of the harmonics one can construct the vectors

\[ \overline{\dot U}_k= \begin{vmatrix} \dot U_{k0}\\ \dot U_{k1}\\ \vdots\\ \dot U_{ks} \end{vmatrix} \quad \text{and} \quad \overline{\dot I}_k= \begin{vmatrix} \dot I_{k0}\\ \dot I_{k1}\\ \vdots\\ \dot I_{ks} \end{vmatrix}, \tag{4.3} \]

where

\[ \dot U_{kl}=U_{kl}e^{j\alpha_{kl}}, \]

\[ \dot I_{kl}=I_{kl}e^{j\beta_{kl}}. \tag{4.4} \]

The quantity \(\overline{\dot U}_k\) represents the symbolic vector of the periodic voltage \(u_k(t)\), and \(\overline{\dot I}_k\) is the analogous vector of the current \(i_k(t)\). The lengths of the vectors \(\overline{\dot U}_k\) and \(\overline{\dot I}_k\) are equal to the effective values of \(u_k(t)\) and \(i_k(t)\).

To characterize a system determined by \(n\) periodic voltages and currents, we compose the vectors:

\[ \overline{\dot U}= \begin{vmatrix} \overline{\dot U}_1\\ \overline{\dot U}_2\\ \vdots\\ \overline{\dot U}_n \end{vmatrix} \quad \text{and} \quad \overline{\dot I}= \begin{vmatrix} \overline{\dot I}_1\\ \overline{\dot I}_2\\ \vdots\\ \overline{\dot I}_n \end{vmatrix}, \tag{4.5} \]

whose components are the expressions (4.3).

The quantities \(\overline{\overline{U}}\) and \(\overline{\overline{I}}\) represent the symbolic vectors of systems of periodic voltages and currents. In the particular case of a system of sinusoidal functions their symbolic vectors are simplified and take the form:

\[ \overline{\overline{U}}= \begin{vmatrix} \dot U_1\\ \dot U_2\\ \vdots\\ \dot U_n \end{vmatrix} \quad \text{and} \quad \overline{\overline{I}}= \begin{vmatrix} \dot I_1\\ \dot I_2\\ \vdots\\ \dot I_n \end{vmatrix} \tag{4.6} \]

5. Powers of a Circuit in Sinusoidal Operation

Let, for a passive section of an electric circuit connected with its supply part (equivalent generator) by \(n\) conductors, the vectors of the systems of complexes of the rms values of its currents \(\overline{\overline{I}}\) and voltages \(\overline{\overline{U}}\) be specified. The scalar product of the vectors \(\overline{\overline{U}}\) and \(\widehat{\overline{\overline{I}}}\), as is easy to see, will be equal to the symbolic power, that is,

\[ \overline{\overline{U}}\cdot \widehat{\overline{\overline{I}}}=P+jQ. \tag{5.1} \]

It follows from this that the active and reactive powers of the circuit are expressed through the vectors \(\overline{\overline{U}}\) and \(\overline{\overline{I}}\) as follows:

\[ P=\operatorname{Re}\left(\overline{\overline{U}}\cdot \widehat{\overline{\overline{I}}}\right), \tag{5.2} \]

\[ Q=\operatorname{Jm}\left(\overline{\overline{U}}\cdot \widehat{\overline{\overline{I}}}\right). \tag{5.3} \]

Thus, the active (average) and reactive powers are respectively equal to the real and imaginary parts of the scalar product of the vector of the system of complexes of the rms voltage values of the circuit by the conjugate vector of currents.

Replacing in the vectors \(\overline{\overline{U}}\) and \(\overline{\overline{I}}\) the complexes \(\dot U_k\) and \(\dot I_k\) by the moduli \(U_k\) and \(I_k\), we obtain the vectors of rms values of the voltages \(u_k=\sqrt{2}\,U_k\sin(\omega t+\alpha_k)\) and currents \(i_k=\sqrt{2}\,I_k\sin(\omega t+\beta_k)\) of the circuit.

