Basic Equations of Single Phase Transformer and Applications of Single Phase Transformer
Basic Equations of Single Phase Transformer and Applications of Single Phase Transformer
Let the applied voltage V1 applied to the primary of a transformer, with secondary open-circuited, be sinusoidal (or sine wave).
Then the current I1, due to applied voltage V1, will also be a sine wave.
The mmf N1 I1 and core flux Ø will follow the variations of I1 closely.
That is the flux is in time phase with the current I1 and varies sinusoidally.
EMF Equation of Single Phase Transformer:
Let the applied voltage V1 applied to the primary of a transformer, with secondary open-circuited, be sinusoidal (or sine wave).
Then the current I1, due to applied voltage V1, will also be a sine wave.
The mmf N1 I1 and core flux Ø will follow the variations of I1 closely.
That is the flux is in time phase with the current I1 and varies sinusoidally.
Let,
NA = Number of turns in primary
NB = Number of turns in secondary
Ømax = Maximum flux in the core in webers = Bmax X A f = Frequency of alternating current input in hertz (HZ)
As shown in figure above, the core flux increases from its zero value to maximum value Ømax in one quarter of the cycle , that is in ¼ frequency second.
Therefore, average rate of change of flux = Ømax/ ¼ f = 4f ØmaxWb/s
Now, rate of change of flux per turn means induced electro motive force in volts.
Therefore,
average electro-motive force induced/turn = 4f Ømaxvolt
If flux Ø varies sinusoidally, then r.m.s value of induced e.m.f is obtained by multiplying the average value with form factor.
Form Factor = r.m.s. value/average value = 1.11.
Therefore, r.m.s value of e.m.f/turn = 1.11 X 4f Ømax = 4.44f Ømax Now, r.m.s value of induced e.m.f in the whole of primary winding
= (induced e.m.f./turn) X Number of primary turns
Therefore,
EA = 4.44f NAØmax = 4.44fNABmA
Similarly, r.m.s value of induced e.m.f in secondary is
EB = 4.44f NB Ømax = 4.44fNBBmA
In an ideal transformer on no load, VA = EA and VB = EB , where VB is the terminal voltage
Voltage Transformation Ratio:
The ratio of secondary voltage to primary voltage is known as the voltage transformation ratio and is designated by letter K.
i.e.
Voltage transformation ratio, K = V2/V1 = E2/E1 = N2/N1
Current Ratio:
The ratio of secondary current to primary current is known as current ratio and is reciprocal of voltage transformation ratio in an ideal transformer.
Transformer on No Load:
When the primary of a transformer is connected to the source of an ac supply and the secondary is open circuited, the transformer is said to be on no load.
The Transformer on No Load alternating applied voltage will cause flow of an alternating current I0 in the primary winding, which will create alternating flux Ø.
No-load current I0, also known as excitation or exciting current, has two components the magnetizing component Im and the energy component Ie.
Im is used to create the flux in the core and Ie is used to overcome the hysteresis and eddy current losses occurring in the core in addition to small amount of copper losses occurring in the primary only (no copper loss occurs in the secondary, because it carries no current, being open circuited.)
From vector diagram shown in above it is obvious that
- Induced emfs in primary and secondary windings, E1 and E2 lag the main flux Ø by and are in phase with each other.
- Applied voltage to primary V1 and leads the main flux Ø by and is in phase opposition to E1.
- Secondary voltage V2 is in phase and equal to E2 since there is no voltage drop in secondary.
- Im is in phase with Ø and so lags V1 by
- Ie is in phase with the applied voltage V1.
- Input power on no load = V1Ie = V1I0 cos Ø0 where Ø0 = tan-1
Transformer on Load:
The transformer is said to be loaded, when its secondary circuit is completed through an impedance or load.
The magnitude and phase of secondary current (i.e. current flowing through secondary) I2 with respect to secondary terminals depends upon the characteristic of the load
i.e. current I2 will be in phase, lag behind and lead the terminal voltage V+2+ respectively when the load is non-inductive, inductive and capacitive.
The net flux passing through the core remains almost constant from no-load to full load irrespective of load conditions and so core losses remain almost constant from no-load to full load.
Vector diagram for an ideal transformer supplying inductive load is shown
Resistance and Leakage Reactance In actual practice, both of the primary and secondary windings have got some ohmic resistance causing voltage drops and copper losses in the windings.
In actual practice, the total flux created does not link both of the primary and secondary windings but is divided into three components namely the main or mutual flux Ø linking both of the primary and secondary windings, primary leakage flux ØL1 linking with primary winding only and secondary leakage flux ØL2 linking with secondary winding only.
