Current transformers measure current or transfer energy from one circuit to another, so their design requires calculations different from their voltage transformer cousins. The reason for the difference is that current transformer magnetizing current is the load current itself, unlike voltage transformers, where magnetizing current is "separated" from the load current and has a value that's a small fraction of the total current at full load.

We want to supply the load with current I_{L} from a current source, producing primary winding current I_{P}*(Fig. 1)*. We're dealing with a current source, so the load voltage is set up by the load itself if it consumes current (like an incandescent lamp, voltage regulator, or zener diode).

To design the current transformer, we need to know its core shape, size, and material, as well as the number of turns to use. Select a toroidal core because it delivers minimum losses and provides appropriate flux coupling between the primary and secondary windings if the number of turns is high enough to cover most of the core surface.

According to Ampere's Law, current I_{P} flowing through a winding with N_{P} turns *(Fig. 2)* produces a magnetic field depending on the magnetic line length *l*_{M}, which is described by the differential equation:

_{p} × N_{p}*l*_{M}**(1)**

Integrating over the whole magnetic line length and assuming usage of a toroidal core, obtain:

_{p} × N_{p}*l*_{M}**(2)**

where *l*_{M} = median length of a magnetic line, which, for a toroidal core is:

*l*_{M} = _{MED}_{OUT} + D_{INN})/2**(3)**

where D_{MED} = median diameter; D_{OUT} = the toroidal core's outer diameter; and D_{INN} = the toroidal core's inner diameter.

Magnetic field H produces magnetic flux in the core, which has a density of B. This flux density depends on the core material relative permeability μ_{R}:

where B = _{R} × μ_{0} × H,_{0} = ^{–7}

Therefore, Equation 2 may be rewritten as:

_{p} × N_{p}_{R} × μ_{O}) × *l*_{M}**(4)**

To transfer energy with minimal losses, the magnetic core should not impose excessive losses while the energy transfer lasts. In other words, it should not saturate. Hence, the core flux density, B, should not exceed the saturation limit value of B_{SAT}. In addition, the primary side current I_{P}, which produces this B_{SAT}, should always be below some maximum value I_{P}^{MAX}. Therefore, Equation 4 can be rewritten as:

_{p}^{MAX} × N_{p}_{SAT}/(μ_{R} × μ_{O}) × *l*_{M}**(5)**

In our consideration, we're dealing with a constant current, which may be assumed as I_{P}^{MAX}. Therefore, we should either assign a value to N_{P} (to cover as much of the core surface as possible) and calculate *l*_{M}, and then the core dimension D_{MED}, or select a core and derive a proper N_{P}. The B_{SAT} value is a core datasheet item, as is the μ_{R}.

We don't care about the core cross-sectional area. Therefore, we can use a core of any thickness — only the core diameter is of interest.

There's a paradox associated with the relationships depicting current transformer operation:

_{p} × N_{p}_{L} × N_{S}**(6)**

where N_{S} = number of turns on the secondary winding. I_{P} is fixed, so the secondary side load current is also fixed and:

I_{L} = _{p} × (N_{p}/N_{S})**(7)**

This means that if fewer turns are put on the secondary side, we can obtain higher current in the secondary circuit. That is, if we have to replace an electric bulb in the secondary side with one having a higher current rating, we should take off a few turns from the secondary winding.

Or, if we use this transformer to supply some electronic schematic arrangement, we should know its consumption current I_{L} and wind the appropriate number of turns on the secondary side, as stated by Equation 7. A reasonable zener diode should protect the schematic from overvoltage if this occurs or if the secondary-side induced voltage is higher than allowed.

Therefore, at a given N_{P}, we can easily obtain the core size using Equation 5 or determine how many primary turns N_{P} should be wound on the given core having median length *l*_{M}. For example, if a current source produces I_{P} = 1.2 A for the load and we have to feed a 1.7-A incandescent lamp, it would be necessary to design a new transformer for this arrangement.

First, we should get a core suitable for this unit. Let it be a 36- by 23- by 8mm toroidal core with a D_{MED} of 29.5 mm. Assume that the relative permeability for this core is 3000 and B_{SAT} = 0.28 T. From Equation 5, we can easily determine the admissible number of turns on the primary side N_{P}, knowing that *l*_{M} = π × D_{MED}. Therefore:

N_{P} = _{SAT} / μ_{R} × μ_{0}) × *l*_{M}/I_{P}^{MAX}^{–7} × 3000) × 29.5 × 10^{–3}/1.2

Assume N_{P} = 6 turns, which is usually safe because the flux density saturation value was chosen far below the maximum, i.e., there's a safety margin.

Using Equation 7, it's easy to determine the number of turns of the secondary winding as 4.235 turns, which makes the secondary winding 4 turns. Assuming possible losses at the energy transfer, we can verify that this number fits the design pretty well.

It's obvious that in the case when the secondary current should be lower than the primary current, the number of turns on the secondary side would be higher than those on the primary one. That's why current transformers intended for high current measurement have many turns on the secondary side.

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