These vectors will be of the form

\[ \overline{\overline{U}}= \begin{vmatrix} U_1\\ U_2\\ \vdots\\ U_n \end{vmatrix} \quad \text{and} \quad \overline{\overline{I}}= \begin{vmatrix} I_1\\ I_2\\ \vdots\\ I_n \end{vmatrix}. \tag{5.[[unclear: remainder of formula number]]} \]

The apparent power \(S\), as is directly evident from expression (1.3), may be interpreted as the scalar product of the vectors \(\overline{\overline{U}}\) and \(\overline{\overline{I}}\), i.e.,

\[ S=\overline{\overline{U}}\cdot \overline{\overline{I}} =\sum_{k=1}^{n} U_k I_k . \tag{5.5} \]

It is not difficult to see that in the general case the power \(S\) will be greater than the modulus of the symbolic power, that is,

\[ S>|P+jQ|=\left|\overline{\overline{U}}\cdot \widehat{\overline{\overline{I}}}\right|. \tag{5.6} \]

Consequently, in order to link together the powers \(P\), \(Q\), and \(S\), it is necessary to introduce into consideration some additional power, which we shall denote by \(T_{\varphi}\). Let us find an expression for the power \(T_{\varphi}\) in terms of the vectors \(\bar{\dot U}\) and \(\bar{\dot I}\).

Using the equalities

\[ |\bar{\dot U}\cdot \hat{\dot I}|^2+|\bar{\dot U}\times \bar{\dot I}|^2 =|\bar{\dot U}|^2|\dot I|^2, \tag{5.7} \]

\[ |\bar{\dot U}\cdot \bar{\dot I}|^2+|\bar{\dot U}\times \bar{\dot I}|^2 =|\bar{\dot U}|^2|\bar{\dot I}|^2 \tag{5.8} \]

and noting that

\[ |\bar{\dot U}\cdot \hat{\dot I}|^2=P^2+Q^2, \tag{5.9} \]

\[ |\bar{\dot U}\cdot \bar{\dot I}|^2=S^2, \tag{5.10} \]

\[ |\dot U|^2|\bar{\dot I}|^2=|\bar{\dot U}|^2|\bar{\dot I}|^2, \tag{5.11} \]

we can obtain

\[ S^2=P^2+Q^2+T_{\varphi}^{\,2}, \tag{5.12} \]

where

\[ T_{\varphi}=\sqrt{\,|\bar{\dot U}\times \hat{\dot I}|^2-|\bar{\dot U}\times \bar{\dot I}|^2\,}. \tag{5.13} \]

Considering expression (5.12), we conclude that the adopted relation (1.4) between apparent and active powers is confirmed, and the accompanying power \(D\) is expressed in the following way through the vectors \(\bar{\dot U}\) and \(\bar{\dot I}\):

\[ D=\sqrt{\,|\operatorname{Im}(\bar{\dot U}\cdot \hat{\dot I})|^2 +|\bar{\dot U}\times \bar{\dot I}|^2 -|\bar{\dot U}\times \bar{\dot I}|^2\,}. \tag{5.14} \]

Let us denote

\[ |\bar{\dot U}\times \bar{\dot I}|=T, \tag{5.15} \]

\[ |\bar{\dot U}\times \bar{\dot I}|=T_a \tag{5.16} \]

and clarify the physical meaning of the powers \(T_{\varphi}\), \(T\), and \(T_a\). For this purpose let us take a section of a circuit (a load) that is symmetrical with respect to the reference point of the voltages \(\dot U_1, U_2,\ldots,\dot U_n\), and suppose that

\[ \frac{\dot U_1}{\dot I_1}=Z_1,\ldots,\quad \frac{\dot U_k}{\dot I_k}=Z_k,\ldots,\quad \frac{\dot U_n}{\dot I_n}=Z_n . \]

An example of such a section may be a three-phase load whose phases are connected in a star with a neutral wire, in the absence of interaction between the phases.

