Apparent Power, Power Factor, and Current Factor in Single-phase Circuits with Non-negligible Line Impedances
Bibliographic record
Abstract
Abstract The first part of this article reviews the concepts of apparent power and power factor in single-phase circuits with non-negligible line impedances. The study is based on the well-known definition that the apparent power is the maximum load power subject to constant line losses and constant load voltage. In reality, however, all transmission/distribution lines have some finite impedance, and thus, the constant voltage condition cannot be maintained at the load terminals. In this article, the constant load voltage condition is relaxed and allowed to change, as long as it remains within acceptable range of application. This is a scenario that better reflects the realistic situations. Modified expressions for the apparent power and power factor are derived and discussed. The second part of the article presents a new concept called the current factor as the ratio of the minimal current to the actual current while keeping the load power constant. The current factor coincides with the power factor when the line impedances are neglected. The article explains how the current factor can be superior to the power factor when the line impedances are not negligible. The current factor is then extended to non-sinusoidal current factor for a non-linear load and is examined. Keywords: power factorapparent powerreactive powerIEEE Standard 1459current factorpower quantitiesline lossesnon-sinusoidal currentoptimal shunt compensation Notes In this note, the capital bold letters are used for phasor (or complex) quantities, where the regular capital letters show the RMS (real) values. 2For point A, X = −x; then R is equal to the maximum power S is equal to S = RI 2, and thus, the PF is equal to Note that this also satisfies Eq. (Equation10). 3It is worthwhile noticing that the expression just obtained in Eq. (Equation7) and its alternative in the footnote are different from the PF as seen from the source terminals. The conventional PF expression at the load terminals is the one given by Eq. (Equation8) and at the source terminals is given by which is clearly different from Eq. (Equation7). 4The case of unbalanced three-phase circuits with line resistances is studied in [Citation9, Citation12], and it is stated that the value of the PF based on this definition can go beyond unity. The analysis presented in this article sheds light on what may happen in such scenarios. 5Equation (Equation15) considers the square of the current. 6This can be obtained by multiplying the second line in Eq. (Equation16) by R . 7Point A assumes X = −x. This means that the line reactance is compensated at the load terminals. In such a case, R will be obtained from Extension of Eq. (Equation18) in this case is given by This satisfies Eqs. (Equation22) and (Equation23).
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".