Circuits Electrical Voltage 1 M H Miller

ELECTRICAL VOLTAGE

Electrical work is done by the action of Coulomb forces from one set of charges on other electricalcharge as it moves in their presence. An important quantity in electrical circuit analysis is the workdone per unit charge moved between two points. Together with knowledge of the electrical current(from which the total amount of charge transported in a given period of time can be determined) thework per unit charge transported characterizes the energy performance of the circuit. Work done perunit charge transported between two points is called the voltage difference between the two points. It isessential to appreciate that a transport of charge between two points is implicit in the meaning of'voltage'. Work is the application of a force over a distance. This is so even if the two points arephysically the same point, e.g., the charge is moved away from and then back to the same point. Evenso the notion of a charge displacement over a distance is involved.

Because of its importance it is worth emphasizing yet again that the concept of work involves a forceapplied over a distance. Work per unit charge refers to a charge transport between two points, and thereis no meaning to an expression such as 'work at a point'. As it happens however it is quite common inpractice to refer colloquially to the voltage 'at a point'. But without exception this is done after a priorunderstanding that what is actually meant is the voltage difference between the specified point andanother implied reference point previously agreed upon. Jargon aside voltage difference is not someentirely new concept but rather the familiar concept of mechanical work as applied to the electrical forceacting on an electrically charged object.

The Coulomb force is an inverse square-law force, i.e.; the force between electrical charges variesinversely as the square of the distance separating the charges. It is an important property of such forcelaws that the net work done in moving between two points is independent of the path taken between thetwo points. Details of the work done in transit are path-dependent, but the overall result is that the sameamount of work is done in moving between two points whatever the specific path taken. This pathindependence is particularly important because it says something (although not everything) which is notdependent on geometric detail. We do not need a literal picture of an electric circuit in order todetermine important properties of the circuit. Thus the sketch to theright is sufficient to convey the information that there is an electricalconnection between points A and B; the geometrical shape of thecurve is not electrically significant. Any other curve connecting Aand B conveys the same information. Later we shall insert an icononto the connection curve to indicate certain electrical aspects of theconnection. It is often generally necessary for some purposes to distinguish different paths (withdifferent icons) between A and B. Nevertheless whatever the specific physical character of theconnection path the voltage difference between the end-points is the same for any and all pathsconnecting them.

We have need of a convention by which to describe the voltage difference between two points. Thereare two aspects of the electrical work that must be included in a description. One is the amount of workdone; obviously it is of some importance whether a little bit of work is involved or whether a great dealof work is involved. Equally important is the matter of whether work is being done by the movingelectricity against forces from other charges as it moves, or whether work is being done on the electricityby other charges to force it to move. A gravitational analogy might be the act of 'pressing' weights. Itcertainly makes a difference if you have to lift the weight from the floor over your shoulders ascompared to returning the weight to the floor after lifting it. The amount of work involved theoreticallyis the same for either case. In the one case work is done against the force of gravity, and in the othergravity does the work. If you were holding the weights the distinction would be quite evident.

Circuits Electrical Voltage 2 M H Miller

One complication in the description of electrical work is that quite often the amount of work andwhether it is done on or by the moving electricity is not known initially; that information is obtainedfrom an appropriate circuit analysis in which the description is used. It follows then that the form of thedescription can not depend on the specifics of the work being described. As is the case with a currentdescription here also it is necessary to distinguish between what is being described and the manner inwhich the description is made.

An earlier sketch, repeated to the right, indicates an electrical path connecting points A and B. Thegeometrical details of the path are suppressed, i.e., the curve only represents a connection and is not theconnecting path itself. The alphabetic character V (to honor the Italian physicist Volta) is a commonsymbol used to represent a voltage difference. The magnitude of the value assigned to V is the amountof work involved in a unit charge transport between A and B.Indicating whether work is done by or on the charge is a bit moreinvolved. First it is necessary to distinguish a traversal of theconnection from A to B from the converse transversal B to A. Themost common means for making the distinction is to assign differentmarkers to the two ends of the connection; '+' and '-' signs are usedmost often. The assignment of these end markers to one or the other end of the connection is completelyarbitrary. They have no arithmetic meaning; they simply are convenient well-known distinguishablesymbols.

Associated with the assignment of the ± end markers are some common colloquialisms. The voltagedifference is referred to as a 'drop' going from '+' to '-', and it is a 'rise' going from '-' to '+'. This jargon Isuspect is based on the feeling that '+' is in some sense higher than '-'. It is regularly used as aconvenient contraction for unwieldy phrasing, for example, instead of 'voltage difference from the '+'end of the connection to the '-' end of the connection'; one simply refers to the 'voltage drop'. Similarlyfor 'voltage rise'. Note that use of these colloquialisms requires or presumes a prior assignment of the ±signs. There is no particular physical significance to the use of 'voltage drop' or 'voltage rise'; thesecolloquialisms simply indicate a polarity in a conventional manner.

