1. Transformers

4. Single Phase Transformers

5. Principles of Operation – Single Phase

6. Flux and Voltage 90o out of phase
Relationship of induced voltage and flux:
Also Eqn 1.148
Fundamental Transformer equation
Only accurate if leakage impedance of coil is negligible
Primary winding flux divided into leakage flux and mutual flux

7. Ideal Transformer Relationships

12. Exercise 2-1

13. 𝑒 1 𝑒 2 = 𝑁 1 𝑁 2 ≡𝑎
𝐼 2 𝐼 1 = 𝑁 1 𝑁 2 ≡𝑎
𝑍 1 𝑍 2 = 𝑁 1 𝑁 ≡ 𝑎 2

14. Non-ideal Transformer

15. Equivalent Circuits of a non-ideal Transformer

16. Equivalent Circuits of a non-ideal Transformer

17. 𝑎≡𝑡𝑢𝑟𝑛𝑠 𝑟𝑎𝑡𝑖𝑜 >1
𝐸 1 ≡𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
𝐸 2 ≡𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
𝑉 1 ≡𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
𝑉 2 ≡𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
𝐼 1 ≡𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝐼 2 ≡𝑠𝑒𝑐𝑜𝑛𝑒𝑎𝑟𝑦 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝐼 0 ≡𝑛𝑜−𝑙𝑜𝑎𝑑 𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑅 1 ≡𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑤𝑖𝑛𝑑𝑖𝑛𝑔
𝑅 2 ≡𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑤𝑖𝑛𝑑𝑖𝑛𝑔
𝑋 1 ≡𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑙𝑒𝑎𝑘𝑎𝑔𝑒 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒
𝑋 2 ≡𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑙𝑒𝑎𝑘𝑎𝑔𝑒 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒
𝐼 𝑚 , 𝑋 𝑚 ≡𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑧𝑖𝑛𝑔 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑛𝑑 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒
𝐼 𝑐 , 𝑅 𝑐 ≡𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑛𝑑 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑜𝑟𝑒 𝑙𝑜𝑠𝑠𝑒𝑠

19. Open-Circuit (or No-Load) Test
Here one winding is open-circuited and voltage---usually, rated voltage at rated frequency--is applied to the other winding. Voltage, current, and power at the terminals of this winding are measured. The open circuit voltage of the second winding is also measured, and from this measurement a check on the turns ratio can be obtained. It is usually convenient to apply the test voltage to the winding that has a voltage rating equal to that of the available power source. In step-up voltage transformers, this means that the open-circuit voltage of the second winding will be higher than the applied voltage, sometimes much higher. Care must be exercised in guarding the terminals of this winding to ensure safety for test personnel and to prevent these terminals from getting close to other electrical circuits, instrumentation, grounds, and so forth.
In presenting the no-load parameters obtainable from test data, it is assumed that voltage is applied to the primary and the secondary is open-circuited. The no-load power loss is equal to the wattmeter reading in this test; core loss is found by subtracting the ohmic loss in the primary, which is usually small and may be neglected in some cases. Thus, if Po, Io and Vo are the input power, current, and voltage, then the core loss is given by:
𝑃 𝑐 = 𝑃 𝑜 − 𝐼 𝑜 2 𝑅 1

20. The primary induced voltage is given in phasor form by:
𝑬 𝟏 = 𝑉 𝑜 ∠ 0 𝑜 −( 𝐼 𝑜 ∠ 𝜃 𝑜 )( 𝑅 1 +𝑗 𝑋 1 )
where 𝜃 𝑜 ≡ no-load power-factor angle = 𝑐𝑜𝑠 −1 𝑃 𝑜 𝑉 𝑜 𝐼 𝑜 <0.
Other circuit quantities are found from:
𝑅 𝑐 = 𝐸 𝑃 𝑐
𝐼 𝑐 = 𝑃 𝑐 𝐸 1
𝐼 𝑚 = 𝐼 − 𝐼 𝑐 2
𝑋 𝑚 = 𝐸 1 𝐼 𝑚
a≈ 𝑉 𝑜 𝐸 2

