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 IL from a current source, producing primary winding current IP(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 IP flowing through a winding with NP turns (Fig. 2) produces a magnetic field depending on the magnetic line length lM, which is described by the differential equation:
Integrating over the whole magnetic line length and assuming usage of a toroidal core, obtain:
where lM = median length of a magnetic line, which, for a toroidal core is:
where DMED = median diameter; DOUT = the toroidal core's outer diameter; and DINN = 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 μR:
where B =
Therefore, Equation 2 may be rewritten as:
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 BSAT. In addition, the primary side current IP, which produces this BSAT, should always be below some maximum value IPMAX. Therefore, Equation 4 can be rewritten as:
In our consideration, we're dealing with a constant current, which may be assumed as IPMAX. Therefore, we should either assign a value to NP (to cover as much of the core surface as possible) and calculate lM, and then the core dimension DMED, or select a core and derive a proper NP. The BSAT value is a core datasheet item, as is the μR.
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:
where NS = number of turns on the secondary winding. IP is fixed, so the secondary side load current is also fixed and:
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 IL 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 NP, we can easily obtain the core size using Equation 5 or determine how many primary turns NP should be wound on the given core having median length lM. For example, if a current source produces IP = 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 DMED of 29.5 mm. Assume that the relative permeability for this core is 3000 and BSAT = 0.28 T. From Equation 5, we can easily determine the admissible number of turns on the primary side NP, knowing that lM = π × DMED. Therefore:
Assume NP = 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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