From basic physics it can be determined that electrical work is done by the charge involved with a +current flowing through a + voltage drop, and work is done on the charge if the current flows through avoltage rise. As noted several times before one must be very careful here to distinguish between thearbitrary choice allowed for the means of describing voltage difference and current, and specific valuesfor a voltage or a current in a particular circuit. The ± voltage polarity signs can be assigned arbitrarily,and the current polarity arrow direction chosen arbitrarily, because these choices do not declare specificvalues either for the voltage difference or for the current. Such values will depend on the physicalproperties of the circuit path ('branch') between A and B, and also the physical properties of all the otherbranches comprising the complete circuit. All that is involved here is the means for referring to anddescribing (eventually) the branch current and voltage.

It is convenient before continuing further to introduce the concept of electrical power. In general poweris simply the rate of doing work; 'electrical ' power simply refers to the rate at which work is done byelectrical forces. If V is the work done per unit of charge transported across a branch, and if I is the rateof charge transport though that branch then

VI = (work/charge)(charge/time) = power

Circuits Electrical Voltage 3 M H Miller

The product of VI is the electrical power, the work done per unit time, in the charge transport. Thequestion as whether the power corresponds to work done by or on the charge transported turns out to be

resolvable very easily. If voltage and current polarity references areapplied and used as illustrated in the figure to the left then the questionof whether the work involved is done by or on the charge is resolvedautomatically by the sign of the product of the voltage drop and thecurrent. All this can be worked out fairly easily from basic principles byconsidering the movement of a test charge in a Coulomb electric forcefield. However here all we do is assert (and apply) the conclusion.

Since they do not indicate actual voltage and current polarities the polarity markers can be chosenarbitrarily and independently if desired. But the arbitrary nature of the choice allows for a convenientchoice to be made. If a choice of relative polarities is made as shown in the figure, i.e., the currentarrow is directed from + to -, then a positive VI product means work is being provided by the currentand a negative product means work is being exerted on the current.

This simplification is not automatic; it comes about because the opportunity afforded by anarbitrary–choice option is not wasted by actually making an arbitrary choice. Instead this freedom isused constructively to provide a simplification in the analysis procedure. For future discussion we shallalways use the convention noted.

Let us return to the important property of the Coulomb force that was asserted previously. The net workdone in transporting a charge between two points is independent of the path between the two points.Details of the work done along one path differ in general from those along another path, but the net workover the entire path depends only on the start and end points. This path independence of the workinvolved with a charge transport is the content of a very important circuit theorem called Kirchoff'sVoltage Law, or KVL for short. In generalized form the theorem simply states that the net workinvolved with a charge transport depends only on the beginning and ending points of the transport. Thecumulative effect of the details of the work done over a specific path always is such as to make the netoverall result path-independent.

Less formally KVL states that the net voltage drop (or equivalently net voltage rise if you prefer)between any two points is the same whatever the path taken between the points. If VAB is the voltage

drop from A to B, VBC is the voltage drop from B to C, and VAC is the voltage drop from A to C then

KVL assertsVAC = VAB + VBC

Similarly if a different path between A and C, through node D, is takenVAC = VAD + VDC

It is worth emphasizing that it does not follow that VAB = VAD and VBC = VDC.

An important variant statement of this theorem states that the sum of the voltage drops (or voltage rises)around a closed path is zero; a closed path is one for which the start and end points are the same. Notethat this does not preclude work being done along the way; it only says that as much work is done on thecharge as is done by the charge in the completed round trip. (If this closed loop condition were not truethen the work done in moving between two points would depend on whether the path used involved aclosed loop or not, i.e., it would not be path-independent.)

A simple but important special case is the loop indicated by the equationVA->B + VB->A = 0

Circuits Electrical Voltage 4 M H Miller

It follows that VA->B = - VB->A, i.e., if work is done on the charge moving from A to B then an equal

amount of work is done by the charge for the opposite transit, and vice versa.

KVL and KCL (discussed earlier) are two of the three tools of basic electric circuit analysis. Like KCLthe importance of KVL is that it enables general conclusions to be drawn about a circuit withoutrequiring specific knowledge of the details of that circuit. It is not necessary, for example, to know theparticular electrical components used in a circuit to apply either KCL or KVL. Clearly howeverindependence of circuit component details means these two theorems are not sufficient to complete acircuit analysis. Details of what happens in a circuit depend in general on its specific physicalconstitution, i.e., different circuits will have different properties depending on the electrical componentswith which they are assembled. The third 'tool' needed to provide necessary and sufficient means toperform a circuit analysis is knowledge of the specific composition of a circuit, i.e., the circuitcomponents.