The difference between voltage and potential difference Slavko Vujević, Tonći Modrić, Dino Lovrić University of Split Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture Split, Croatia
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[email protected] Abstract—In this paper some basic terms such as voltage and potential difference are presented. In many cases they are regarded as identical which leads to confusion with understanding of the fundamental concept of electromagnetic field. Related to this topic, some authors in their books and papers on electromagnetic theory have discussed what the voltmeter actually measures, which is resolved here in a simple way. In this paper it is shown that there is a difference between the terms voltage and potential difference depending on what is the observation point - static fields or time-varying fields. Also in the transmission line model, the voltage between two points depends on the path of integration and, therefore, is ambiguous. What is commonly referred to as voltage, is transversal voltage that is a special case of voltage equal to the potential difference that is unique. Similarly, in electrical circuit analysis, branch voltages are unique and equal to difference of nodal voltages (nodal potentials). Keywords—voltage; potential difference; induced electromotive force; static fields; time-varying fields; electric field intensity
I. INTRODUCTION There are different definitions of potential difference, voltage and electromotive force in various textbooks [1] and this leads to confusion with some basic notions. The definitions, however, are often self-contradictory and contrary to common usage, although they should be unambiguous and easily understood. There must be a difference between these terms in static fields and in time-varying fields [2-3]. In this paper it will be shown that there is a difference between voltage and potential difference, except in static fields, where the two concepts are equivalent. Maxwell equations for static electric fields describe the conservative nature of an electrostatic field which implies that the electromotive force for any closed curve is zero. However, in time-varying fields there exists an induced electric field and therefore, the electromotive force induced in the closed curve can be expressed in terms of partial time derivative of the magnetic flux. There are three different ways in which the partial time derivative of the magnetic flux can be considered [4-6]. Timevarying electric field is not a conservative field anymore and the voltage between two points in a time-varying electric and magnetic field depends on the choice of the integration path between these two points.
Moreover, in numerous papers [7-9] that speak about voltmeters readings, concepts such as voltage drop between two points and other terms were used, other than the above mentioned terms. The authors have discussed whether the position of observed points or position of the voltmeter leads affects the voltmeter readings. The conclusions obtained in literature and used principles are not always convincing. It can be shown that solution of this problem follows from Maxwell equations and the most elementary properties of vector fields and their line integrals. In transmission line model voltage between conductors and current in conductors depends on the position of the observation point along the line. That voltage between two points is ambiguous, and it is correct to call it transversal voltage, as some authors do [10]. This will be explained on simple examples and figures of the transmission line model. II. STATIC FIELDS Static fields are the simplest kind of fields, due to the fact they do not change with time. Electrostatic fields are produced by static electric charges; stationary currents are associated with free charges moving along closed conductor circuits and magnetostatic fields are due to motion of electric charges with uniform velocity (direct current) or static magnetic charges (magnetic poles). The electric field generated by a set of fixed charges can be written as the gradient of a scalar field, known as the electric scalar potential φ, which can be defined as the amount of work per unit charge required to move a charge from infinity to the given point: r E = −∇ϕ
(1)
r where E is electric field intensity.
r The negative sign in (1) shows that the direction of E is opposite to the direction in which φ increases.
In Cartesian coordinates, (1) is equivalent to:
Ex = −
∂ϕ ∂x
(2)
Ey = −
∂ϕ ∂y
(3)
Ez = −
∂ϕ ∂z
(4)
∫
Applying the Stokes theorem to (6), it can be obtained:
∫ E ⋅ d l = ∫∫ (∇ × E )⋅ dS = 0
Equation (1) can be written inversely as: B r r u AB = E ⋅ d l = ϕ A − ϕ B
The Stokes theorem states that the circulation of a electric field intensity around a closed curve C is equal to the surface integral of the curl of electric field intensity over ther open surface Sr bounded by closed curve C, provided that E and curl of E are continuous on S.
r
r
Ci
; ∀Ci
(5)
A
Equation (5) expresses the fact that a unique voltage uAB can be defined for any pair of points A and B independent of the path of integration between them (Fig. 1).
r
r
; ∀C i
(7)
Si
Equation (5) can be written as:
r ∇× E = 0
(8)
It is well known that any vector field that satisfies (7) or (8) is conservative, or irrotational. Equation (7) or (8) is referred to as Maxwell equation for static electric fields. Whether in integral form (7) or in differential form (8), they depict the conservative nature of an electrostatic field. The work done on the particle when it is taken around a closed curve is zero, so the voltage around any contour Ci can be written as: u AA =
r
r
∫ E ⋅ dl = ϕ
A
− ϕ A = 0 ; ∀Ci
(9)
Ci
Figure 1. Set of curves between two points.
