Formula Ee

DC Generators

Unit 1 DC Machines

e = Em sinωt. Emf induced Em = Blbω Maximum value of induced emf  EMF induced in the DC generator  Eg = ΦZNP/60 A volts (lap winding: A=P; wave winding: A=2)  Number of conductors = number of slots x conductors/slot. Z = 2T  Number of conductors

Separately excited DC generator

Armature current Ia = Load current IL V = Eg – IaRa - Vbrush Terminal voltage Eg = V + IaRa + Vbrush Generated emf 

Self excited DC generators Series generator:

Armature current Ia = Load current IL = field current Ise. V = Eg – IaRa - IaRse - V brush Terminal voltage Eg = V + Ia(Ra+Rse) + Vbrush Generated emf 

Shunt generator:

Armature current Ia = IL + Ish. Shunt field current Terminal voltage Generated emf 

Ish = V / Rsh. V = Eg – IaRa - Vbrush Eg = V + IaRa + Vbrush

Compound generator: Long shunt compound generator

Armature current Ia = Ise = IL + Ish. Ish = V / Rsh. Shunt field current V = Eg – IaRa - IaRse - V brush Terminal voltage Eg = V + Ia(Ra+Rse) + Vbrush Generated emf 

Short shunt compound generator

Series field current Ise = IL Armature current Shunt field current Terminal voltage Generated emf 

Ia = Ish + Ise Ish = (V + Ise Rse) / Rsh. V = Eg – IaRa - IseRse - Vbrush Eg = V + IaRa + IseRse + Vbrush

Power developed in armature = Eg Ia Power delivered to load = V IL

DC Motors Back emf Eb = ΦZNP/60 A volts F = B I l Newtons

Separately excited DC motor

Armature current Ia = Line current IL V = Eb + IaRa + Vbrush Terminal voltage Eb = V - IaRa - Vbrush Generated emf 

Self excited DC motor Series motor:

Armature current Ia = Line current IL = field current Ise. V = Eb + IaRa + IaRse + Vbrush Terminal voltage Eb = V - Ia(Ra+Rse) - Vbrush Generated emf 

Shunt motor: Line current Shunt field current Terminal voltage Generated emf 

IL = Ia + Ish. Ish = V / Rsh. V = Eb + IaRa +Vbrush Eb = V - IaRa -Vbrush

Compound motor: Long shunt compound motor

Armature current Ia = Ise Line current IL = Ia + Ish. Shunt field current Terminal voltage Generated emf 

Ish = V / Rsh. V = Eb + IaRa + IaRse + Vbrush Eb = V - Ia(Ra+Rse) - Vbrush

Short shunt compound motor

Series field current Ise = IL Line current Shunt field current Terminal voltage Generated emf 

IL = Ise = Ish + Ia Ish = (V - IL Rse) / Rsh. V = Eb + IaRa + IseRse + Vbrush Eb = V - IaRa - IseRse - Vbrush

Angular velocity Power developed Armature torque Armature torque Shaft torque

ω = 2ΠN / 60 rad/sec P = Tω watts Ta = 0.159 Φ Ia PZ /A Nm Ta = Tsh + Tf  Tsh = 9.55 Pout / N  Nm

Copper losses Series field copper losses

Pcu =Ia2 Ra Pse cu = I2se Rse

Losses

Psh cu = I2sh Rsh Shunt field copper losses Iron loss = eddy current loss + hysteresis loss We = K B2m f 2 t2 V2 Eddy current loss Wh = ηB1.6m f V Hysteresis loss Total loss = variable loss + constant loss Variable loss = armature copper loss Constant loss = shunt field copper loss + stray loss. Mechanical loss = friction + windage loss

Brake test:

Torque developed by the motor  Efficiency

T = (F1 ~ F2) x 9.81 x r Nm η = output power /input power x 100

Constant loss  No load armature current

Pc = V Io – Iao2 Ra Iao = Io – Ish

Swinburne’s test:

Efficiency of machine as motor:

Pin = V IL Input power  Pc = VIo – (Io – Ish)2 Ra Constant loss Pcu = (IL – Ish)2 Ra Armature copper loss PT = Pc + Pcu Total loss Output power = input power – total loss Efficiency

Po = V IL - (Io – Ish)2 Ra - Pc η = (V IL - PT) / V IL x 100

Efficiency of machine as generator: Po = V IL Output power  Pc = VIo – (Io - Ish)2 Ra Constant loss Pcu= (IL + Ish)2 Ra Armature copper loss PT = Pc + Pcu Total loss Input power = output power + total loss η = V IL Efficiency B – Flux density in wb/m2  b – Breadth of the coil in metres. l - Length of the coil co il in metres. Φ – Flux per pole in webers P – Number of poles Z – Total number of conductors in the armature. T – Number of turn per coil. Ra, Ra, Ia – resi resist stan ance ce & curr curren entt of the the armature conductor. Rse, Ise – resistance & current of series field winding. Rsh, Ish – resistance & current of shunt field winding.

