Option 4 : 4000/π rpm

**Concept:**

Induced emf E_{f} = K_{a}ϕω

ω = angular speed (rad/sec)

ϕ = flux (weber)

K_{a} = constant

\({K_a} = \frac{{PZ}}{{2\pi A}}\)

**Calculation:**

Given that terminal voltage V_{t} = 220 V,

armature current I_{a} = 20 A

armature resistance R_{a} = 1 Ω and K_{a}ϕ = 1.5 rad/sec

**For separately excited DC motor**

E_{f} = V_{t} – I_{a} R_{a} = 220 – 20 × 1 = 200 V

Now E_{f} = Kaϕω

ω = 200/1.5

\(\omega = \frac{{2\pi N}}{{60}}\)

\(N = \frac{{200 \times 60}}{{1.5 \times 2\pi }} = \frac{{4000}}{\pi }\;rpm\)

Option 1 : 154 V

__Concept:__

EMF equation of a DC Generator:

As the armature rotates, a voltage is generated in its coils, which is called Generated EMF or Armature EMF and is denoted by Eg.

\({E_g} = \frac{{ϕ ZNP}}{{60A}}\)

Where,

Eg = Generated EMF

P = Number of poles of the machine

ϕ = flux per pole in weber

Z = total number of armature conductors

N = speed of armature in revolution per minute (rpm)

A = number of parallel paths in the armature winding

Also,

A = P × m

Where,

m = multiplexity (simplex/duplex)

In wave winding, multiplexity is always 2 (two)

Therefore, A = 2

While in lap winding, there are two types:

- Simplex Lap winding: m = 1

∴ A = P

- Duplex Lap winding: m = 2

∴ A = 2P

__Calculation:__

Given that,

Number poles P = 4

Conductors Z = 462

Speed N = 1000 rpm

The flux per pole (ϕ) = 0.02 Wb

As winding is wave type

Number of parallel paths A = 4

\(E_g = {(0.02)(462)(1000)(4) \over 60(4)}\) = 154 V

Option 4 : 210 V

__Concept:__

EMF equation of a DC Generator:

As the armature rotates, a voltage is generated in its coils, which is called Generated EMF or Armature EMF and is denoted by Eg.

\({E_g} = \frac{{ϕ ZNP}}{{60A}}\)

Where,

Eg = Generated EMF

P = Number of poles of the machine

ϕ = flux per pole in weber

Z = total number of armature conductors

N = speed of armature in revolution per minute (rpm)

A = number of parallel paths in the armature winding

Also,

A = P × m

Where,

m = multiplexity (simplex/duplex)

In wave winding, multiplexity is always 2 (two)

Therefore, A = 2

While in lap winding, there are two types:

- Simplex Lap winding: m = 1

∴ A = P

- Duplex Lap winding: m = 2

∴ A = 2P

__Calculation:__

Given that,

Number poles P = 6

Conductors Z = 600

Speed N = 700 rpm

The flux per pole (ϕ) = 0.01 Wb

As winding is wave type

Number of parallel paths A = 2

\(E_g = {(0.01)(600)(700)(6) \over 60(2)}\) = 210 V

Option 4 : \({E_a} = \dfrac{{\phi {\omega _m}{N_c}P}}{\pi }\)

__Concept:__

E.M.F. generated by a generator is

\(E_a = \frac{{\phi ZNP}}{{60A}}\)

Where,

ϕ = flux

Z = number of conductors

N = speed in RPM

P = number of poles

A = number of parallel paths

**Application:**

\(E_a = \frac{{\phi ZNP}}{{60A}}\)

As the number of parallel paths is not given take A = 1

\(E_a = \frac{{\phi ZNP}}{{60}}\)**.....(1)**

ω_{m }= speed in rad/sec

ω_{m} = 2πN/60

N = 30ω_{m }/ π **......(2)**

Z = 2 × N_{c} **.......(3)**

N_{c} = number of coil turns

From equations (1), (2) and (3)

\(E_a = \frac{{\phi \times 2N_c\times 30 w_m\times P}}{{60\times \pi}}\)

\({E_a} = \dfrac{{\phi {ω _m}{N_c}P}}{\pi }\)

Option 1 : 0.04 Wb

**Concept:**

The emf equation for a DC generator is given as

\(E = \frac{{ϕ ZNP}}{{60A}}\) .........(1)

ϕ = flux per pole in weber

Z = number of conductors

N = speed of the rotor

P = number of poles

A = number of parallel paths.

**Calculation:**

Given

E = 600 V, Z = 250, N = 1200 rpm, P = 6.

For wave winding A = 2

From equation(1)

\(ϕ = \frac{{60EA}}{{ZNP}}\)

\(ϕ = \frac{{60 \times 600 \times 2}}{{250 \times 1200 \times 6}}\)

**ϕ = 0.04 Wb**

Option 4 : \( \frac{{\phi ZNP}}{{120}} V\)

__Concept:__

EMF equation of a DC Generator:

As the armature rotates, a voltage is generated in its coils, which is called Generated EMF or Armature EMF, and is denoted by Eg.

\({E_g} = \frac{{\phi ZNP}}{{60A}}\)

Where,

Eg = Generated EMF

P = Number of poles of the machine

ϕ = flux per pole in weber

Z = total number of armature conductors

N = speed of armature in revolution per minute (r.p.m)

A = number of parallel paths in the armature winding

Also,

A = P ⋅ m

Where,

m = multiplexity (simplex/duplex)

In wave winding, multiplexity is always 2 (two)

Therefore, A = 2

While in lap winding, there are two types:

- Simplex Lap winding: m = 1

∴ A = P

- Duplex Lap winding: m = 2

∴ A = 2P

NOTE:

If multiplexity is not mentioned, then always take simplex lap winding i.e. m = 1

__Application:__

EMF generated per path in a simplex wave-wound DC generator

∴ A = 2, m = 1

∴ \(E_g = \frac{{\phi ZNP}}{{120}} V\)

The generated emf per parallel path in armature of a DC Generator is:

1. Directly proportional to flux

2. Inversely proportional to number of poles

3. Directly proportional to rotational speed of armature

Which of these is / are correct?