The electric field generated by stationary charges is an example of a conservative field. Equation (6) shows that the r line integral of E along any closed curve Ci (Fig. 2) must be zero. Physically, this implies that no work is done in moving a charge along a closed curve in an electrostatic field.
r
r
∫ E ⋅ dl = 0
; ∀C i
Ci
(6)
III. TIME-VARYING FIELDS Time-varying fields can be generated by accelerated charges or time-varying current. In the discussion of static fields, voltage is defined to be the same as the potential difference. Actually, the voltage between two points is defined as the line integral of the total electric field intensity, from one point to the other. In distinction from static fields described by (8), in time-varying fields, the following equation is valid:
r r r r r dB ∂B ∇× E =− =− +∇× v × B dt ∂t
(
)
(10)
r where v is relative velocity between magnetic field and r medium while B is the magnetic flux rdensity and it can be expressed as a curl of the other vector, A, which is known as a magnetic vector potential: r r B =∇× A
Figure 2. Set of closed curves joined to point A.
(11)
Equation (10) is one of the Maxwell equations for timevarying fields, which states that the curl of the electric field intensity is equal to the time rate of decrease of the magnetic flux density. It also shows that the time varying electric field is not conservative. The work done in taking a charge around a closed curve in a time-varying electric field is a consequence
of the energy from the time-varying magnetic field. An induced current in the contour would produce a magnetic field in the opposite direction to the direction of increasing magnetic field. Thus the induced current would reduce the rate of change of the magnetic flux in the contour. From (10) and (11) it is clear that the scalar electric potential φ is now by itself insufficient to completely describe the time-varying electric field because there is also direct dependence on the magnetic field variations. r By recalling the definition of magnetic vector potential A, the electric field intensity for time-varying fields is given by:
r r ∂A r r E = − ∇ϕ − +v×B ∂t
(12)
In (18), integral of the static part of the electric field intensity along any closed curve Ci is zero:
∫
r r E stat ⋅ d l = ϕ A − ϕ A = 0 ; ∀C i
Ci
and voltage around any contour Ci can be written as: u=
∫
r r E ⋅ dl =
Ci
(13)
where static part of the electric field intensity is defined by:
r E stat = − ∇ϕ
(14)
whereas the induced part of the electric field intensity can be written as a sum of transformer and motional parts:
r r r Eind = E tr + E m
(15)
where transformer part is defined as:
r r E ind ⋅ d l = e = etr + e m
∫
(20)
Ci
where e is the induced electromotive force, etr is the transformer electromotive force and em is the motional electromotive force. According to (15) - (17) and (20), following equations can be written:
Total electric field intensity is defined by the sum of the static part produced by charges and the induced part. In electrostatics, the induced electric field does not exist, and voltage does not depend on the integration path between these points. This is not the case in a time-varying electric and magnetic field [5, 6]:
r r r E = E stat + E ind
(19)
e=
r
∫E
ind
r ⋅ dl
(21)
Ci
etr =
r
∫E Ci
tr
r ∂ ⋅ dl = − ∂t
r
r
∫ A ⋅ dl
∫
∫ (v × B )⋅ d l
Ci
Ci
r r em = E m ⋅ d l =
(22)
Ci
r
r
r
(23)
Therefore, for any contour Ci, voltage u is equal to induced electromotive force e: r r r r u CAAi = e CAAi = E ⋅ d l = E ind ⋅ d l
∫
∫
Ci
Ci
(24)
where voltage and induced electromotive force depend on the integration path.
r r ∂A Etr = − ∂t
(16)
and motional part is defined as:
r r r Em = v × B
(17)
A. Closed curves According to (13), the voltage around any contour Ci (Fig. 2) can be written as: u=
r
r
r
∫ E ⋅ dl = ∫ E Ci
Ci
stat
r ⋅ dl +
r
∫E Ci
ind
r ⋅ dl
(18) Figure 3. Transformer electromotive force etr induced in the contour Ci.