Substituting into formulas (5.15), (5.16), and (5.13) the relation

\[ \frac{\dot U_k}{\dot I_k}=Z_k e^{j\varphi_k},\quad (k=1,2,\ldots,n), \tag{5.17} \]

where \(\varphi_k=\alpha_k-\beta_k\), we obtain

\[ T=\sqrt{\sum \left|z_p e^{j\varphi_p}-z_q e^{j\varphi_q}\right|^2 I_p^2 I_q^2}, \tag{5.18} \]

\[ T_a=\sqrt{\sum |z_p-z_q|^2 I_p^2 I_q^2}, \tag{5.19} \]

\[ T_\varphi=\sqrt{\sum \left\{\left|z_p e^{j\varphi_p}-z_q e^{j\varphi_q}\right|^2-\left|z_p-z_q\right|^2\right\} I_p^2 I_q^2} \tag{5.20} \]

(the indices \(p\) and \(q\) are chosen in accordance with 2.4).

Hence we see that, when \(z_p=z_q\) (the moduli of the impedances between the reference terminal and the other conductors are equal), the power \(T_a\) becomes zero. When \(\varphi_p=\varphi_q\) (the impedance angles are equal), the power \(T_\varphi\) will be equal to zero. Finally, when both conditions hold, the power \(T\) also becomes equal to zero. Thus, the power \(T_a\) is naturally called the power of amplitude (modulus) unbalance, \(T_\varphi\) the power of phase unbalance, and \(T\) the power of total unbalance of the circuit.

It is obvious that the accompanying power is expressed in the following way through the powers \(Q\) and \(T_\varphi\):

\[ D=\sqrt{Q^2+T_\varphi^2}. \tag{5.21} \]

Thus, in the sinusoidal regime of a passive portion of an electric circuit, the difference between the consumed active power and its maximum possible value under the given conditions, i.e., the apparent power—a difference characterized by the accompanying power \(D\)—arises, first, because of the phase mismatch between voltages and currents, which is accounted for by the reactive power \(Q\), and, second, because of the unbalance of the circuit with respect to the chosen voltage reference point, which is accounted for by the phase-unbalance power \(T_\varphi\). The square of the accompanying power is equal to the sum of the squares of the reactive power and the phase-unbalance power. If the amplitude-unbalance power is zero, then the total unbalance power and the phase-unbalance power turn out to be equal. In this case

\[ D=\sqrt{Q^2+T^2}. \tag{15.22} \]

6. Powers of a two-terminal network in a nonsinusoidal regime

Before passing to the general case of the nonsinusoidal regime of a multi-terminal network, let us see how to express the various kinds of power of a two-terminal network through the symbolic vectors of voltage \(\vec U\) and current \(\vec I\) (see expressions 4.3).

Since the lengths of the vectors \(\vec U\) and \(\vec I\) are equal to

\[ |\vec U|=\sqrt{\sum_{k=0}^{s}|U_k|^2} \quad\text{and}\quad |\vec I|=\sqrt{\sum_{k=0}^{s}|i_k|^2} \tag{6.1} \]

and represent the effective values of the voltage \(u(t)\) and the current \(i(t)\), the apparent power is obtained as the product of the lengths of the vectors \(\vec U\) and \(\vec I\), that is,

\[ S=|\vec U|\,|\vec I|. \tag{6.2} \]

Computing the scalar product of the voltage vector \(\bar U\) and the conjugate current vector \(\hat I\), we find that it is equal to the symbolic power of the two-terminal network. Indeed,

\[ \bar U \cdot \hat I = U_0 I_0 + \dot U_1 \hat{\dot I}_1 + \ldots + \dot U_s \hat{\dot I}_s = P + jQ . \tag{6.3} \]

Consequently,

\[ P = Re\,(\bar U \cdot \hat I), \tag{6.4} \]

\[ Q = Im\,(\bar U \cdot \hat I). \tag{6.5} \]

