Equivalent Circuit of an Electrical Transformer

 

The equivalent circuit of an electric transformer is its theoretical representation by using standard linear active and passive electrical elements, while retaining all the electrical properties of an actual transformer. 

 

An equivalent circuit of a transformer, or any other electrical machine, is the simplest circuit of that machine, which is used to make various calculations easier and to analyze the performance of machine without actually loading it

 

In transformers, the equivalent circuit helps us to calculate their core losses, copper losses, magnetizing current and other performance parameters in a straightforward manner.

 

Since, the equivalent circuit of any electrical machine retains all the electrical properties of that machine, Therefore, it is assumed that one should know all the basics of that particular machine. 

 

So, before moving on to the main concept of the equivalent circuit of an electrical transformer, it is recommended that first go through the basics of electric transformer, such as: What is a transformer? Working principle of transformer, types of transformers, and constructional features of transformer, etc. Studying these topics will make it easier to grasp the equivalent circuit of an electric transformer.

Equivalent Circuit of an Electrical Transformer

 

In the basics of transformers, we study that there are two sets of windings, the primary and the secondary, which are mounted on the magnetic core, as shown in the given figure below. 

In that article, we first explain the working of transformers for an ideal transformer and then move on to the practical one. 

There we discussed that in an ideal transformer, there are no losses, meaning the resistance of its windings is zero, and the reluctance of the magnetic core is also zero. Therefore, all the magnetic flux passes through the core, resulting in no leakage of flux. However, in practical transformers, these conditions do not hold true. A practical transformer has losses, leakage flux and also its magnetic core offers some reluctance to the magnetic flux.

In practical transformers, both the primary and secondary winding have finite resistance which is associated with the copper losses (I²R) of the transformer. Let us say for the given example, R₁ and R₂ is the resistance for both the primary and secondary winding respectively, which are distributed over the length.

 

In an ideal transformer, the magnetic core has zero reluctance, meaning all the magnetic flux passes through the core. However, it is not possible in practical transformers. In reality, while a major part of a total flux is confined to the core as mutual flux ɸ linking both the primary and secondary, a small amount of flux does leak through paths, which lie mostly in air and link separately to the individual winding. 

 

Below the given figure shows the schematic of a practical transformer on load. In the given figure ɸ_L1 and ɸ_L2 is the leakage flux caused by primary MMF and secondary MMF respectively and the path of the leakage is shown by the green dotted line.

ɸ_L1 Is the leakage flux caused by the primary MMF which links with the primary winding itself, similarly ɸ_L2 is the leakage flux caused by secondary MMF which links to secondary winding, thereby causing self-linkages of two windings. The self-linkage caused by leakage flux and the winding MMF are linearly related with each other in each winding. Therefore, contributing constant leakage inductance or leakage reactance corresponding to the frequency at which the transformer operates. Let us say the leakage reactance offered by the primary winding is X_L1 and the leakage reactance offered by the secondary winding is X_L2.

 

In practical transformers, both the resistance and the leakage reactance of the winding are considered series effects. Therefore, the practical transformer can now be represented with the help of their winding resistance and leakage reactance as shown in the given figure. Theoretically, this representation is referred to as a semi-ideal transformer because it accounts for the effect of winding resistance (i.e. copper losses) and leakage reactance, but does not include the effect of resistance and reluctance of the magnetic core. 

Since the magnetic core practically has some resistance and reluctance, so when the transformer is on no load, connected to the AC supply then the primary winding draws some no-load current (I_o). This no-load current is the combination of the two components of current: the one component of current (I_c) is dissipated in the magnetic core resistance (core losses) and the other component of current (I_m) is used to establish the magnetic field in the magnetic core. 

 

And when the transformer is loaded, the load draws the current I₂ from the transformer's secondary winding. This secondary current I₂ produces its own MMF that opposes the main flux. So, in order to maintain main flux constant and independent of load, the primary winding draws more current which counterbalances the secondary current requirement. 

 

So, in a practical transformer under load condition, the current drawn by primary winding is the sum of no load current and the secondary current referred to the primary side.

 

I₁  =  I_o  +  I₂’

 

where,

 

• I₁  =  Primary Current
• I_o  =  No load current
• I₂’  =  Secondary current referred to primary side 

 

The concept of referring is discussed further in this article.

 

So based on the above discussion, the above equivalent circuit of transformer can now be redrawn as,

Above the given figure shows the equivalent circuit of transformer, in which:

• R₁ and R₂ represent the resistance of primary and secondary winding, respectively.
• X_L1 and X_L2 represent the leakage reactance of primary winding and secondary winding, respectively.
• R_c represents the core loss resistance 
• And, X_m represents the magnetizing reactance of the core.

In the given figure above, we clearly see that the current drawn by the primary winding I1 is divided into two components, i.e. the no load current Io and the current I₂’ current which counterbalances the effect of secondary current. The no load current Io is further divided into two components: one is passed through the branch R_C, which represents the core loss branch of the transformer and the other component of the current is passed through the branch X_m, which represents magnetizing reactance of the transformer. 

 

However, there is a discrepancy in the above equivalent circuit that is transformer action, complicates the calculations. So, to remove this transformer action or that ideal transformer in the above equivalent circuit there is a certain step of referring the secondary side element to the primary side.  

 

The equations supporting the process of referring the secondary side elements to the primary side is shown below.

Secondary side quantities referred to primary

 

• X_L2’  =  a² * X_L2
• R₂’  =  a² * R₂
• V₂’  =  a² * V₂
• I₂’  =  (1/a)² * I₂ 

where, a is the transformation ratio of the transformer

These secondary side quantities referred to the primary side represent the image of the secondary side quantities which can be used in the primary side to make calculations easier without altering their characteristics. So, after referring secondary side quantities to the primary side, the equivalent circuit now can we redrawn as follows:

In the equivalent circuit of the transformer shown above, there are no parameters on the secondary side therefore, there is no need to show that ideal transformer. After removing that ideal transformer, the equivalent circuit of the transformer can now be redrawn as shown in the given figure.

Above the given figure represents the final equivalent circuit of a transformer referred to the primary side. This equivalent circuit can also be redrawn by referring the primary elements to the secondary side, resulting in the equivalent circuit of a transformer referred to the secondary side.

 

The equations supporting the process of referring the primary element to the secondary side are as follows:

 

• X_L1’  =  (1/a)² * X_L2
• R₁’  =  (1/a)² * R₂
• V₁’  = (1/a)² * V2
• I₁’  =  a² * I₂ 
• R_c’  =  (1/a)² * R_c
• X_m’  =  (1/a)² * X_m

 

With the understanding that all the quantities have been referred to a particular side; either primary side or secondary side, Equivalent circuit of the transformer can be drawn referred to either side, depending on the convenience.

 

The circuit deduced in the above discussion is the final equivalent circuit of the transformer and is adequate for the most power and radio frequency transformers. In transformers operating at higher frequencies, such as pulse transformers and other electronics transformers, the interwinding capacitances are often significant and must be included in the equivalent circuit.