The difference between voltage and potential difference-1

INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić The difference between voltage and potential differenceSlavko Vujević1, Tonći Modrić1 and Dino Lovrić1 1University of Split, Faculty of electrical engineering, mechanical engineering and naval architecture Split, Croatia different definitions of potential difference, voltage and electromotive force →confusion with some basic notions there is a difference between voltage and potential difference, depending on what is ourobservation point static electric fields →conservative field s→ the electromotive force for any closed curve is zero time-varying electric field is not a conservative field → the electromotive force induced in the closed curve can be expressed in terms of partial time derivative of the magnetic flux and it is different from zeroINDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Introduction (1) electrical circuit analysis → branch voltages are unique and equal to difference of nodal voltages (nodal potentials)what voltmeter measures , whether the position of observed points or position of the voltmeter leads affects the voltmeter readings ? transmission line model → the voltage depends on the path of integrationINDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Introduction (2) transversal voltage is a special case of voltage equal to the potential difference INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Static fields (1) static fields do not change with time →the simplest kind of fields electrostatic fields →produced by static electric charges stationary currents →associated with free charges moving along closed conductor circuits magnetostatic fields →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 →electric scalar potential φ E intensity field electricE potential scalar electric INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić i B AB AAB C dE u ;  Static fields (2) unique voltage uABcan be defined for any pair of points Aand B independent of the path of integration between them Stokes theorem → Maxwell equation for static electric fields :  00  ESdE dE i i C S  INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Static fields (3) the work done on the particle when it is taken around a closed curve is zero, so the voltage around any contour Cican be written as: i CA A AA C dE u i;0  INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Time -varying fields (1) can be generated by accelerated charges or time -varying current BvtB dtBdE  medium and field magnetic beetween velocity relativev densityflux magneticB → Maxwell equation for time-varying fields BvtAE  → the electric field intensity for time -varying fields ind stat E EE  → total electric field intensity statE BvtAE E Em tr ind  A B -magnetic vector potential A INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Time -varying fields (2) Closed curves :    m tri i i eeeCind Cstat Cd E d E dE u   0 force ive electromot inducede  i ii i Cind CC AAC AA d E dE e u   voltage and induced electromotive force depend on the integration pathfor any contour Ci,voltage uis equal to induced electromotive force e: transformer electromotive force , etr, can be expressed as negative of partial time derivative of the magnetic flux Φthrough the contour Ciover the surface Si: tSdBtdAte i i S Ctr   INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Time -varying fields (3) Open curves : voltage between any pair of points Aand Bcan be defined as:    mAB trAB AB B A e e eB AindB AstatB AAB d E d E dE u  AB B A AB e u  difference between time -varying voltage and potential difference is evident and these two concepts are not equivalent potential difference between any two points is independent of the integration path voltage and induced electromotive force between any two points are not equal and depend on the integration path INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić AC voltmeter reading (1) conventional circuit analysis without time -varying fields → Ohm law and Kirchhoff voltage law time-harmonic electromagnetic field → Ohm law and Kirchhoff voltage law extend with Faraday law the voltmeter readings are path dependent time-harmonic electrical network currents and current through the voltmeter, connected between points Aand B, will induce a transformer electromotive force :  j force ive electromot induced theof phasor  contour gh the flux throu magnetic theof phasor the measured voltage depends on the rate of change of magnetic flux through the surface defined by the voltmeter leads and the electrical network INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić AC voltmeter reading (2) Thevenin equivalent consists of Thevenin electromotive force and Thevenin impedance andrepresents the electrical network between points Aand B → Thevenin electromotive force ET, induced electromotive force ε, magnetic flux Φ and current through the voltmeter are phasors with magnitudes equal to effective values voltmeter reading is equal to effective value of voltage on voltmeter impedance V L V TT V V V ZZ Z ZEZI U U  INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Transmission line model (1) two-conductor transmission line model → voltage uand current ialong the line : tiLiRxu  tuCuGxi  in time -varying electromagnetic field, voltage between two points depends on integrating path trans versal voltage is a special case of voltage equal to the potential difference 4 14 1 14 u dE u  dxxuu dE u 3 23 223  INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Transmission line model (2) single -conductor representation of the two -conductor transmission line of length ℓ, with uniformly distributed per -unit-length parameters R, L, Cand G: transversal voltages u1and u2are equal to the potentials φ1and φ2 INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Electrical circuit theory (1) is an approximation of electromagnetic field theory that can be obtained from Maxwell equations active circuit elements: current and voltage sources passive circuit elements: resistance, inductance and capacitance in direct current, time -harmonic and transient electrical circuit analysis, voltage is unique and equal to difference of nodal voltages (nodal potentials ) INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Summary only in the 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 time-varying fields → voltage and potential difference are not identical ; potential difference between two points is unique ; voltage and induced electromotive force depend on the integration path in the transmission line model → the time -varying voltage between two points depends on the path of integration → voltage is ambiguous trans versal voltage is a special case of voltage equal to the potential difference in electrical circuit analysis → voltage is unique and equal to difference of nodal voltages (nodal potentials ) INDS ′11 & ISTET ′11─ S. Vujević, T. Modrić, D. Lovrić Thank you!