In order to design current sense transformers, you need to determine the DCR of the SEC, the inductance of the SEC, the SEC wire area/gauge to use, the magnetic area of the core, the excitation current, the % error of the CT and the desired output voltage for a given input current.
Let’s start by assuming you will use a toroid as the form.
To find the approximate DCR of the SEC winding you can use the following formula:
This above equation sets the path length around the toroid using the OD, ID and height, H. It is a function of the number of turns, N, the resistivity of copper, rho, and is inversely a function of the wire’s cross-sectional area, Aw.
Let’s make another assumption here. Let’s set the cross-sectional area equal to the current divided by 2,000,000. We do this because this sets the current density in the wire to 2A/mm^2. This is a safe assumption to use when determining the needed cross-sectional area of the wire.
Here Is is the SEC current, with the PRI current being that going through the 1T of wire inserted through the ID of the toroid.
A few more formulas need to be developed…..
Let’s find one that determines the needed core area to avoid core saturation…..
Let’s also assume the current through the PRI is sinusoidal in nature. This means the flux and flux density within the core are also sinusoidal in nature. The time derivative of a sinusoid is just the angular frequency times another sinusoid. So we can re-write the above as:
Now to re-write the peak voltage in terms of Vrms, frequency and pi…
Therefore the core area is:
In the above equation, Bm is the MAX flux density of the core. You can set this to whatever you’d like below the cores actual MAX flux density.
Now let’s get a formula for finding the inductance of this toroid:
The relative permeability of the core material can be found on the core datasheet. Ok now let’s determine the excitation current for our CT…
Therefore…
This is the RMS value of the excitation current needed to “satisfy” the core. It is also the main source of error with the CT. The error can be found as follows:
Ok… time for an example…
Design a 1:100 CT that takes in 1A – 10A and outputs 1V for every 1A of PRI current…
STEP # 1:
The SEC current is a function of the turns ratio, which means it will range from 10mA to 0.1A with a 1:100 turns ratio. Since 0.1A would be the MAX current we will choose a SEC wire area/gauge that can handle this based on our formula given initially.
A wire with a 0.05mm^2 area has a diameter of:
The DCR is then:
STEP # 2:
Let’s find the needed core area. Let’s use a frequency of 60Hz. For this frequency, we can use a tape wound GOSS toroid. Let’s set the MAX flux density to say 0.6T. This core can handle more but we will just use this value.
STEP # 3:
Choose toroid dimensions. Let’s go with say:
This toroid has an area of:
So this is fine in this respect. Now before we move on let’s be sure it can fit 100T’s of 0.25mm wire inside of it leaving room for the PRI wire to be inserted.
The available window area is:
The bare copper area is:
Utilized area with bare copper is:
This leaves a fair amount of space for the PRI wire to be inserted and so this core seems good as far as both the flux and the window area available to wind the SEC wire…
STEP # 4:
Find an inductance….let’s say we found out this core has a relative permeability of 2,500. The inductance then can be found using the formula:
STEP # 5:
Determine the excitation current….
The % error is:
Not too bad….
STEP # 6:
Choose your burden resistor. You know the SEC current ranges from 10mA to 100mA and you want the voltage to be 1V to 10V, so the burden resistor must be:
Now you can spec all of this and either make the CT yourself or know better what to look for off the shelf.
CT DEFINED:
A 1:100 CT that takes in 1A – 10A and outputs 1V for every 1A of PRI current……
NOW THIS CURRENT SENSE TRANSFORMER CAN LIKELY SENSE UP TO ABOUT 30A BECAUSE WE SET THE FLUX DENSITY TO 0.6T WHEN THE CORE CAN LIKELY HANDLE UP TO ABOUT 1.8T. THE SEC WINDING MAY GET WARM/HOT THOUGH IF CURRENTS THIS HIGH ARE SENSED FOR ANY SIGNIFICANT PERIOD OF TIME.