21. Short-Circuit Test
In this test, one winding is short-circuited across its terminals, and a reduced voltage is applied to the other winding. This reduced voltage is of such a magnitude as to cause a specific value of current---usually, rated current--to flow in the short-circuited winding. Again, the choice of the winding to be short-circuited is usually determined by the measuring equipment available for use in the test. However, care must be taken to note which winding is short-circuited, for this determines the reference winding for expressing the impedance components obtained by this test. Let the secondary be short-circuited and the reduced voltage be applied to the primary.
With a very low voltage applied to the primary winding, the cord-loss current and magnetizing current become very small, and the equivalent circuit reduces to that of Fig Thus, if Ps, Is, and Vs are the input power, current, and voltage under short circuit, then, referred to the primary,
𝑍 𝑠 = 𝑉 𝑠 𝐼 𝑠
𝑅 1 + 𝑎 2 𝑅 2 ≡ 𝑅 𝑠 = 𝑃 𝑠 𝐼 𝑠 2

23. 𝑋 1 + 𝑎 2 𝑋 2 ≡ 𝑋 𝑠 = 𝑍 𝑠 2 − 𝑅 𝑠 2
Given R1and a, R2 can be found. It is usually assumed that the leakage reactance is divided equally between the primary and the secondary; that is:
𝑋 1 = 𝑎 2 𝑋 2 = 1 2 𝑋 𝑠

41. Short Circuit Test:
(Impedance or copper-loss test)

42. Open Circuit Test:
Core-loss test
Usually done on the side of the transformer with the lower rated voltage

44. Transformer Characteristics:
Inrush
8 to 12 times rated current,
duration about 100 ms

45. Efficiency

46. Regulation

47. kVA Rating Apparent power or kVA is always inscribed on the nameplate
As long as the transformer delivers rated or reduced kVA, it will operate with nominal heat
If it is cooled, it can safely deliver higher-than-rated kVA
A transformer of 1000 kVA supplied with fans can deliver 1333kVA

48. Protective equipment such as circuit breakers and fuses, must be capable of safely withstanding the forces produced by the short-circuit currents.
It is customary in industry to rate the short-circuit capacity of equipment in terms of short-circuit MVA.

50. Three-Phase, Two-Winding Transformers

52. Review of Three-Phase Systems
P= 3 V L−L I L cos θ
Q= 3 V L−L I L sin θ
S= 3 V L−L I L *

53. Delta-Connected Load

54. Star-Connected Load

55. Electric Circuit Analysis

56. Star-Delta

57. Star-Star

59. Auto Transformers
Understanding and analyzing autotransformers is simplified by noting the following:
The ampere-turns of each coil are the same whether the transformer is connected as an autotransformer or as a two-winding transformer. That is,
(NI)= constant
In other words, the magnitude of the current through each coil remains the same, regardless of whether the transformer operates as an autotransformer or as a conventional two-winding transformer.

60. Since the current through each autotransformer winding is the same as in the conventional two-winding transformer, the winding loss remains the same. However, the efficiency of the autotransformer is increased if the output power is increased.
Neglecting losses, the complex power (kVA) at the input is equal to the complex power to the output. The kVA transformation capability of the autotransformer is the same as that of the two-winding transformer. An autotransformer, however, delivers higher kVA than the conventional transformer because of the direct electrical connections between the primary and secondary windings. In other words, part of the output kVA is conducted from the primary to the secondary winding. The conducted kVA is referred to as the untransformed kVA.

61. 4. The two-winding conventional transformer has its primary and secondary circuits electrically isolated, while in the autotransformer electrical disturbances in the primary can be easily passed to the secondary through their direct electrical connections.

64. Disadvantages of Autotransformers
The disadvantages of autotransformers are as follows:
They are not economical for voltage ratios larger than 2.
They require primary protective devices of higher capacity.
Because of the electrical conductivity of the primary and secondary windings, the lower voltage circuit is liable to be impressed upon by higher voltage. To avoid breakdown in the lower voltage circuit, it becomes necessary to design the low-voltage circuit to withstand higher voltage.
The autotransformer has a common terminal between the primary and the secondary windings and when the secondary is shorted, the voltage applied to the primary is much higher than its rated voltage. This results in higher short-circuit currents and thus requires more expensive protective devices.

65. 3. The connections on the primary and secondary sides must necessarily be the same, except when using interconnected starring connections. This introduces complications due to changing primary and secondary phase angles, particularly in the case-by-case of the delta-delta connection.
4. Because of a common neutral in a star-star connected autotransformer, it is not possible to ground the neutral of one side only. Both if its sides must have its neutrals either grounded or isolated.
5. It is more difficult to preserve the electromagnetic balance of the winding when voltage adjustment tappings are provided. It should be known that the provision of adjusting tapping on an autotransformer increases the frame size of the transformer considerably.

66. Parallel Operation of Transformers:
Equal impedance
Equal turns ratio
Equal phase shift between
primary and secondary
open-circuited voltages
Same phase rotation