Electrical Machines · Transformers

Efficiency of a transformer

Dr. Mithun Mondal BITS Pilani, Hyderabad Campus Electrical Machines

Demonstrative Video

Condition for the maximum efficiency
\[\begin{aligned} \eta & =\dfrac{\mbox{output power}}{\mbox{input power}}=\dfrac{\mbox{output power}}{\mbox{output power+losses}}\\ & =\dfrac{\mbox{output power}}{\mbox{output power+iron losses+copper losses}}\\ & =\dfrac{V_{2}I_{2}\cos\Phi_{2}}{V_{2}I_{2}\cos\Phi_{2}+P_{i}+P_{c}} \end{aligned}\]
where
\[\begin{array}{cc} V_{2} & \mbox{Secondary terminal voltage}\\ I_{2} & \mbox{Full-load secondary current}\\ \cos\Phi_{2} & \mbox{Power factor of the load}\\ P_{i} & \mbox{Iron losses =hysterises +eddy current losses}\\ P_{c} & \mbox{Full load copper losses = }I^{2}R_{T} \end{array}\]
If \(x\) is the fraction of the full-load, then efficiency is given as
\[\eta_{x}=\dfrac{x\times\mbox{Output}}{x\times\mbox{Output}+P_{i}+x^{2}P_{c}}=\dfrac{xV_{2}I_{2}\cos\Phi_{2}}{xV_{2}I_{2}\cos\Phi_{2}+P_{i}+x^{2}I_{2}^{2}R_{T}}\]

Condition for the maximum efficiency

\[\eta = \dfrac{V_2I_2\cos\Phi_2}{V_2I_2\cos\Phi_2+P_i+I_2^2R_T} = \dfrac{V_2\cos\Phi_2}{V_2\cos\Phi_2+P_i/I_2+I_2R_T}\]
For maximum efficiency . Then is the fraction of full load KVA at which If . Hence, efficiency will be maximum when depends upon , is constant. Thus for a given Now,
All-Day Efficiency

  • Power transformer (at generating station) and Distribution transformer (distribution substation)

  • Power transformers are switched in or out of the circuit depending upon the load to be handled

  • Distribution transformer is never switched off remain irrespective of the load

  • In such case, constant loss continues to be dissipated

  • \[\begin{aligned} \%\mbox{all-day}\eta & =\dfrac{\mbox{Output energy in KWh during a day}}{\mbox{Input energy in KWh during a day}}\times100\\ & =\dfrac{\mbox{Output energy in KWh during a day}}{\mbox{Output energy +Energy spent for total losses}}\times100 \end{aligned}\]
    Concept of energy based efficiency arise - "all day efficiency"
Saving of Cu
SECTION 02

Auto-Transformer

auto
Auto

  • Transformer with one winding only

  • Part of the winding is common to both primary and secondary

  • No electrical isolation as in 2-winding transformer

  • Theory and operation remains similar

  • Because of one winding, uses less copper and hence cheaper

  • It is used where transformation ratio varies little from unity

    step autotransformer
    Step Autotransformer
    autotransformer
    Autotransformer

  • Volume (or weight) \(\propto\) length and area of cross-section length of conductor \(\propto\) number of turns cross-section depends on current Weight \(\propto\) product of current and number of turns

    Wt. of Cu in section AC is \(\propto\) \((N_1-N_2)I_1\) Wt. of Cu in section BC is \(\propto\) \(N_2(I_2-I_1)\) \(\therefore\) Total Wt. of Cu \(\propto\) \((N_1-N_2)I_1+N_2(I_2-I_1)\)

  • \[\therefore \dfrac{\mbox{Wt. of Cu in auto-transformer}}{\mbox{Wt. of Cu in ordinary transformer}} = \dfrac{(N_1-N_2)I_1+N_2(I_2-I_1)}{N_1I_1+N_2I_2} = (1-K)\]
    Total Wt. of Cu\(N_2I_2\)\(\propto\)Wt. of Cu in primary If a 2-winding T/F perform the same task, then

    \(\Rightarrow\) \(W_a\) = \((1-K) \times W_0\)

    \[\boxed{\text{Saving}=W_0-W_a=W_0-(1-K)W_0=KW_0}\]

    Hence, saving will increase as \(K\) approaches unity \(\Rightarrow\) Power transferred inductively is

    \[\boxed{P_{ind}=Input\times(1-K)}\]

    \(\Rightarrow\) The rest of the power is conductively transferred

    \[\boxed{P_{cond}=\text{Input}\times K}\]

    Advantages of Auto transformer

    • Less costly

    • Better regulation

    • Low losses as compared to 2-winding T/F of the same rating

    Disadvantages of Auto transformer

    • The secondary winding is not insulated from primary so if a low supply voltage is used from a high voltage and if a break occurs in secondary winding then full primary voltage comes across the secondary terminal which is dangerous to the operator and the equipment.

    • used only in limited places where a slight variation of the output voltage from input voltage is required.

    Applications of Auto transformer

    • Used as a starter to give 50-60% of full voltage to the stator of a squirrel cage induction motor during starting

    • used to give a small boost to a distribution cable, to correct the voltage drop

    • used as a voltage regulator

    • in power T & D system and also in audio system and railways