Formulae Ee

Joule's Law When a current I is passed through a resistance R, the resulting power P dissipated in the resistance is equal to the square of the current I multiplied by the resistance R: P = I2R By substitution using Ohm's Law for the corresponding voltage drop V (= IR) across the resistance: P = V2 / R = VI = I2R

Ohm's Law When an applied voltage E causes a current I to flow through an impedance Z, the value of the impedance Z is equal to the voltage E divided by the current I. Impedance = Voltage / Current Z = E / I Similarly, when a voltage E is applied across an impedance Z, the resulting current I through the impedance is equal to the voltage E divided by the impedance Z. Current = Voltage / Impedance I = E / Z Similarly, when a current I is passed through an impedance Z, the resulting voltage drop V across the impedance is equal to the current I multiplied by the impedance Z. Voltage = Current * Impedance V = IZ Alternatively, using admittance Y which is the reciprocal of impedance Z: Voltage = Current / Admittance V = I / Y

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Voltmeter Multiplier The resistance RS to be connected in series with a voltmeter of full scale voltage VV and full scale current drain IV to increase the full scale voltage to V is: RS = (V - VV) / IV The power P dissipated by the resistance RS with voltage drop (V - VV) carrying current IV is: P = (V - VV)2 / RS = (V - VV)IV = IV2RS

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Ammeter Shunt The resistance RP to be connected in parallel with an ammeter of full scale current IA and full scale voltage drop VA to increase the full scale current to I is: RP = VA / (I - IA) The power P dissipated by the resistance RP with voltage drop VA carrying current (I - IA) is: P = VA2 / RP = VA(I - IA) = (I - IA)2RP

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Power The power P dissipated by a resistance R carrying a current I with a voltage drop V is: P = V2 / R = VI = I2R Similarly, the power P dissipated by a conductance G carrying a current I with a voltage drop V is: P = V2G = VI = I2 / G The power P transferred by a capacitance C holding a changing voltage V with charge Q is: P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C The power P transferred by an inductance L carrying a changing current I with magnetic linkage Y is: P = VI = LI(di/dt) = Y(di/dt) = Y(dy/dt) / L Energy The energy W consumed over time t due to power P dissipated in a resistance R carrying a current I with a voltage drop V is: W = Pt = V2t / R = VIt = I2tR Similarly, the energy W consumed over time t due to power P dissipated in a conductance G carrying a current I with a voltage drop V is: W = Pt = V2tG = VIt = I2t / G The energy W stored in a capacitance C holding voltage V with charge Q is: W = CV2 / 2 = QV / 2 = Q2 / 2C The energy W stored in an inductance L carrying current I with magnetic linkage Y is: W = LI2 / 2 = YI / 2 = Y2 / 2L

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Reactance Inductive Reactance The inductive reactance XL of an inductance L at angular frequency w and frequency f is: XL = wL = 2pfL For a sinusoidal current i of amplitude I and angular frequency w: i = I sinwt If sinusoidal current i is passed through an inductance L, the voltage e across the inductance is: e = L di/dt = wLI coswt = XLI coswt The current through an inductance lags the voltage across it by 90°. Capacitive Reactance The capacitive reactance XC of a capacitance C at angular frequency w and frequency f is: XC = 1 / wC = 1 / 2pfC For a sinusoidal voltage v of amplitude V and angular frequency w: v = V sinwt If sinusoidal voltage v is applied across a capacitance C, the current i through the capacitance is: i = C dv/dt = wCV coswt = V coswt / XC The current through a capacitance leads the voltage across it by 90°.

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Admittance An impedance Z comprising a resistance R in series with a reactance X can be converted to an admittance Y comprising a conductance G in parallel with a susceptance B: Y = Z -1 = 1 / (R + jX) = (R - jX) / (R2 + X2) = R / (R2 + X2) - jX / (R2 + X2) = G - jB G = R / (R2 + X2) = R / |Z|2 B = X / (R2 + X2) = X / |Z|2 Using the polar form of impedance Z: Y = 1 / |Z|Ðf = |Z| -1Ð-f = |Y|Ð-f = |Y|cosf - j|Y|sinf Conversely, an admittance Y comprising a conductance G in parallel with a susceptance B can be converted to an impedance Z comprising a resistance R in series with a reactance X: Z = Y -1 = 1 / (G - jB) = (G + jB) / (G2 + B2) = G / (G2 + B2) + jB / (G2 + B2) = R + jX R = G / (G2 + B2) = G / |Y|2 X = B / (G2 + B2) = B / |Y|2 Using the polar form of admittance Y: Z = 1 / |Y|Ð-f = |Y| -1Ðf = |Z|Ðf = |Z|cosf + j|Z|sinf The total impedance ZS of impedances Z1, Z2, Z3,... connected in series is: ZS = Z1 + Z1 + Z1 +... The total admittance YP of admittances Y1, Y2, Y3,... connected in parallel is: YP = Y1 + Y1 + Y1 +...

