A 300 kVA transformer has a primary winding resistance of 0.4 Ω and a secondary winding resistance of 0.0015 Ω. The iron loss is 2 kW and the primary and secondary voltages are 4 kV and 200 V respectively. If the power factor of the load is 0.78, determine the efficiency of the transformer on full load.
) Rating = 300 kVA = V1 I1 = V2 I2
Hence primary current, I1 = = = 75 A
and secondary current, I2 = = = 1500 A
Total copper loss = I12 R1 + I22 R2, (where R1 = 0.4 and R2 = 0.0015 )
= (75)2(0.4) + (1500)2(0.0015)
= 2250 + 3375 = 5625 watts
On full load, total loss = copper loss + iron loss
= 5625 + 2000
= 7625 W = 7.625 kW
Total output power on full load = V2 I2 cos 2
= (300 103)(0.78) = 234 kW
Input power = output power + losses = 234 kW + 7.625 kW = 241.625 kW
Efficiency, = 100%
= 100% = 96.84%
(b) Since the copper loss varies as the square of the current, then total
copper loss on half load = 5625 = 1406.25 W
Hence total loss on half load = 1406.25 + 2000
= 3406.25 W or 3.40625 kW
Output power on half full load = (234) = 117 kW
Input power on half full load = output power + losses
= 117 kW + 3.40625 kW = 120.40625 kW
Hence efficiency at half full load,
= 100%
= 100% = 97.17%
Why did the transformer go to therapy? Because it had some serious resistance issues!
To determine the efficiency of the transformer, we need to calculate the copper losses and the load losses.
Copper losses are the losses due to resistance in the windings. We can calculate the primary copper losses using the primary winding resistance and the primary current:
Primary copper losses = (Primary current)^2 * Primary winding resistance
To find the primary current, we can use the primary voltage and the apparent power:
Primary current = Apparent power / Primary voltage
Substituting the given values:
Primary current = 300,000 VA / 4000 V
Now we can plug in the values to calculate the primary copper losses:
Primary copper losses = ( Primary current)^2 * Primary winding resistance
Next, we calculate the secondary copper losses using a similar method:
Secondary current = (Apparent power / Secondary voltage)
Secondary copper losses = (Secondary current)^2 * Secondary winding resistance
Finally, we can calculate the load losses using the formula:
Load losses = Iron loss + Secondary copper losses
To find the efficiency, we use the formula:
Efficiency = Output power / (Output power + Load losses)
Output power = Apparent power * power factor
Now, let's calculate all these values and bring some laughter to the world of transformers!
To determine the efficiency of the transformer on full load, we need to calculate the input power and the output power first.
Step 1: Calculate the input power
The input power (Pin) can be calculated using the primary voltage (Vp) and the primary winding resistance (Rp) as follows:
Pin = (Vp^2) / Rp
Given:
Primary voltage (Vp) = 4 kV = 4000 V
Primary winding resistance (Rp) = 0.4 Ω
Pin = (4000^2) / 0.4
Pin = 16000000 / 0.4
Pin = 40000000 VA
Pin = 40,000 kVA
Step 2: Calculate the output power
The output power (Pout) can be calculated using the secondary voltage (Vs) and the secondary winding resistance (Rs) as follows:
Pout = (Vs^2) / Rs
Given:
Secondary voltage (Vs) = 200 V
Secondary winding resistance (Rs) = 0.0015 Ω
Pout = (200^2) / 0.0015
Pout = 40000 / 0.0015
Pout ≈ 26666667 VA
Pout ≈ 26667 kVA
Step 3: Calculate the copper losses
Copper losses can be calculated as:
Copper losses = (Primary winding resistance^2) * Load current
Given:
Primary winding resistance (Rp) = 0.4 Ω
Power factor (pf) = 0.78
Load current can be calculated as:
I = Pout / (sqrt(3) * Vp * pf)
I = 26667 / (sqrt(3) * 4000 * 0.78)
I ≈ 6.082 A
Copper losses = (0.4^2) * 6.082
Copper losses ≈ 0.9696 kW
Step 4: Calculate the loss component of the output power
The total losses can be calculated as:
Total losses = Copper losses + Iron losses
Given:
Iron losses = 2 kW
Total losses = 0.9696 + 2
Total losses ≈ 2.9696 kW
Step 5: Calculate the efficiency
Efficiency (η) can be calculated as:
η = (Output power - Losses) / Input power
η = (Pout - Total losses) / Pin
η = (26667 - 2.9696) / 40000
η ≈ 0.6668
The efficiency of the transformer on full load is approximately 66.68%.
To determine the efficiency of the transformer on full load, we need to calculate the copper losses and the combined total losses.
First, let's calculate the copper losses in the primary and secondary windings. The copper loss formula is given by:
Copper Loss = (I1^2 * R1) + (I2^2 * R2),
where I1 and I2 are the primary and secondary currents, and R1 and R2 are the primary and secondary winding resistances, respectively.
To find the primary and secondary currents, we can use the following formulas:
I1 = (VA) / (√3 * V1),
I2 = (VA) / (√3 * V2),
where VA is the apparent power in kVA, V1 is the primary voltage in volts, and V2 is the secondary voltage in volts.
Plugging in the given values, we have:
I1 = (300 kVA) / (√3 * 4000 V) = 0.0866 A,
I2 = (300 kVA) / (√3 * 200 V) = 0.1733 A.
Now we can calculate the copper losses:
Copper Loss = (0.0866^2 * 0.4 Ω) + (0.1733^2 * 0.0015 Ω)
= 0.003876 + 0.00004497
= 0.00392197 Ω.
Next, let's calculate the total losses. The total losses are the sum of the copper losses and iron losses, given by:
Total Losses = Copper Loss + Iron Loss
= 0.00392197 Ω + 2 kW (convert to watts)
= 0.00392197 Ω + 2000 W
= 2000.00392197 W.
Finally, we can calculate the efficiency. The efficiency formula is given by:
Efficiency = (Output Power) / (Input Power),
where Output Power = VA - Iron Loss and Input Power = VA.
Substituting the given values, we have:
Output Power = 300 kVA - 2 kW (convert to kVA)
= 300 kVA - 2 kVA
= 298 kVA,
Input Power = 300 kVA,
Efficiency = (298 kVA) / (300 kVA)
= 0.9933 (approximately 99.33%).
Therefore, the efficiency of the transformer on full load is approximately 99.33%.