Option 4 : 1 and 3

The generated emf per parallel path in the armature of a DC Generator is given by

E = \(\frac{ϕ \ P\ N}{60}\times\frac{Z}{A}\)

where,

ϕ = flux per pole

P = number of poles

N = Rotational speed in rpm

Z = number of conductors

A = number of parallel paths

Therefore, Generated EMF(E) is **directly proportional** to **flux(ϕ ),**

**rotational speed(N),** and **Number of poles(P).**

__Additional Information__

**Back EMF in DC Motor:**

- When the current-carrying conductor placed in a magnetic field, the torque induces on the conductor, the torque rotates the conductor which cuts the flux of the magnetic field.
- According to the Electromagnetic Induction Phenomenon “when the conductor cuts the magnetic field, EMF induces in the conductor”.
- It is seen that the direction (Right hand rule) of the induced emf is opposite to the applied voltage. Thereby the emf is known as the counter emf or back emf.

Determine the generated EMF of the given generator if the armature resistance is 0.1 Ω.

Option 4 : 275.46 V

Terminal voltage (V) =230 volts

Generated voltage (E_{g}) = V + (I_{a} × R_{a})

Armature current = I_{a }= load current + I_{sh}

Field current = I_{sh}

Armature resistance = R_{a} = 0.1 ohms.

Field resistance = R_{sh} = 50 ohms.

\({I_{sh}} = \frac{{\rm{V}}}{{{{\rm{R}}_{{\rm{sh}}}}}}\)

\({I_{sh}} = \frac{{230}}{{50}}\;\)

I_{sh} = 4.6 Amps

I_{a} =load current + I_{sh }

I_{a} = 450 + 4.6 = 454.6 Amps.

E_{g} = V + (I_{a} × R_{a})

E_{g} = 230 + 454.6 × 0.1

**E _{g} = 275.46 Volts.**

Option 4 : 200 V

EMF equation of DC Machine:

When the armature conductor rotates in a stationary magnetic field then EMF is induced in the armature.

Assumption:

ϕ = useful flux per pole in webers

P = number of poles

Z = total number of armature conductor

n = speed of rotation in rms

A = number of parallel paths

Hence, Z/A = number of armature conductors in series for each parallel path

Since the flux per pole is ϕ, therefore conductor cuts the flux Pϕ in one revolution,

Hence, Generated Voltage (E_{g}) = \(\frac{Pϕ}{T}\)

Where, (T = 1/n) is the time taken for one revolution in sec.

∴ (Eg) = Pϕn

The generated voltage is determined by the number of armature conductors in series in any one path between the brushes.

Therefore Total Voltage generated is given by,

E_{g} = (Average voltage per conductor) × (Number of conductor per parallel path)

Hence, E_{g} = Pnϕ × Z/A .... (1)

If N is speed in RPM,

∴ n = N/60 .... (2)

From equation (1) & (2),

\(E_g=\frac{P\phi ZN}{60A}\)

For Wave Winding: A = 2

For Lap Winding: A = P

Application:

Given,

P = 4

slot = 60 & slot/conductor = 20

∴ Condctor (Z) = 60 × 20 = 1200

N = 1000 RPM

ϕ = 5 mWb

A = 2 (Given wave winding)

From above concept,

\(E_g=\frac{P\phi ZN}{60A}=\frac{4\times 5\times 10^{-3}\times 1200\times 1000}{120}\)

Hence, Generated voltage (Eg) = 200 V

A 4 pole DC shunt generator running at 500 rpm has a simplex wave wound armature containing 48 coils of 6 turns each. The flux produced per pole is 0.02 Wb. Calculate the induced emf in the armature.

Option 1 : 192 V

__Concept:__

EMF equation of a DC Generator:

As the armature rotates, a voltage is generated in its coils, which is called Generated EMF or Armature EMF, and is denoted by Eg.

\({E_g} = \frac{{\phi ZNP}}{{60A}}\)

Where,

Eg = Generated Emf

P = Number of poles of the machine

ϕ = flux per pole in weber

Z = total number of armature conductors

N = speed of armature in revolution per minute (r.p.m)

A = number of parallel paths in the armature winding

Also,

**A = P ⋅ m**

Where,

m = multiplexity (simplex/duplex)

In wave winding, multiplexity is always 2 (two)

Therefore, A = 2P

While in lap winding, there are two types:

- Simplex Lap winding : m = 1

∴A = P

- Duplex Lap winding : m = 2

∴ A = 2P

NOTE:

If multiplexity is not mentioned, then always take simplex lap winding i.e. m = 1

__Calculation:__

Given:

P = 4, ϕ = 0.02 Wb per pole

N = 500 rpm, Coils = 48, Turns = 6 each

∴ **Number of conductors (Z) = 2× (Number of coils) × (number of turns in each coil)**

Z = 2× 48 × 6 (one coil contain 2 conductors that is why multiplied by 2)

Z = 576

A = 2 (wave winding simplex)

We know:

\({E_g} = \frac{{\phi ZNP}}{{60A}}\)

\({E_g} = \frac{{0.02 \times 576 \times 500 \times 4}}{{60 \times 2}}\)

Eg = 192 V