# Background

I have few old power transformers laying around and right now I need one for a project I am working at. The only problem is that they were not labeled properly (or at all) by the manufacturer so I have no reference point regarding which wires are for the primary coil and which are for the secondary coils.

A well designed transformer would normally have a label that, besides the input/output voltages and/or maximum supported amperage/power, would also show a map between the transformer tap numbers and their corresponding voltage.

Since that's not the case I have somehow to figure out which is what by using an invasive method being careful not to destroy the device nor risking my life (yes, main AC voltages can kill you). There are different types of transformers, here we deal only with single-phase isolated transformers. For RF transformers you are better off watching this YouTube video.

## What is a voltage transformer

A voltage transformer is an electrical device that allows us to step-up or step-down an **AC** voltage supply. It works based on the principle of Faraday's Law of induction by converting electrical energy from one value to another. Please note that a transformer works only with Alternating Current (AC). So if the input current is Direct Current (DC) then the transformer would output nothing (0V).

A step-down voltage transformer converts a higher AC voltage (eg. 110/220V) to a lower usable AC voltage (eg. 5V,9V,12V). A step-up voltage transformer converts a lower AC voltage (eg. 110V) to a higher AC voltage (eg. 220V).

A single-phase transformer consists of a magnetic core, a primary coil and one or more secondary coils. The primary coil represents the input and is connected to the source voltage and each secondary coil represents the converted output voltage and is normally connected to a load (which could be anything, including an complex electrical circuit). In order to transform the voltage the coil sizes must be different. When they are the same then no transformation occurs (such device it's normally useless with an exception).

**Please note that the primary coil is isolated electrically from the secondary coil**. Such a transformer is also called **isolation transformer**.

The coil consists of an electrical conductive wire (normally copper but I guess we could use also aluminum, silver, gold, etc) which is wrapped around the magnetic core. Let's note with N_{P} the number of turns necessary to form the primary coil and with N_{S} the number of turns necessary to form the secondary coil.

When AC current alternates in the primary coil it creates an alternate magnetic field around the primary coil. The primary coil input energy is stored in the magnetic field. The transformer magnetic core function is to increase this magnetic field by a factor of thousands such that more energy could be stored within that magnetic field. When the magnetic field changes the energy stored in the magnetic field is transferred via electromagnetic induction into the secondary coil(s), thus a current will start flowing through the secondary coil(s).

The coil size is directly proportional with its wire cross-section area and its length. The current that flows through each coil is proportional with the coil size: the larger the coil the more electrons flows through it.

**The more electrons flows through a wire the larger wire size is required to support that current** (R_{1}).

The voltage in the primary coil is directly proportional with the number of turns of the primary coil N_{P} and the change of the magnetic flux in time. The same is true for the secondary coil with respect to the N_{S}. Generally we can write this as:

V=N*d_{phi}/dt.

Since the magnetic flux (phi) is directly proportional with the magnetic field (B) and the magnetic core effective area (A), ie.Â phi=B*A, and since the B and A are the same (common) for both the primary and the secondary coil(s) it means that the magnetic flux phi is the same for all coils. We can rewrite the above equation with respect to the primary (P) and secondary (S) coils:

V_{P}=N_{P}*d_{phi}/dt <=> d_{phi}/dt=V_{P}/N_{P}

and

V_{S}=N_{S}*d_{phi}/dt <=> d_{phi}/dt=V_{S}/N_{S}

and thus

V_{P}/N_{P}=V_{S}/N_{S} <=> **V _{P}/V_{S}=N_{P}/N_{S}** - this is called the

**transformer turns ratio**(R

_{2}).

A transformer for which V_{P}/V_{S}>1 is called **step-down transformer**. A transformer for which V_{P}/V_{S}<1 is called **step-up transformer**.

Let's assume that we connect a load to a secondary coil. In an ideal transformer the input power must be equal with the output power:

P_{P}=V_{P}*I_{P}=P_{S}=V_{S}*I_{S} <=> **V _{P}/V_{S}=I_{S}/I_{P}** (R

_{3})

The relation between the transformer turns ratio and primary/secondary inductance is calculated as:

V_{P}/V_{S}=SQRT(L_{P}/L_{S}) (R_{4}), where L_{P} and L_{S} are the inductance of the primary and respectively secondary coil.

