Advantage of Auto Transformer:
The weight of conductor for any winding depends upon two parameters:
 Current carried by the winding
 Number of turns required in the winding
Thus we can say that weight of conductor in a winding is directly proportional to its Ampere Turns i.e. MMF.
Now, we will focus our attention on the weight of conductor required for Auto Transformer and a Two Winding Transformer.
As discussed in earlier post Basic principle of Auto Transformer, the current carried by winding AC i.e. I_{AC}= I_{1} and that of winding BC i.e. I_{BC} = (I_{2}I_{1}). Also the number of turns for winding AC is (N_{1}N_{2}) and that for winding BC is N_{2}.
Therefore,
The weight of conductor required for Auto Transformer, W_{CAT}
∝Ampere turns of winding AC + Ampere turns of winding BC
∝ (N_{1}N_{2})xI_{1}+ N_{2}x(I_{2}I_{1})
∝ N_{1}I_{1}+N_{2}I_{2}2N_{2}I_{1}
But N_{2} / N_{1}= I_{1} / I_{2}
So, N_{2}I_{2}= N_{1}I_{1}
Therefore,
The weight of conductor required for Auto Transformer, W_{CAT}
∝ N_{1}I_{1}+N_{2}I_{2}2N_{2}I_{1}
∝ 2N_{1}I_{1 }– 2N_{2}I_{1}
∝ 2(N_{1} – N_{2})xI_{1} …………………….(1)
If we want to compare the weight of conductor required for Auto Transformer and Two Winding Transformer, then both the Transformers must have same voltage ratio (V_{2} / V_{1}), current ratio (I_{2} / I_{1}), input VA i.e. V_{1}I_{1}and output VA i.e. V_{2}I_{2}.
Assuming all the above parameters to be same for Two Winding and Auto Transformer,
Weight of conductor in Two Winding Transformer W_{CTWT,}
∝Ampere Turns of Primary + Ampere Turns of Secondary
∝ N_{1}I_{1}+ N_{2}I_{2}
But N_{1}I_{1}= N_{2}I_{2}
So,
Weight of conductor in Two Winding Transformer W_{CTWT},
∝ 2N_{1}I_{1} ……………………………………….(2)
Therefore from equation (1) and (2),
W_{CAT} / W_{CTWT}= [2(N_{2} – N_{1})xI_{1}] / 2N_{1}I_{1}
= 1 – N_{2}/N_{1}
= (1 – k)
As for step down Auto Transformer k<1, hence the weight of conductor required for Auto Transformer is less than that required for Two Winding Transformer.
Saving in Conductor = 1 – W_{CAT}/ W_{CTWT}
= 1 – (1 – k)
= k
Hence there is net saving of conductor in case of Auto Transformer. Let us assume that k =0.1, thus the saving in conductor for Auto Transformer will be only 10% but if k = 0.9 then saving in conductor will be 90% which is quite lucrative. Thus we can conclude that Auto Transformer is more economical when the voltage ratio k is more near to unity.
Another important aspect is core. As the conductor required for Auto Transformer is less than that required for Two Winding Transformer that means for Auto Transformer lower window dimension will be required. Thus by using Auto Transformer, there is net saving in Core material as well as conductor material, the saving will be more and more as we tend toward voltage ratio unity.
Owing to reduction in conductor and core material, the Ohmic loss in conductor and core loss is reduced considerably. Therefore an Auto Transformer has higher efficiency than Two Winding Transformer of same output.
Reduction in conductor material means lower value of winding resistance. Since a part of winding is common to both the Primary and Secondary circuit in Auto Transformer, leakage reactance will be less. Because of lower value of leakage reactance, a superior voltage regulation is achieved with Auto Transformer.
Disadvantage of Auto Transformer:

As seen earlier in this post that saving of conductor in Auto Transformer is k so saving in conductor decreases as the value of voltage ratio k decreases.

Another biggest disadvantage of Auto Transformer is direct electrical connection between the Primary and Secondary circuit. If Primary is supplied with high voltage then any open circuit in common winding will lead to dangerously high voltage on LV side which may damage load as well as this dangerously high voltage will be very harmful of working personnel. Thus special protection must be provided to prevent such an occurrence.