Transformer electromotive force (22) can be expressed as negative of partial time derivative of the magnetic flux Φ through the contour Ci over the surface Si (Fig. 3):
∂ ∂t
etr = −
r
∂
r
r
r
∂Φ
∫ A ⋅ d l = − ∂t ∫∫ B ⋅ dS = − ∂t
Ci
(25)
Si
B. Open curves In the case of time-varying electromagnetic field, voltage uAB between any pair of points A and B (Fig. 1) can be defined as:
From (31) is evident that there exists a difference between time-varying voltage and potential difference and these two concepts are not equivalent:
u AB ≠ ϕ A − ϕ B
Moreover, potential difference between any two points A and B is independent of the integration path Ci. Otherwise, voltage and induced electromotive force between any two points A and B are not equal and depend on the integration path Ci:
u CABi ≠ e CABi ≠ ϕ A − ϕ B B r r B r r B r r u AB = E ⋅ d l = E stat ⋅ d l + E ind ⋅ d l
∫
∫
∫
A
A
A
(26)
Integral of the static part of the electric field intensity is equal to the potential difference between points A and B: B
r
∫E
stat
r ⋅ dl = ϕ A − ϕB
; ∀C i
(27)
A
which is independent of the integration path Ci. Integral of the induced part of the electric field intensity between points A and B can is equal to induced electromotive force between these points: B
r
∫E
ind
r ⋅ d l = e AB = etrAB + e mAB
(28)
A
where transformer electromotive force between points A and B can be written as: B r r ∂ etr = E tr ⋅ d l = − ∂t
B
∫
∫
A
A
r r A ⋅ dl
(32)
(33)
IV. AC VOLTMETER READING An alternating current (AC) voltmeter is device of high impedance, which gives an indication, a deflection of a meter needle, proportional to the current that passes through it. The voltmeter reading was a topic of numerous papers, where authors tried to explain what voltmeter measures. In some examples [7-9], two identical voltmeters, both connected to the same two points in the electrical network, didn't show identical results. In conventional circuit analysis without timevarying fields, one can apply Ohm law and Kirchhoff voltage law. But, if time-harmonic electromagnetic field is present, one must extend Ohm law and Kirchhoff voltage law with Faraday law. The presence of a time-harmonic field produces two different voltmeter results, even if the voltmeters are equal and both are connected to the same nodes in the electrical network. The authors in [7-9] concluded that the position of the voltmeter leads affect the voltmeter readings, which are path dependent. Therefore, the voltage between two points in a time-harmonic electric and magnetic fields depends on the choice of integration path between these two points. The measured voltage depends on the rate of change of magnetic flux through the surface defined by the voltmeter leads and the electrical network. This effect is particularly pronounced at high frequencies.
(29)
whereas motional electromotive force between points A and B can be written as: B
∫
∫(
A
A
)
r r B r r r em = E m ⋅ d l = v × B ⋅ d l
(30)
According to (26) - (28), the following equation is obtained for voltage between points A and B:
u AB = ϕ A − ϕ B + e AB
(31)
Figure 4. Voltmeter connected between two points of electrical network.
Time-harmonic electrical network currents and current through the voltmeter, which is connected between points A and B (Fig. 4), will induce a transformer electromotive force:
ε = − j ⋅ω⋅ Φ
(34)
where ε is phasor of the induced electromotive force, Φ is phasor of the magnetic flux through the contour formed by voltmeter, voltmeter leads and electrical network (Fig. 4), j is imaginary unit, whereas ω represents the angular frequency. The simplest way to explain what voltmeter measures is to use a Thevenin equivalent, which consists of Thevenin electromotive force ET and Thevenin impedance Z T (Fig. 5). Thevenin equivalent represents the electrical network between points A and B.
Figure 5. Voltmeter connected to the Thevenin equivalent.
AC voltmeter reading depends on the Thevenin equivalent parameters, induced electromotive force ε , voltmeter impedance ZV and internal impedance of voltmeters leads Z L . Electromagnetic influence of the current through the voltmeter on Thevenin equivalent parameters can be neglected because current through the voltmeter is low in magnitude. Thevenin electromotive force ET , induced electromotive force ε , magnetic flux Φ and current through the voltmeter are phasors with magnitudes equal to effective values.
Internal impedance of the voltmeters leads Z L has negligible value. If this impedance is ignored, voltmeter reading can be written as:
UV = UV =
ET + ε ⋅ ZV Z T + ZV
(38)
V. TRANSMISSION LINE MODEL In the case of two-conductor transmission line model, voltage u and current i along the line are described by following equations [11, 12]:
−
∂u ∂i = R ⋅i + L ⋅ ∂x ∂t
(39)
−
∂i ∂u =G⋅u + C ⋅ ∂x ∂t
(40)
Equations (39) and (40) follow from Maxwell equations for the differential two-conductor transmission line segment (Fig. 6). In time-varying electromagnetic field, voltage between two points depends on integrating path. Voltage u used in (39) and (40) can be called transversal voltage [10], which is equal to potential difference between two joined points (Fig. 6). Transversal voltage and current along the transmission line depend on the x coordinate. Therefore, what is referred to as transmission line voltage u is a special case of voltage equal to the potential difference. Transversal voltage is equal rto potential difference because magnetic vector potential A, which only has the x component, is perpendicular to the used straight integration path.
According to Fig. 5, current through the voltmeter can be written as:
I =
ET + ε ZT + ZV + Z L
(35)
and voltage on voltmeter impedance can be written as:
U V = I ⋅ ZV =
ET + ε ⋅ ZV Z T + ZV + Z L
(36)
Voltmeter reading is equal to effective value of voltage on voltmeter impedance:
UV = UV =
ET + ε ⋅ ZV Z T + ZV + Z L
(37)
Figure 6. Voltage and current distribution along the two-conductor transmission line differential segment.