/ (V IL + PT) x 100 V brush – voltage  – voltage drop at the contacts of the  brush. IL – load current (generator) ; Line current (motor) Tf  – lost torque in Nm. K – Constant depending on material Bm – maximum flux density in Tesla f – Frequency of magnetic reversals in Hz t – Thickness of laminations V – Volume of armature core in m3 η – Hysteresis coefficient r – Radius of pulley Io – no load input current

Unit 2 Transformers RMS value of emf induced in the primary winding RMS value of emf induced in the secondary winding Transformation ratio Voltage ratio Current ratio

Conditions: N2 >N1 ; K>1 ; step up transformer N2
Transformer on no-load: Active component Reactive component  No-load primary current  No-load input power 

Transformer on load:

Iw = Io cosΦo Iμ = Io sinΦo Io = √(Iw2 + Iμ2 ) Po = V1 Io cosΦo

Primary current

I1 = Io + I2  

Voltage drop across resistance = Voltage drop across reactance = Primary voltage

I1 R 1 j I1 X1 V1 = E1 + I1 Z1

In primary side

In secondary side

E1 = 4.44 f Φm N1 volts E1 = 4.44 f Bm A N1 volts E2 = 4.44 f Φm N2 volts E2 = 4.44 f Bm A N2 volts K = E2 / E1 = N2 / N1 = I1 / I2 E2 / E1 = K  I1 / I2 =K 

I2 R 2 j I2 X2 V2 = E2 + I2 Z2 Secondary windings referred to primary:

Voltage drop across resistance = Voltage drop across reactance = Secondary voltage

Equivalent resistance of the transformer referred to primary Equivalent reactance of the transformer referred to primary Equivalent impedance of the transformer referred to primary

Primary windings referred to Secondary: Equivalent resistance of the transformer referred to secondary Equivalent reactance of the transformer referred to secondary Equivalent impedance of the transformer referred to secondary

R 01 01= R 1 + R 2   X01= X1 + X2   2 2 Z01= √( R 01 01 + X01 ) R 2   = R 2 / K 2 X2   = X2 / K 2 R 02 02= R 1   + R 2 X02= X1   + X2 2 2 Z02= √( R 02 02 + X02 ) R 1   = R 1 K 2 X1   = X1 K 2

Equivalent circuit of the transformer referred to Primary: R 2   = R 2 / K 2 X2   = X2 / K 2 ZL  = ZL / K 2

I2   = I2 K  V2   = V2 / K 

Ro = V1 / Iw Xo = V1 / Iμ

Voltage regulation of the transformer: (V2(NL) - V2(L) ) / V2(NL) x 100 % regulation = (I1 R 01 % regulation = 01 cosΦ ± I1 X01 sinΦ) / V1 x 100 (+ for lagging pf; - for leading pf) (I1 R 01 % regulation = 01) / V1 x 100 (unity pf) Efficiency of a transformer: Maximum efficiency = full load KVA x √(iron loss / full load copper loss) Open circuit test:

 No- load power factor  cosΦo = Wo/ V1 Io

(Open circuit test gives no load loss Pi, Iw, Iμ, Ro, Xo: refer the formula already mention above)

Short circuit test: Total copper loss Pcu = I12 R 1 + I12 R 2   = I12 R 01 01 Total impedance referred to primary Z01 = Vsc / I1 2 Total leakage reactance referred to primary X01= √( Z012 - R 01 01 ) Short circuit power factor cosΦsc = Pcu / Vsc I1 (Short circuit test gives full load cu loss, R 01 01 , X01 , cosΦsc ) Efficiency from OS & SC test: Efficiency = (full load KVA x pf)/ ((full load KVA x pf) + Pi + Pcu) For any load (n) Efficiency = (n x full load KVA x pf)/ ((n x full load KVA x pf) + Pi + n 2Pcu) η all day = Kwh output in 24 hrs / Kwh input in 24 hrs Bm – maximum flux density in the core in Tesla f – Frequency of AC supply in Hz Φm – maximum value of flux in the core in webers A – area of the core in m2  N1, N  N2 – number of primary and secondary turns E1, E2 – Emf induced in the primary and secondary in volts cosΦo – no load power factor  V1, V2 – primary voltage & secondary voltage I2   - load component of primary current Z1, Z2- total impedance in the primary & secondary