It is known that, for a two-terminal network, the square of the apparent power is equal to the sum of the squares of the active power, the reactive power, and the square of the distortion power \(T_u\). The power \(T_u\), as is evident from the expression

\[ |\bar U \cdot \hat I|^2 + |\bar U \times \bar I|^2 = |\bar U|^2 \cdot \bar I^2, \tag{6.6} \]

which can be rewritten in the form

\[ P^2 + Q^2 + |\bar U \times \bar I|^2 = S^2, \tag{6.7} \]

is equal to the absolute value of the vector product of the vectors \(\bar U\) and \(\bar I\), that is

\[ T_u = |\bar U \times \bar I|. \tag{6.8} \]

The distortion power becomes zero when the vectors \(\bar U\) and \(\bar I\) are proportional, which is possible only for a circuit consisting of active resistances. Indeed, in this case

\[ \bar U = r\bar I, \]

where \(r\) is the active resistance of the two-terminal network, and

\[ T_u = |\bar U \times \bar I| = |r(\bar I \times \bar I)| = 0. \tag{6.9} \]

7. Powers of a Multi-terminal Network under Nonsinusoidal Conditions

For an \(n\)-terminal network, the symbolic vectors of voltage and current, as we have seen, are equal to

\[ \bar U = \begin{vmatrix} \bar U_1\\ \bar U_2\\ \vdots\\ \bar U_n \end{vmatrix} = \begin{vmatrix} \dot U_{10}\\ \dot U_{11}\\ \vdots\\ \dot U_{1s}\\ \dot U_{20}\\ \vdots\\ \dot U_{2s}\\ \vdots\\ \dot U_{n0}\\ \vdots\\ \dot U_{ns} \end{vmatrix}, \qquad \bar I = \begin{vmatrix} \bar I_1\\ \bar I_2\\ \vdots\\ \bar I_n \end{vmatrix} = \begin{vmatrix} \dot I_{10}\\ \dot I_{11}\\ \vdots\\ \dot I_{1s}\\ \dot I_{20}\\ \vdots\\ \dot I_{2s}\\ \vdots\\ \dot I_{n0}\\ \vdots\\ \dot I_{ns} \end{vmatrix}, \tag{7.1} \]

where, in the components \(\dot U_{ik}\) and \(\dot I_{ik}\), the first subscript indicates the number of the terminal of the multi-terminal network, and the second subscript is the harmonic number.

Vectors of rms values of a system consisting of \(n\) voltages and \(n\) currents will be

\[ \overline{U}= \begin{vmatrix} \overline{U}_1\\ \overline{U}_2\\ \vdots\\ \overline{U}_n \end{vmatrix} = \begin{vmatrix} \sqrt{\displaystyle\sum_{k=0}^{s} U_{1k}^{2}}\\ \sqrt{\displaystyle\sum_{k=0}^{s} U_{2k}^{2}}\\ \vdots\\ \sqrt{\displaystyle\sum_{k=0}^{s} U_{nk}^{2}} \end{vmatrix}, \qquad \overline{I}= \begin{vmatrix} \overline{I}_1\\ \overline{I}_2\\ \vdots\\ \overline{I}_n \end{vmatrix} = \begin{vmatrix} \sqrt{\displaystyle\sum_{k=0}^{s} I_{1k}^{2}}\\ \sqrt{\displaystyle\sum_{k=0}^{s} I_{2k}^{2}}\\ \vdots\\ \sqrt{\displaystyle\sum_{k=0}^{s} I_{nk}^{2}} \end{vmatrix}. \tag{7.2} \]

