📐 Basic Parameters
Transformation Ratio (K)
\[K = \frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{I_1}{I_2}\]
Current Relations
\[I_{AB} = I_1\]
\[I_{BC} = I_2 - I_1\]
Turns Relations
\[N_1 = N_{AB} + N_{BC}\]
\[N_2 = N_{BC}\]
⚙️ Power Relations
Input KVA
\[\text{KVA}_{\text{input}} = V_1 I_1\]
Output KVA
\[\text{KVA}_{\text{output}} = V_2 I_2\]
Inductive (Transformed) Power
\[\text{KVA}_{\text{ind}} = (V_1 - V_2) I_1\]
Conductive Power
\[\text{KVA}_{\text{cond}} = V_2 I_1 = K \cdot V_1 I_1\]
Power Transformation Ratio
\[\frac{\text{KVA}_{\text{ind}}}{\text{KVA}_{\text{input}}} = 1 - K\]
Power Conduction Ratio
\[\frac{\text{KVA}_{\text{cond}}}{\text{KVA}_{\text{input}}} = K\]
💰 Material Savings & Weight
Copper Weight in Section AB
\[W_{AB} \propto (N_1 - N_2) I_1\]
Copper Weight in Section BC
\[W_{BC} \propto (I_2 - I_1) N_2\]
Total Copper Weight (Autotransformer)
\[W_{\text{auto}} \propto I_1(N_1 - N_2) + (I_2 - I_1)N_2\]
Total Copper Weight (Two-Winding)
\[W_{\text{2-wdg}} \propto I_1 N_1 + I_2 N_2\]
Weight Ratio
\[\frac{W_{\text{auto}}}{W_{\text{2-wdg}}} = 1 - K\]
Copper Saving
\[\text{Saving} = K \times W_{\text{2-wdg}}\]
\[\text{Saving (\%)} = K \times 100\%\]
📊 Efficiency & Losses
Full-Load Copper Losses
\[P_{cu(\text{auto})} = (1 - K) \times P_{cu(\text{2-wdg})}\]
Core/Iron Losses
\[P_i = \text{Same as 2-winding}\]
Core losses remain unchanged as flux density is same
Efficiency
\[\eta = \frac{\text{Output}}{\text{Output + Losses}} \times 100\%\]
\[\eta = \frac{V_2 I_2 \cos\phi_2}{V_2 I_2 \cos\phi_2 + P_i + P_{cu}} \times 100\%\]
Maximum Efficiency Condition
\[P_i = P_{cu}\]
\[x = \sqrt{\frac{P_i}{P_{cu(\text{FL})}}}\]
where x = fraction of full load
🔧 Impedance & Voltage Regulation
Equivalent Impedance
\[Z_{\text{eq(auto)}} = (1 - K) \times Z_{\text{eq(2-wdg)}}\]
Percentage Impedance
\[(\% Z)_{\text{auto}} = (1 - K) \times (\% Z)_{\text{2-wdg}}\]
Voltage Regulation
\[\text{VR} = \frac{V_2(\text{NL}) - V_2(\text{FL})}{V_2(\text{FL})} \times 100\%\]
Approximate Voltage Regulation
\[\text{VR} \approx \frac{I_2 R_{eq} \cos\phi + I_2 X_{eq} \sin\phi}{V_2} \times 100\%\]
Short Circuit Current
\[I_{SC} = \frac{I_{\text{rated}}}{\% Z / 100}\]
📈 KVA Rating Relationships
Autotransformer Rating from 2-Winding
\[\text{KVA}_{\text{auto}} = \frac{\text{KVA}_{\text{2-wdg}}}{1 - K}\]
Rating Multiplication Factor
\[\text{Factor} = \frac{1}{1 - K} = \frac{V_1}{V_1 - V_2}\]
🔍 Additional Important Relations
Turns Ratio (Step-Up)
\[a = \frac{N_1}{N_2} = \frac{V_1}{V_2} = \frac{1}{K}\]
No-Load Current
\[I_0 = \sqrt{I_w^2 + I_m^2}\]
Iw = working component, Im = magnetizing component
Equivalent Resistance (Primary)
\[R_{eq1} = R_1 + K^2 R_2\]
Equivalent Reactance (Primary)
\[X_{eq1} = X_1 + K^2 X_2\]
✅ Advantages of Autotransformers
- Lower Cost: Requires (1-K) times less copper compared to two-winding transformer
- Higher Efficiency: Lower copper losses result in better efficiency
- Compact Size: Smaller physical dimensions due to reduced material
- Better Regulation: Lower impedance leads to improved voltage regulation
- Lower Losses: Copper losses are (1-K) times that of two-winding type
- Higher KVA Rating: Can handle larger power with same core and copper
❌ Disadvantages of Autotransformers
- No Electrical Isolation: Primary and secondary are electrically connected
- Higher Short Circuit Current: Lower impedance results in higher fault currents
- Safety Concerns: Transfer of high voltage to low voltage side during faults
- Limited Transformation Range: Economical only when K > 0.5 (voltage ratio close to unity)
- Unstable Neutral: Issues with grounding in three-phase systems
🔌 Common Applications
Typical Uses:
- Starting of induction motors (reduced voltage starting)
- Interconnection of power systems with small voltage differences
- Voltage regulation in distribution systems (boosters)
- Laboratory voltage variations (variable autotransformers/Variacs)
- Audio systems and impedance matching
💡 Key Points to Remember
- Autotransformer is economical when K > 0.5
- Saving increases as K approaches 1 (voltage ratio near unity)
- All quantities (weight, losses, impedance) scale by factor (1-K)
- KVA rating increases by factor 1/(1-K) compared to 2-winding
- Not suitable where electrical isolation is required
- Core losses remain same as two-winding transformer