Rules of thumb:

- the greater the coil length, the higher its inducted voltage (wire length = distance an electron must travel in the electrical field), and vice-versa;
- the greater the coil length is the more it resists to the current flow and thus the greater its impedance is, and vice-versa;
- given the fact that the total transferred power between the primary and secondary coils is the same it means that the greater the coil length the smaller the current flow (would normally require a thin wire), and vice-versa.

## How to figure out an unknown transformer

We assume that we are investigating a **single-phase isolated transformer** which has the input-output tap wires already paired by their isolation color or somehow else (otherwise we should figure out how to pair them first by checking the continuity between them). We assume we don't know anything else about it. In order to figure out which wire is what we are going to use only the theoretical notions above.

Ideally we would have an variable AC transformer (Variac, called also autotransformer) capable of supplying low **AC voltages** and low currents (eg. 5VA) which would be linked in series with a current limiter resistor (its power drain should be greater than the transformer power drain). That way we could apply a low voltage at one coil then read the output voltage of every other coil. By repeating the process for all coils we would finally be able to sketch-up the transformer windings and their respective voltages. But we don't have such aparatus so we are going to try a different approach.

So, theoretically the unknown transformer could be either a step-up transformer (A), a step-down transformer (B) or both (C). We are going to analyze each case in order to identify a method that would allow us to identify the coils.

#### (A) step-down transformer

In case of a step-down transformer (V_{P}/V_{S}>1) we would expect also that the transform turns ratio N_{P}/N_{S}>1, ie. N_{P}>N_{S}. By using (R_{3}) we deduce that V_{P}/V_{S}=I_{S}/I_{P }> 1. This means that the resistivity of the secondary coil is smaller than the one in the primary coil (R_{5}).

So by looking for the most resistive coil we find actually the transformer's primary coil.

#### (B) step-up transformer

In case of a step-down transformer (V_{P}/V_{S}<1) we would expect also that the transform turns ratio N_{P}/N_{S}<1, ie. N_{P}<N_{S}. By using (R3) we deduce that V_{P}/V_{S}=I_{S}/I_{P }< 1. This means that the resistivity of the secondary coil is greater than the one in the primary coil (R_{6}).

So by looking for the least resistive coil we find actually the transformer's primary coil.

## Examples

In order to test the above methodology we should run few examples. Unfortunately I have only step-down transformers so I cannot test the step-up transformer case. However, the principle should be verified for at least one category transformers and this is what we are going to do in the following.

Regarding the step-down transformers I have few of these with different design so hopefully our tests would cover as many as possible variations:

- a 220V to a single 12V output
- a 220V to 2 x 12V center-tap (CT)
- a 110V to multiple outputs: 10V, 26V and 30V
- a 230V to dual multi-tap outputs: 10V, 12V, 15V

You may say "but wait, these are not unknown transformers, at least you know their input/output voltages!" and that would be actually right. What I want to the next is to test the above theory. If it verifies then we can use it as a methodology to test even real-life unknown transformers. So let's get started!

#### Example 1

We have a known 220V to 12V step-down transformer. Normally I would expect that the thinnest most resistive wire to be linked to the primary coil. Since the input-output connection wires have the same gauge (2 red wires and 2 yellow wires having the same gauge) we can only verify their resistivities. Since the ratio between the primary to secondary coils voltages is 220/12=18.3 I would expect - according to (R_{4}) - that the squared root of the ratio between the primary and secondary coils inductance to be about 18.3. Let's measure first the coils' inductance:

- the inductance of the coil connected to the red wire is L
_{r}=7.62H - the inductance of the coil connected to the yellow wire is L
_{y}=27.7mH

So judging by their inductivity I can safely assume the red wire gives the primary coil and the yellow wire gives the secondary coil. If that's true then I would expect that SQRT(L_{r}/L_{s}) ~ 18.3. We poke the SQRT(L_{r}/L_{s}) in the calculator and we get ~ 16.6 which is quite close to the theoretical value (-9.3%).

#### Example 2

We have a known 220V to 2 x 12V center-tap step-down transformer. This one has 2 thick red wires, 2 thin blue wires and one thin black wire. Since we have two 12V outputs the total output voltage is 24V. I would expect that - according to (R_{4}) - the squared root of the ratio between the primary and secondary coils inductance to be about 220/12=18.3. Let's measure first the coils' inductance:

- the inductance of the coil connected to the red wire is L
_{r}=12.85H - the inductance of the coil connected between the blue-black wire is L
_{bl1}=47.25mH - the inductance of the coil connected between the other blue-black wire is L
_{bl2}=47.25mH

So judging by their inductivity I can safely assume the red wire is the primary coil and the blue is the the secondary center-tapped coil. If that's true then I would expect that SQRT(L_{r}/L_{bl1}) and SQRT(L_{r}/L_{bl2}) to be around 18.3. A simple math shows us that SQRT(Lr/Lbl1) ~ 16.5.Â which is quote close to the theoretical value (-9.8%).