Using the vectors \(\dot{U}, \dot{I}, \overline{U}\), and \(\overline{I}\), by the method indicated in Sec. 6, we arrive at the following relations, which define the various kinds of power of an \(n\)-terminal network:

\[ P = Re(\dot{U}\cdot \dot{I}), \tag{7.3} \]

\[ S = \overline{U}\cdot \overline{I}, \tag{7.4} \]

\[ D=\sqrt{\left|Im(\dot{U}\cdot \dot{I})\right|^{2} +\left|\overline{U}\times \overline{I}\right|^{2} -\left|\dot{U}\times \dot{I}\right|^{2}}. \tag{7.5} \]

Comparing these expressions with those obtained in Sec. 6, we see that, formally, all kinds of power under nonsinusoidal conditions are determined in the same way as under sinusoidal conditions; only the symbolic vectors \(\dot{U}\) and \(\dot{I}\) must be constructed with allowance for the complexes of the effective values of all harmonics of the currents and voltages.

As an example, let us consider the definition of the various kinds of power for a three-phase circuit with a neutral conductor.

8. Powers of a three-phase circuit whose phases are connected in a star with a neutral conductor

In the circuit under consideration, in practice it is customary to take the neutral conductor as the reference point for the voltages. Denoting the phase voltages with respect to the neutral conductor by \(u_A, u_B\), and \(u_C\), the phase currents by \(i_A, i_B\), and \(i_C\), and the current in the neutral conductor by \(i_0\), we construct, assuming sinusoidal conditions, the symbolic vectors

\[ \dot{U}= \begin{vmatrix} \dot{U}_A\\ \dot{U}_B\\ \dot{U}_C\\ 0 \end{vmatrix}, \qquad \dot{I}= \begin{vmatrix} \dot{I}_A\\ \dot{I}_B\\ \dot{I}_C\\ \dot{I}_0 \end{vmatrix}. \tag{8.1} \]

The fourth component of the vector \(\overline{U}\) is taken to be zero, since the voltage between the neutral conductor and the reference point, which is also taken to be the neutral conductor, is obviously zero.

On the basis of expressions (5.2 and 5.6) we find the active and apparent powers

\[ P=\operatorname{Re}(\overline{U}\cdot \dot{I})=\sum_{k=A,B,C} U_k I_k \cos \varphi_k, \tag{8.2} \]

\[ S=\overline{U}\cdot \overline{I}=\sum_{k=A,B,C} U_k I_k, \tag{8.3} \]

where

\[ \overline{U}= \begin{vmatrix} U_A\\ U_B\\ U_C\\ 0 \end{vmatrix}, \quad \overline{I}= \begin{vmatrix} I_A\\ I_B\\ I_C\\ I_D \end{vmatrix}. \tag{8.4} \]

From relations (5.13, 5.15 and 5.16) we determine the components of the accompanying power

\[ T=\sqrt{ \sum_{p,q=A,B,C} |U_p I_q-U_q I_p|^2 + \left(\sum_{k=A,B,C} U_k^2\right) I_D^2 }, \tag{8.5} \]

\[ T_a=\sqrt{ \sum_{p,q=A,B,C} |U_p I_q-U_q I_p|^2 + \left(\sum_{k=A,B,C} U_k^2\right) I_D^2 }, \tag{8.6} \]

\[ T_i=\sqrt{T^2-T_a^2} = \sqrt{ \sum_{p,q=A,B,C} \left( |U_p \dot{I}_q-U_q \dot{I}_p|^2 - |U_p I_q-U_q I_p|^2 \right) }, \tag{8.7} \]

\[ Q=\operatorname{Im}(\overline{U}\cdot \dot{I})=\sum_{k=A,B,C} U_k I_k \sin \varphi_k. \tag{8.8} \]

