The voltages and polarity of an inductor

One may be perplexed about the voltage across an inductor, since sometimes it is  −LdI/dt, but more commonly it is LdI/dt. We explain why this is so, and point out that in fact, each of them is fine to be used if under its own sign convention (listed in table in the last section of this paper), and the latter, mostly used in circuit equations, is for the purpose of intentionally grouping L, C and R as passive components (consuming electric potential energy), enabling circuit analyses more conveniently and results more understandable. As well, we re-examine a closely related subject matter: the determination of the polarities in an a.c. circuit.


Introduction
Inductors work on the principle of electro magnetic induction. The selfinduced (or called back) e.m.f. across an inductor of inductance L is where I is the current through it. The negative sign is a manifestation of Lenz's law. Very nor mally, one may think that the voltage across an inductor appearing wherever in the texts is the selfinduced e.m.f. ε, and hence equation (1) has to be employed. However, this is not always the case. Indeed, in circuits, it always refers to the no negativesign form, i.e. V L = LdI/dt. The choice, whether or not there is a negative sign, purely depends on one basic consideration: treating the inductor as an active or a passive component (sec tion 3). Therefore, we start off with revisiting the distinction between the electromotive force (the voltage of an active component) and the potential difference (the voltage of a passive component) in the next section. Whichever the voltage form is, the actual polarity of an inductor can be deter mined from a sign convention of its own (sections 4 and 5). The inductor discussed here is assumed to be pure, that is, it has no internal resistance. Besides inductors, most of the concepts discussed in this paper are applicable to other components, especially the capacitors.

E.m.f. and P.d.
In general, a voltage can mean an electromotive force (e.m.f.), or a potential difference (p.d.). An e.m.f. source, e.g. an ideal battery, of one volt is defined as one joule of electric poten tial energy (e.p.e.) converted from other form is gained by one coulomb of positive charge pass ing through the source, implying that the current leaving terminal of the source has a more positive electric potential with respect to its currententer ing terminal [1]. In other words, whenever stating a positive e.m.f. of a source, it is understood that the currentleaving terminal of the source is the positive polarity ( figure 1(a)).

Abstract
One may be perplexed about the voltage across an inductor, since sometimes it is −LdI/dt, but more commonly it is LdI/dt. We explain why this is so, and point out that in fact, each of them is fine to be used if under its own sign convention (listed in table 2 in the last section of this paper), and the latter, mostly used in circuit equations, is for the purpose of intentionally grouping L, C and R as passive components (consuming electric potential energy), enabling circuit analyses more conveniently and results more understandable. As well, we reexamine a closely related subject matter: the determination of the polarities in an a.c. circuit. What is a p.d.? In electrostatics, a p.d. between any two specified points is the difference of the electric potentials at these two points, but in current electricity, the p.d. across a passive com ponent, e.g. a resistor, must be the electric poten tial at its currententering terminal minus the electric potential at its cur rentleaving terminal. Hence, whenever stating a positive p.d. across a component, it is understood that the current entering terminal of the comp onent is the positive polarity ( figure 1(b)). A passive component has a p.d. of one volt implies that one joule of e.p.e.is dissipated when one coulomb of positive charge passes through it [2].
Nowadays, it seems that the voltage drop (or known as potential drop) tends to be more popularly used than the p.d., especially on the web. According to Wikipedia, the voltage drop is defined as the decrease of electric potential along the path of a current flowing in an electric circuit [3]. Obviously, it is synonymous with the circuits' p.d., which however has been used for decades. If a reader is used to using the voltage drop or the potential drop, just 'replaces' the p.d. in the rest of this paper with it.

Inductor is dual-status
Although an inductor produces electric potential energy (e.p.e) from magnetic energy, it trans forms e.p.e. back to magnetic energy as well. An inductor creates and consumes e.p.e. in halfand half of the time in an a.c. cycle, so it is, at least a semipassive component, nevertheless qualifying for using the term 'p.d.'. Hence, an inductor can legitimately be treated as an active (e.p.e. gener ating) or a passive (e.p.e. consuming) comp onent [4]. The voltage of the former is the e.m.f. ε, while that of the latter is the p.d. V L .
But actually, an inductor is a combination of an active and a passive component, how can the e.m.f. ε or the p.d. V L itself work singly to reflect the alter nate shifts between these two statuses? Simply, this is achieved through its sign (or the sign of its power in a.c. circuits), as summarised in table 1: One form of Kirchhoff's voltage law is Σe.m.f.s = Σp.d.s in any loops of a circuit. Therefore, in an RLC series a.c. circuit, we have, if the inductor is treated as an active component, where ξ is the source e.m.f., ε is the selfinduced e.m.f. due to L, V C and V R are the p.d.s across the capacitor and resistor, respectively. After the selfinduced e.m.f. ε is moved from the lefthand side to the righthand side, the above expression becomes where is the p.d. across the inductor. From equations (1) and (4), we have Equation (2) is rewritten as equation (3), marking a change of the status of the inductor. Under what circumstances should the self induced e.m.f. ε (equation (1)), or the p.d. V L (equation (5)) be used? Simply put, if the action of the electromagnetic induction is what we focus on, most probably the selfinduced e.m.f. ε should be used, but in circuits, quite surely the p.d. V L is the right one.
In d.c. and a.c. circuits, an inductor is usually connected in series or parallel with other passive components, such as resistors and/or capacitors. As to compare their individual effects conveniently, the meaning of the 'voltage' used in each of them need to be the same. The only option is that their 'voltages' all mean p.d.s, because it is nonsense to say 'e.m.f. of a resistor'.
Countless conveniences will be brought about if treating L, C and R are all passive comp onents, including: in a series resonance, V L and V C are added up to zero, symbolising the effects of L and C cancel each other out; if L is not pure, its external p.d. is the sum of its internal V L and the p.d. due to its internal rsistamce; the same manipulation and understanding of their indi vidual and combined phasor diagrams; the same interpretation of their voltage signs (see figure 4). Just think that how odd or counterintuitive all these would become if they are reexpressed in terms of the selfinduced e.m.f. ε (do not forget ε = V L ).