#### Example 3

We have a known 110V to multiple outputs (10V, 26V and 30V) step-down transformer. This one has 2 thick red wires, 2 thin blue wires, two thin yellow wires and 2 thin red wires. This one is probably by far the most complicated transformer since it has multiple output coils which can be in series or in parallel connected, etc. However, the principle stated above apply even here.

What I want to do first is to check the coils inductance. By doing that I might guess which wires are connected to the primary coil:

- the inductance of the thick red wire pair L
_{R}=430mH - the inductance of the blue wire pair L
_{bl}=29.55mH - the inductance of the yellow wire pair is L
_{y}=4.965mH - the inductance of the thin red wire pair is L
_{r}=39.4mH

We are going to assume that the thick red wire with the inductance of L_{R}=430mH is the one connected to the primary coil. What I want to do next is to check if there is any continuity between the other wires with a distinct insulator color. After I checked the cross-continuities I found no continuity between wires that don't belong to the same pair (given by wire color/thickness). So these secondary coils are totally isolated from each other.

Ok, let's start methodically. First we figure it out what are the turns ratio between the supposedly primary and secondary coils:

- for the 30V output: V
_{R}/V_{r}=110/30=3.67 - for the 26V output: V
_{R}/V_{bl}=110/26=4.23 - for the 10V output: V
_{R}/V_{y}=110/10=11

If the theory is right we can expectat that the inductance ratios SQRT(L_{R}/L_{r}) ~ 3.67, SQRT(L_{R}/L_{bl}) ~ 4.23 and SQRT(L_{R}/L_{y}) ~ 11.

Let's check it:

- SQRT(L
_{R}/L_{r}) = SQRT(430mH/39.4mH) =3.3 which is close to the expected 3.67 (10% less) - SQRT(L
_{R}/L_{bl}) = SQRT(430mH/29.55mH) =3.82 which is close to the expected 4.23 (9.7% less) - SQRT(L
_{R}/L_{y}) = SQRT(430mH/4.965mH) =9.3 which is close to the expected 11 (15.4% less)

#### Example 4

This case is a combination ofÂ examples 2 and 3 above. I've done the same steps as above and I've got values quite close to the theoretical expected one, with a tolerance of -5%. This transformers (see the Fig 2. above) seems to give the most close results than any other tested transformer (see Fig. 1 above). However, it was also the most expensive one so a better quality and a greater accuracy is somehow expected.

**Note**: if yo don't know how to measure the inductance of a coil then keep in mind that there are at least three simple methods that help:

- by using a DMM that can measure the inductance (L) or a LCR meter (can measure L,C and R)
- by using a function generator (SINE signal), 2 DMMs and a circuit of a known 1% tolerance resistor
*(X*in series with the inductor which inductance value (_{L})*L*) is unknown- measure the voltage across the resistor (V
_{R}) and the voltage across inductor (V_{L}) - adjust the signal frequency until you find the cutoff frequency
*f*for which V_{cutoff}_{R}= V_{L} - use the resistor value
*(X*, the found cutoff frequency_{L})*f*and then find the unknown inductance value (_{cutoff}*L*) from the formula: X_{L}=2*pi*f_{cutoff}*L

- measure the voltage across the resistor (V
- by using a function generator (SINE signal), an oscilloscope and a RLC series-circuit with a known resistor/capacitor and an unknown inductor; by finding and using the resonance frequency
*f*for which the inductive reactance (_{r}*X*=2*pi*f*L) of inductor becomes equal in value with the capacitive reactance (_{L}*X*=1/(2*pi*f*C)) of the capacitor we are able to determine the unknown inductance_{C}*f*=1/(2*pi*SQRT(L*C)) <=> L=1/[(2*pi*f)_{r}^{2}*C]

## Conclusion

The method described above works with a margin of error that, depending on the transformer design and quality, might vary - like in my example above - around 5% to 10%.

Now, if you think that this article was interesting don't forget to rate it. It shows me that you care and thus I will continue write about these things.

#### Eugen Mihailescu

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