# Transformers confusion with output voltage and number of iron cores used

I have been wondering about an experiment which can help me observe that the output voltage of a transformer increases as we increase the number of cylindrical iron cores to an experimental transformer in normal high school labs. So I have indeed conducted this experiment, and I notice that as I put more and more iron cores, the output voltage seems to go up given that I do not change the number of turns of the coils, and I keep the input voltage the same.

My question is why does it does it do this?

My thoughts: If we increase the number of iron cores, then this establishes a stronger magnetic field when the primary current enters the coils. A stronger magnetic field means a greater change in magnetic flux, and therefore the greater emf induced on the secondary coils (how about primary coils)?

There was a suggestion from other stack exchange sites that greater iron cores inserted implies a stronger current, but I do not understand why this is the case. For instance I would argue that:

$$\frac{I_s}{I_p}=\frac{V_p}{V_s}$$

Since, Ip increases (under the assumption that current increases, although I am not sure if primary current or secondary current increases), Is will get smaller over time as there will be some losses as it is moving through the iron core, then it means given Vp remains constant, Vs will therefore increase?

(Is = secondary current, Ip = primary current, Vs= secondary voltage, Vp = primary voltage).

My confusions:

i) Why do I observe an increase in output voltage when I add more iron cores to my transformer? How do I explain this using the laws of physics?

ii) Is there a relationship between the number of iron cores used, and the output voltage in the transformer?

I would appreciate the clarification as I am genuinely very confused with transformers at the moment.

• Could you add a photo? – John Birckhead May 22 '19 at 16:41
• ... or draw a picture. – Andy aka May 23 '19 at 7:41

Why do I observe an increase in output voltage when I add more iron cores to my transformer? How do I explain this using the laws of physics?

Think about it it must be better than air, why would this happen? One is because there is less leakage of magnetic fields. The other is that magnetic fields flow better through iron than air.

Is there a relationship between the number of iron cores used, and the output voltage in the transformer?

You would really have to describe what the total construction of the transformer looks like. Since this is an academic exercise I'll give you a hint:

Also: Air is less efficient, iron is more efficient. The more iron you have the better magnetic permeability you have between the primary and secondary. You could think of air as more of a magnetic resistor than iron. The more Iron you add the better the magnetic permeability you have between primary and secondary cores.

In short:

to find the magnetic field through the core use this.

$$\ V_p = N_p\frac{d\Phi}{dt}\$$

Then find the B field:

$$\ B=\frac{\Phi}{A} \$$

Then insert the magnetic permeability of the material:

• I am still very confused why there is an increase output voltage, I can not seem to picture why this might happen even after reading your explanation. – Aurora Borealis May 23 '19 at 16:00
• Air is less efficient, iron is more efficient. The more iron you have the better magnetic permeability you have between the primary and secondary. You could think of air as more of a magnetic resistor than iron. The more Iron you add the better the magnetic permeability you have between primary and secondary cores. – Voltage Spike May 23 '19 at 16:04
• Thank you, is there any way we can try to understand this using mathematical formulas in physics? – Aurora Borealis May 23 '19 at 16:06
• Yep, it gets complicated, you need to know the equations for an ideal transformer iopscience.iop.org/article/10.1088/1361-6404/aaabc4 The $\frac{d\Phi}{dt}$ term is then replaced by the magnetic equations found here: mysite.du.edu/~jcalvert/tech/transfor.htm – Voltage Spike May 23 '19 at 16:10
• Its because more of the area is iron instead of air – Voltage Spike May 23 '19 at 16:26

An ideal transformer has this realtion: $$\frac{V_P}{V_S}=\frac{N_P}{N_S}$$

Which for your transformer is not the case due to large leakage inductance. When you increase the core size makes this transformer more better in terms of coupling factor between primary and secondary coil (less leakage).

Real transformer $$\X_P\$$ is the primary coil leakage reactance and $$\a^2\cdot X_S\$$ is the reflected secondary coil reactance on the primary side. You can see, that this leakage is causing a voltage drop.