Determination of polarity-D.C.
If we put aside the consideration of which one is more convenient, both the e.m.f. ε and the p.d. V L can be used to determine the actual polarity of an inductor.
There are two steps. First, if ε (V L ) is used, assigning the currentleaving (entering) terminal as the positive polarity defines the sign conven tion for the positive ε (V L ), as shown in figure 2. Secondly, the actual polarity at a particular instant is determined by referring the then sign of ε (V L ) to its own sign convention.
Without loss of generality, we consider a time-increasing d.c. I which flows in the direction as shown in figure 2. Hence, we find ε = −LdI/ dt < 0, indicating that the then polarity is oppo site to that defined from positive ε, therefore P is positive and Q is negative. Absolutely, the same result will be got if the p.d. V L is used instead. In this case, V L = LdI/dt > 0, therefore the polarity is the same as that defined from positive V L . The answer can be easily checked by applying Lenz's law.
In a.c. circuits, the basic idea is the same, but the periodic reversals of the current direction makes the analysis and result a little bit different.

Determination of polarity-A.C.
A problem is raised about the RLC series a.c. cir cuit shown in figure 3: without directly utilising Lenz's law, how can the polarity of the inductor at any instant be figured out? The method basing on V L is explained as follows.
Since side E is earthed, the positive polar ity of the a.c. source is predefined at side S, from where the positive current should leave. Thus, the direction of the current I is defined: when I > 0, I flows clockwise around the circuit, and when I < 0, I flows anticlockwise.
Next, we consider the power of the inductor, P = IV L . When the power is positive, I and V L must be both positive or negative. Whatever the case, a positive power of a p.d. means a consumption of electric potential energy; in other words, the cur rent must flow from a higher potential to a lower potential when crossing the inductor. Hence, in the case of both I and V L are positive (negative), I flows clockwise (anticlockwise) and side A (B), from where the current enters the inductor then, is the positive polarity. Accordingly, the polarity of L is determined: when V L > 0, the polarity is (A + , B−), and when V L < 0, the polarity is (A − , B+). Simply put, the sign of V L gives exactly the sign of polarity at the positive-current-entering terminal. Here, the positive current enters L from side A. We can reach the same result by consider ing a negative power of the inductor. Readers are advised to check this out and work out the case for the selfinduced e.m.f. ε themselves. All the sign conventions for ε and V L are listed at once in table 2 in the next section.
Note that this problem cannot be solved without a prior definition of the positive current direction. If the positive current direction is really unknown, the result will become an ambiguous Table 1. The signs of ε and V L correspond to different statuses of the inductor (for a positive current).
Generating e.p.e. Consuming e.p.e. case. Technically, one output of an a.c. source has to be earthed. Besides V L , the signs of V C and V R are inter preted likewise (one of the pros of treating L, R and C as the same kind of component), as shown in figure 4. Figure 4 serves as a visual summingup of the relationships between the voltage signs and their corresponding polarities in a series a.c. cir cuit. In fact, such relationships are applicable to other a.c. circuits.

Discussion and conclusions
It is well known that a current flows from (−) to (+) when crossing a d.c. source and from (+) to (−) when crossing a resistor. An induc tor possesses this duality, because it can be used as a source of e.m.f. ε = −LdI/dt or a passive component of p.d. V L = LdI/dt. From a practical viewpoint, the difference between ε and V L is nothing more than their opposite default polari ties. All their sign conventions are tabulated in table 2.
Nonetheless, there seems to be a rule: in electric circuits, an inductor is always treated as a passive component, having exactly the same status as a resistor. Other than qualitative discussions, the selfinduced e.m.f. of an induc tor is seldom used in formulating circuit equa tions. In an a.c. circuit, we say 'V L leads I by 90°', rather than 'ε lags I by 90°'; in an RL d.c. circuit, the voltage V L = V o exp(−Rt/L) is, with no exception, a p.d., and many other examples. The reason for this inclination is explained in section 3. This is certainly a point worthy to be emphasised to students for avoiding any mis calculations, since 'inductor', 'induced e.m.f.' and 'Lenz's law (the negative sign)' are all too closely related, often they will come to mind together.
Actually, a phase difference between any two a.c. quantities is meaningful only after the meanings of their own signs are explicitly known. So, in our opinion, a diagram like our figure 4 is an indispensable tool for teaching a.c. circuits, because it tells us what the signs of the a.c. volt ages truly mean in the simplest way. Moreover, it  is unlikely that one can understand a more com plicated system which involves inductances, e.g. an ideal transformer, without a good grasp of the inductor [5].
By the way, the voltage across a capacitor is less confusing, because from the very beginning, the voltage V = Q/C has been defined and used as the p.d. across a capacitor. Therefore, there is lit tle doubt that a capacitor is a passive component, but indeed, like an inductor, a capacitor can be treated as an e.m.f. source [6].