The primary leakage flux ØL1 is produced by primary ampere-turns and is proportional to primary current, number of primary turns being fixed.
The primary leakage flux ØL1is in phase with I1 and produces self induced emf ØL1 is in phase with I1 and produces self induced emf EL1 given as 2f L1 I1 in the primary winding.
The self induced emf divided by the primary current gives the reactance of primary and is denoted by X1.
i.e. X1 = EL1/I1 = 2πfL1I1/I1 = 2FL1,
Similarly leakage reactance of secondary X2 = EL2/E2 = 2fπL2I2/I2 = 2πfL2
Equivalent Resistance and Reactance.
The equivalent resistances and reactance’s of transformer windings referred to primary and secondary sides are given as below Referred to primary side Equivalent resistance,
Equivalent resistance, = X’1 = Referred to secondary side Equivalent resistance,
The equivalent resistance, = X2 + K2X1 Where K is the transformation ratio.
EQUIVALENT CIRCUIT OF Single Phase Transformer
The equivalent impedance of transformer is essential to be calculated because the electrical power transformer is an electrical power system equipment for estimating different parameters of electrical power system which may be required to calculate total internal impedance of an electrical power transformer, viewing from primary side or secondary side as per requirement.
This calculation requires equivalent circuit of transformer referred to primary or equivalent circuit of transformer referred to secondary sides respectively.
Percentage impedance is also very essential parameter of transformer.
Special attention is to be given to this parameter during installing a transformer in an existing electrical power system.
Percentage impedance of different power transformers should be properly matched during parallel operation of power transformers.
The percentage impedance can be derived from equivalent impedance of transformer so, it can be said that equivalent circuit of transformer is also required during calculation of % impedance.
Equivalent Circuit of Transformer Referred to Primary
For drawing equivalent circuit of transformer referred to primary, first we have to establish general equivalent circuit of transformer then, we will modify it for referring from primary side.
For doing this, first we need to recall the complete vector diagram of a transformer which is shown in the figure below.
Let us consider the transformation ratio be,
In the figure right, the applied voltage to the primary is V1 and voltage across the primary winding is E1.
Total current supplied to primary is I1.
So the voltage V1 applied to the primary is partly dropped by I1Z1 or I1R1 + j.I1X1 before it appears across primary winding.
The voltage appeared across winding is countered by primary induced emf E1.
The equivalent circuit for that equation can be drawn as below,
From the vector diagram above, it is found that the total primary current I1 has two components, one is no – load component Io and the other is load component I2′.
As this primary current has two a component or branches, so there must be a parallel path with primary winding of transformer.
This parallel path of current is known as excitation branch of equivalent circuit of transformer.
The resistive and reactive branches of the excitation circuit can be represented as
The load component I2′ flows through the primary winding of transformer and induced voltage across the winding is E1 as shown in the figure right.
This induced voltage E1transforms to secondary and it is E2 and load component of primary current I2′ is transformed to secondary as secondary current I2. Current of secondary is I 2.
So the voltage E2 across secondary winding is partly dropped by I2Z2 or I2R2 + j.I2X2 before it appears across load. The load voltage is V2.
From above equation, secondary impedance of transformer referred to primary is,
So, the complete equivalent circuit of transformer referred to primary is shown in the figure below,
Approximate Equivalent Circuit of Transformer
Since Io is very small compared to I1, it is less than 5% of full load primary current, Iochanges the voltage drop insignificantly.
Hence, it is good approximation to ignore the excitation circuit in approximate equivalent circuit of transformer.
The winding resistanceand reactance being in series can now be combined into equivalent resistance and reactance of transformer, referred to any particular side. In this case it is side 1 or primary side.
Equivalent Circuit of Transformer Referred to Secondary
In similar way, approximate equivalent circuit of transformer referred to secondary can be drawn.
Where equivalent impedance of transformer referred to secondary, can be derived as
VOLTAGE REGULATION
The voltage regulation is the percentage of voltage difference between no load and full load voltages of a transformer with respect to its full load voltage.
Explanation of Voltage Regulation of Transformer
Say an electrical power transformer is open circuited, means load is not connected with secondary terminals.
In this situation, the secondary terminal voltage of the transformer will be its secondary induced emf E2.
Whenever full load is connected to the secondary terminals of the transformer, rated current I2 flows through the secondary circuit and voltage drop comes into picture.
At this situation, primary winding will also draw equivalent full load current from source.
The voltage drop in the secondary is I2Z2 where Z2 is the secondary impedance of transformer.
Now if at this loading condition, any one measures the voltage between secondary terminals, he or she will get voltage V2 across load terminals which is obviously less than no load secondary voltage E2 and this is because of I2Z2 voltage drop in the transformer.
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