In the case of a nonsinusoidal regime of the circuit under consideration, the course of the calculations will be exactly the same; only instead of the vectors (8.1) one should operate with the vectors

\[ \overline{\overline{U}}= \begin{vmatrix} \overline{U}_A\\ \overline{U}_B\\ \overline{U}_C\\ 0 \end{vmatrix}, \quad \overline{I}= \begin{vmatrix} \overline{I}_A\\ \overline{I}_B\\ \overline{I}_C\\ \overline{I}_D \end{vmatrix}, \tag{8.9} \]

where

\[ \overline{U}_A= \begin{vmatrix} \dot{U}_{A0}\\ \dot{U}_{A1}\\ \vdots\\ \dot{U}_{AS} \end{vmatrix}, \quad \overline{U}_B= \begin{vmatrix} \dot{U}_{B0}\\ \dot{U}_{B1}\\ \vdots\\ \dot{U}_{BS} \end{vmatrix}, \quad \overline{U}_C= \begin{vmatrix} \dot{U}_{C0}\\ \dot{U}_{C1}\\ \vdots\\ \dot{U}_{CS} \end{vmatrix}, \tag{8.10} \]

\[ \overline{I}_A= \begin{vmatrix} \dot{I}_{A0}\\ \dot{I}_{A1}\\ \vdots\\ \dot{I}_{AS} \end{vmatrix}, \quad \overline{I}_B= \begin{vmatrix} \dot{I}_{B0}\\ \dot{I}_{B1}\\ \vdots\\ \dot{I}_{BS} \end{vmatrix}, \quad \overline{I}_C= \begin{vmatrix} \dot{I}_{C0}\\ \dot{I}_{C1}\\ \vdots\\ \dot{I}_{CS} \end{vmatrix}, \quad \overline{I}_D= \begin{vmatrix} \dot{I}_{D0}\\ \dot{I}_{D1}\\ \vdots\\ \dot{I}_{DS} \end{vmatrix}. \]

Here \(U_{kl}\) and \(I_{kl}\) are the complexes of the effective values of the voltage and current of the \(k\)-th phase (including the neutral conductor) for harmonic \(l\).

9. On the power factor

If by the active-power factor one understands a quantity characterizing the ratio of the active power consumed by a multi-terminal network to its maximum, i.e., to the apparent power, then the following expression may serve to determine the power factor:

\[ \lambda_a=\frac{P}{S}=\frac{P}{\sqrt{P^2+D^2}}=\frac{Re\,(\overline{U}\cdot \widehat{I})}{U\cdot I}. \tag{9.1} \]

Since the components of the accompanying power are the reactive power and the powers of asymmetry, the value of the power factor is affected both by phase shifts between currents and voltages and by various kinds of asymmetry (nonidentity of the shapes of the current and voltage curves, phase and amplitude asymmetries relative to the voltage reference origin, etc.).

Similarly to (9.1), one may introduce the concept of the accompanying-power factor

\[ \lambda_p=\frac{D}{S}. \tag{9.2} \]

It is obvious that

\[ \lambda_a^2+\lambda_p^2=1. \tag{9.3} \]

In its external form this relation is completely analogous to the usual expression

\[ \cos^2\varphi+\sin^2\varphi=1, \tag{9.4} \]

where \(\cos\varphi\) and \(\sin\varphi\) are, respectively, the active- and reactive-power factors of a two-terminal network under sinusoidal conditions.

10. On Rosenzweig’s theory

In 1939 I. Rosenzweig proposed a theory of power for polyphase currents, the fundamental proposition of which is the definition of apparent power as the maximum active power for given sums of squares of effective voltages and currents. Rosenzweig’s theory is easily extended also to the general case of a multi-terminal circuit. The author of the present work at one time (in 1949) also believed that the indicated proposition should underlie the general theory of power. Recently, in 1951, L. S. Lur’e advanced an almost analogous theory.

However, specifying the sums of squares of the effective currents and voltages obviously characterizes the state of a multi-terminal network much more weakly than specifying the currents and voltages themselves. For this reason we believe that the definition of apparent power adopted in the present work is more correct than the initial proposition of Rosenzweig’s theory.