Unit 3 Induction Motors Three phase induction motor: Phase voltage Synchronous speed Slip Rotor frequency Rotor induced EMF Rotor reactance Rotor current Rotor power factor 

Torque equation: Maximum torque

Vph = VL / √ 3 (star connection) Ns = 120f/P Slip speed = Ns – N s = (Ns - N) / Ns % Slip = (Ns - N) / Ns x 100 f r = s f  E2r = s E2 X2r = s X2 I2r = s E2 / √( R 22 + (sX2)2 ) cosΦ2r = R 2 / √( R 22 + (sX2)2 ) T = K s E22 R 2 /√( R 22 + (sX2)2 ) Nm K = 3 / 2 Πns Tmax = K E22/2X2

Losses:

Pi = √ 3 VL IL cosΦ P2 = Pi - PSL Pcu = 3 I22 R 2 Pm = P2 - Pcu Rotor efficiency = rotor output / rotor input = Pm/ P 2 Motor efficiency = mechanical power output at shaft / Electrical power input to the stator = Po/ P i

Input power to the stator  Rotor input Rotor copper loss Mechanical power developed in the rotor 

rotor input : mechanical power developed by rotor : rotor copper loss P2 : Pm : Pcu = 1 : (1-s) : s P2 / Pm = 1/ (1-s (1-s)) ; Pm / Pcu Pcu = (1(1- s) / s ; Pcu Pcu / P2 = s  No-load current

Equivalent circuit referred to stator: R 2   = R 2 / K 2 X2   = X2 / K 2 R L  = R L / K 2 I2r   = I2r K = K s E2 / √( R 22 + (sX2)2 )

Io = √ (Iw2 + Iμ2 ) Ro = V1 / Iw Xo = V1 / Iμ E2   = E2 / K  Ro = V1 / Iw Xo = V1 / Iμ

2 R 01 Equivalent resistance referred to stator  01= R 1 + R 2   = R 1 + R 2 / K  X01= X1 + X2   = X1 + X2 / K 2 Equivalent reactance referred to stator  Emf induced in the stator winding of the induction motor  V = 2Πf T1Φ Kw

Cascade control: Main motor alone: Auxiliary motor alone: Cumulative cascade connection: Differential cascade connection:

Single phase induction motor: Forward slip Backward slip

Ns = 120 f /P1 Ns = 120 f /P2 N = 120f / (P1 + P2) N = 120f / (P1 - P2) ; (P1 > P2) Sf  = (Ns - N) / Ns Sb = (2 - s)

f – Supply frequency P – Number of poles for which stator is wound  Ns – Synchronous speed in rpm  N – Rotor speed in rpm ns - Rotor speed in rps = Ns /60 R 2,2, X2 – rotor resistance and reactance per phase under standstill R 2  , X2   - rotor resistance and reactance referred to stator  R L - load resistance referred to stator  K – Constant of proportionality PSL - stator losses Ro, Xo – no load resistance & reactance per phase. T1 – number of turns in the primary p rimary f- Frequency of the stator supply Kw – winding factor  P1, P2 – number of poles in main motor & auxiliary motor 

Unit 4 Synchronous & Special Machines Frequency of induced emf  Pitch factor  Distribution factor  Angular displacement between the slots Emf equation of an alternator 

Alternator on load: Synchronous impedance / phase Generated Emf per phase

f = PN /120 Hz Kp = cos (α/2) Kd = (sin (mβ/2)) / (m sin (β/2)) β = 180 / n E= 4.44 Kp Kd f ΦT volts Zs = √( R a2 + Xs2 ) E = V + Ia (R a + jXs) = V + Ia Zs

Voltage Regulation: % Regulation = (Eo – V) / V x 100 Eo = √ (V + IR a)2 + (IXs)2 Eo = √ (V cosΦ + IR a)2 + (V sinΦ ± IXs)2

(unity pf) (+ for lagging pf; - for leading pf)

Stepper motor: β = (Ns ~ N) / Ns . Nr x 360 Step angle β = 360 / m Nr (m – number of stator phases) Resolution = number of steps/ revolution = 360 / β  N – Rotor speed in rpm P – Number of rotor poles m – Number of slots per pole per phase (Distribution factor) n – Number of slots per pole Φ – Flux per pole T – Number of turns per phase Ra – armature resistance Xs – synchronous impedance Eo, V – no load & full load rated terminal voltage per phase Ia - armature current per phase  Ns, Nr – number of stator poles & rotor poles I – full load current