The first conductor with nodes 1 and 2 is the same as the second with nodes 3 and 4. Fig. 6 shows transversal voltages which can be written as: 4 r r u14 = E ⋅ d l = ϕ1 − ϕ 4 = u
∫ 1
(41)
3 r r ∂u u 23 = E ⋅ d l = ϕ 2 − ϕ3 = u + ⋅ dx ∂x
∫
(42)
The following equations for resistance per-unit-length R and inductance per-unit-length L are valid:
2
because: 4
∫
r r 3 r r A ⋅ dl = A ⋅ dl = 0
∫
1
(43)
r r 2 r r 3 r r 4 r r 1 r r E ⋅ dl = E ⋅ dl + E ⋅ dl + E ⋅ dl + E ⋅ dl
C
∫ C
(48)
L = Lint 1 + Lint 2 + Lext
(49)
According to (41), (42) and Fig. 7, (44) can be written as:
2
For closed rectangular contour C defined by nodes 1, 2, 3 and 4 (Fig. 7) the following equations can be written:
∫
R = R1 + R2
∫
∫
∫
∫
1
2
3
4
r r ∂Φ ext ∂i E ⋅ dl = − ⋅ dx = − Lext ⋅ dx ⋅ ∂t ∂t
(44)
(45)
where Φ ext is external magnetic flux per-unit-length and Lext is external inductance per-unit-length.
r r ∂u E ⋅ d l = (E x1 ⋅ dx ) + u + ⋅ dx + (E x 2 ⋅ dx ) − u (50) ∂x C
∫
According to (45) – (47) and (50), following equation can be obtained:
−
∂u ∂i = (R1 + R2 ) ⋅ i + (Lint 1 + Lint 2 + Lext ) ⋅ ∂x ∂t
(51)
After substituting (48) and (49) into (51), transmission line equation (39) can be obtained. Applying Kirchhoff law or Maxwell continuity equation on the differential line segment (Fig. 8), the following expression can be obtained:
∂i ∂u i = i + ⋅ dx + (G ⋅ dx ) ⋅ u + (C ⋅ dx ) ⋅ ∂ x ∂t
(52)
where G is conductance per-unit-length, and C is capacitance per-unit-length. From (52) follows the transmission line equation (40).
Figure 7. Longitudinal parameters of differential two-conductor transmission line segment.
In Fig. 7, Ex1 and Ex2 are tangential components of the electric field intensity along the external surface of the conductors and they can be called longitudinal per-unit-length voltages:
E x1 = R1 ⋅ i + Lint 1 ⋅
∂i ∂t
(46)
E x 2 = R2 ⋅ i + Lint 2 ⋅
∂i ∂t
(47)
where R1 and R2 are resistances of the conductors, whereas Lint1 and Lint2 are internal inductances of the conductors perunit-length [13, 14].
Figure 8. Transversal parameters G⋅dx and C⋅dx of differential two-conductor transmission line segment.
Single-conductor representation of the two-conductor transmission line of length ℓ, with uniformly distributed perunit-length parameters R, L, C and G, is shown if Fig. 9. In this case, transversal voltages u1 and u2 are equal to the potentials φ1 and φ2.
voltage equal to the potential difference. This voltage can be called transversal voltage. Similarly, in direct current, time-harmonic and transient electrical circuit analysis, voltage is unique and equal to difference of nodal voltages, which are also called nodal potentials. REFERENCES [1] Figure 9. Single-conductor representation of two-conductor transmission line.
VI. ELECTRICAL CIRCUIT THEORY Electrical circuit theory is not exact. It is an approximation of electromagnetic field theory that can be obtained from Maxwell equations [11]. Electric circuits consist of active elements and passive elements. Active elements are current and voltage sources. The passive circuit elements resistance R, inductance L and capacitance C are defined by the manner in which the voltage and current are related for the individual element. In direct current, time-harmonic and transient electrical circuit analysis, the voltage is unique [15]. One node can be used as the reference node and all the other nodal voltages are referenced to this common node. The voltage across any network branch is equal to difference between nodal voltages at the two branch ends. Nodal voltages are also called nodal potentials [16]. Therefore, in electrical circuit analysis, branch voltages are unique and equal to difference of nodal potentials. VII. CONCLUSION In this paper, some basic notions and equations about potential difference, voltage and induced electromotive force in electromagnetism are presented. Only in the case of static fields, voltage is identical to the potential difference. Due to conservative nature of static fields, voltage does not depend on the integration path between any two points. In the case of time-varying electromagnetic fields, voltage and potential difference are not identical. Potential difference between two points is unique, whereas voltage and induced electromotive force between two points depend on the integration path. In the transmission line model, the time-varying voltage between two points depends on the path of integration. Therefore, voltage is ambiguous. What is commonly referred to as transmission line voltage is actually a special case of
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