Let us note that if this theory is extended to multi-terminal networks of general form, then the expressions for the various kinds of power are obtained in the form:

\[ S=|U|\,|I|,\quad P=Re(\overline{U}\cdot \widehat{I}),\quad P=\sqrt{Q^2+T^2}, \]

where

\[ Q=Im(\overline{U}\cdot \widehat{I}) \quad \text{and} \quad T=|\overline{U}\times \widehat{I}|. \tag{10.1} \]

These formulas are applicable to multiphase circuits and were obtained by I. Rosenzweig (in different notation).

11. Conclusions

In an electric circuit (multipole), in the general case, the following types of power should be distinguished:

1. Active (average) power, equal to the real part of the scalar product of the symbolic voltage vector by the conjugate current vector, i.e.

\[ P = Re(\overline{U}\cdot \widehat{I}). \]

The magnitude of this power does not depend on the choice of the reference point for the voltages applied to the multipole.

1. Apparent power, defined as the maximum active power that can be obtained for given rms values of the currents and voltages of the multipole, and equal to the scalar product of the effective-voltage vector by the current vector, i.e.

\[ S = \overline{U}\cdot \overline{I}. \]

1. Accompanying power, taking into account the difference between the magnitudes of active and apparent power, equal to the square root of the difference of the squares of the apparent and active powers, i.e.

\[ D = \sqrt{S^{2}-P^{2}}. \]

It is important to note that the powers \(S\) and \(D\) acquire a definite meaning only after the reference point for the voltages of the multipole has been specified.

The accompanying power, in turn, consists of a number of parts: reactive power and asymmetry powers, and all these types of powers are related by the quadratic dependence

\[ D^{2}=Q^{2}+T_{\varphi}^{2}=Q^{2}+T^{2}-T_{a}^{2}, \]

where \(Q\) is determined by the imaginary part of the scalar product of the vectors \(\overline{U}\) and \(\widehat{I}\), i.e.

\[ Q = Jm\,(\overline{U}\cdot \widehat{I}), \]

and the powers \(T\), \(T_{a}\), and \(T_{\varphi}\) are determined as the absolute values of vector products of the vectors \(\overline{U}\), \(\overline{I}\), \(\overline{U}\) and \(\overline{I}\), i.e.

\[ T = |\overline{U}\times \overline{I}|,\qquad T_{a}=|\overline{U}\times \overline{I}|, \]

\[ T_{\varphi}=\sqrt{|\overline{U}\times \overline{I}|^{2}-|\overline{U}\times \overline{I}|^{2}}. \]

Literature

1. K. A. Krug. Fundamentals of Electrical Engineering, vol. 2, 1936.

2. I. Rosenzweig.—Symboliczny wielowymiarowy rachunek wektorowy jako metoda analizy układów wielofazowych. Czasopismo techniczne, No. 6, ...11, Lwów, 1939.

3. W. Quade.—Wirk-, Blind- und Scheinleistung bei Wechselströmen mit beliebiger Kurvenform. Arch. f. El., 28, 1934, p. 130.

4. W. Quade.—Zusammensetzung der Wirk-, Blind- und Scheinleistung bei Wechselströmen beliebiger Kurvenform und neue Leistungsdefinitionen für unsymetrische Mehrphasensysteme beliebiger Kurvenform. E.T.Z. 58, 1937, p. 1312.

1. L. S. Lur’e. Apparent power of a three-phase system. Elektrichestvo, No. 1, 1951.

2. L. S. Lur’e. Power factor of an unbalanced load of a three-phase circuit. Elektrichestvo, No. 3, 1952.

3. M. E. Syrkin. Some questions of the theory of resonance in three-phase circuits. Proceedings of the Institute of Power Engineering, Academy of Sciences of the Uzbek SSR, issue 3, 1949.

4. V. I. Smirnov. A Course of Higher Mathematics, vol. 3, 1934.

5. G. E. Pukhov. The theory of energy flow when generalized forces and velocities are periodic functions of time. Scientific Notes of the Lviv Polytechnic Institute, issue VIII, 1949.

6. G. E. Pukhov. The theory of power of periodic multiphase currents. Elektrichestvo, No. 2, 1953.