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Synchronous Machines The synchronous rotational speed ns and synchronous angular speed ws of a machine with p pole pairs running on a supply of frequency fs are: ns = 60fs / p ws = 2pfs / p The output power Pm for a load torque Tm is: Pm = wsTm The rated load torque TM for a rated output power PM is: TM = PM / ws = PMp / 2pfs = 60PM / 2pns Synchronous Generator For a synchronous generator with stator induced voltage Es, stator current Is and synchronous impedance Zs, the terminal voltage V is: V = Es - IsZs = Es - Is(Rs + jXs) where Rs is the stator resistance and Xs is the synchronous reactance Synchronous Motor For a synchronous motor with stator induced voltage Es, stator current Is and synchronous impedance Zs, the terminal voltage V is: V = Es + IsZs = Es + Is(Rs + jXs) where Rs is the stator resistance and Xs is the synchronous reactance Note that the field excitation of a parallelled synchronous machine determines its power factor: - an under-excited machine operates with a leading power factor, - an over-excited machine operates with a lagging power factor. The field excitation of an isolated synchronous generator determines its output voltage.

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Induction Machines The synchronous rotational speed ns and synchronous angular speed ws of a machine with p pole pairs running on a supply of frequency fs are: ns = 60fs / p ws = 2pfs / p = 2pns / 60 The per-unit slip s of an induction machine of synchronous rotational speed ns running at rotational speed nm is: s = (ns - nm) / ns Rearranging for rotational speed nm: nm = (1 - s)ns Using angular speed w instead of rotational speed n: wm = (1 - s)ws The rated load torque TM for a rated output power PM is: TM = PM / wm = 60PM / 2pnm For an induction machine with Ns stator turns and Nr rotor turns running at slip s on a supply of voltage Es and frequency fs, the rotor induced voltage and frequency Er and fr are: Er = sEsNr / Ns fr = sfs For a rotor current Ir, the equivalent stator current Irs is: Irs = IrNr / Ns Note that the rotor / stator ratios are Ns / Nr for current, sNr / Ns for voltage and s for frequency. For an induction machine with rotor resistance Rr and locked rotor leakage reactance Xr, the rotor impedance Zr at slip s is: Zr = Rr + jsXr The stator circuit equivalent impedance Zrf for a rotor / stator frequency ratio s is: Zrf = Rrs / s + jXrs For an induction motor with synchronous angular speed ws running at angular speed wm and slip s, the airgap transfer power Pt, rotor copper loss Pr and gross output power Pm for a gross output torque Tm are related by: Pt = wsTm = Pr / s = Pm / (1 - s) Pr = sPt = sPm / (1 - s) Pm = wmTm = (1 - s)Pt The power ratios are: Pt : Pr : Pm = 1 : s : (1 - s) The gross motor efficiency hm (neglecting stator and mechanical losses) is: hm = Pm / Pt = 1 - s An induction machine can be operated as a generator, a motor or a brake: - for negative slip (speed above synchronous) the machine is a generator, - for positive slip between 0 and 1 (speed below synchronous) the machine is a motor, - for positive slip greater than 1 (speed negative) the machine is a brake, In all cases the magnetizing current (at lagging power factor) is provided by the supply system. No Load Test If an induction machine with its rotor unloaded is energised at rated voltage, then the input power represents the sum of the iron loss and mechanical loss of the machine. Locked Rotor Test If an induction machine with its rotor locked is energised at a reduced voltage which causes rated current input, then the input power represents the sum of the full load copper loss and stray loss of the machine. Stator Resistance Test The resistance of the stator winding is measured using a small direct current