### Magnetics (Part 5) - Current Transformers

About the writer: Harvey Morehouse is a contractor/consultant with many years of experience using circuit analysis programs. His primary activities are in Reliability, Safety, Testability and Circuit Analysis. He may be reached at harvey.annie@verizon.net. Simple questions for which I know the answer are free. Complex questions, especially where I am ignorant of the answers, are costly!!!

Summary: Current transformers are usually used to measure currents, especially where large values of current are expected, and a voltage drop across a series resistive element might represent large losses, the voltages might be hazardous, or a 'clamp-on' sensing element might be desired. A 'current' transformer (CT) and its model is no different in topology than a conventional 'voltage' transformer. However its intended usage is somewhat different causing some conceptual difficulties.

Current Transformer basics:

A typical current transformer application is to apply a current to the primary of a transformer low-turn primary winding and to measure or otherwise sense the voltage across a resistive load of the multi-turn transformer secondary.

Because the transformer is a 1:N device, the current at the secondary is the primary current divided by 'N' and the secondary voltage is the primary voltage multiplied by 'N'. 'N' being the number of secondary turns divided by the number of primary turns.

Now in most cases we wish to minimize the insertion effects of the transformer on the current being measured. Consequently, by construction of the transformer, either as a 'clamp-on' or permanent insertion into a circuit, the number of primary turns is often unity. Refer to Figure 1 following:

Figure 1
Primitive CT model & Application

In the primitive circuit and model of Figure 1 an ideal transformer is used to represent the CT. Resistor RB is the 'burden resistor' while the primary current is established by V1 and some load RL. Now the load might be complex, but the intent is to choose 'n' and RB in such a manner as to minimize the effect of the transformer and RB upon the effects of the measured current.

We know (or should know) that ideally the effect of the transformer is to present at the primary RB/n2 ohms in series with RL. Suppose that V1 represented a high voltage AC powerline of 10,000 volts, and that RL was 10 ohms. The primary current in the absence of the CT and RB insertion into the circuit would be 1,000 amps, and the power delivered by V1 would be 10 megaWatts.

Let us suppose now that 'n' equaled 1000. The secondary current would be approximately 1 amp, equal to the primary current divided by 'n'. This current would cause the secondary voltage to be approximately 10 volts. The secondary power would be about 10 watts. Now the secondary power has to come from somewhere, and in this case the power delivered to the primary load would be slightly diminished.

With the reflected load being equal to RB/n^2 ohms, or 10^-5 ohms, the load on V1 would be decreased ever so slightly, and the voltage across the primary load also diminishing slightly due the voltage divider of RL and RB/n^2 ohms.

One can perform tests on the circuit, bearing in mind that the 'sense' of the transformer currents are such that when current enters the primary 'dot' it exits the secondary dot. One finds that the peak primary current is 999.999A, the secondary current is 999.999mA, and the secondary voltage a peak of 9.167 volts.

However one things that the model reveals is that when the burden resister is not present, the voltage V1 appears across the primary and the secondary voltage can ideally become extremely large. Practically a 'real' transformer, as well as a better CT model would not allow this, but it still could reach hazardous levels.

A more practical CT model:

Figure 2 following shows a somewhat more complete CT model, and this should look familiar.

Figure 2
Primitive CT model2

In Figure 2, Lkp and Lks represent the primary and secondary leakage inductances. Rs is the secondary winding resistance. (Because the primary winding resistance is usually much much smaller than the primary load resistance it is not shown.)

Ri represents an isolation resistance such that one may 'float' the primary or secondary winding. Its value should usually be on the order to 10 to 100 megohms.

Rcl and Lm represent the transformer core. Rcl is a fair representation of the core loss, while Lm represents the transformer magnetization inductance. Now IF the Lm were represented by an equation or function representing an 'average' B-H curve value, this is not a bad representation.

Note that secondary load impedance appears in parallel with Lm and Rcl. If the magnetizing inductance and Rc are large compared to the transformed load impedance, the error is small.

A better representation for the magnetization inductance would include hysteresis and saturation effects, such that the Rcl would vanish as an element, being incorporated into the B-H curve model for Lm.

If the Lm exhibited saturation effects, then when it did become fully saturated, Lm would reflect an air core transformer incrementally, and thus the secondary output voltage would not become as large as in the ideal case, but it still could become excessive.

Now of course if the Lm value represented a 1 turn winding, it could be placed as shown in Figure 2 as part of the primary, but if it were based on 'n' turns it would appear at the secondary.

One might question whether the use of an ideal transformer, which passes DC, would cause problems. But at DC the inductances Lkp and Lm are short circuits, resulting in heavy CT primary currents, but no ideal transformer primary voltage, and no secondary voltage.

Now elements not present in the model are the winding capacitances, as well as the interwinding capacitance. If the model were used for a low frequency application, such as to measure a 60 Hz line current, they might be neglected. But to make the model more accurate one would add winding capacitances to the model as shown in Figure 3 of the next section.

Better CT model:

Figure 3
Better CT model2

Here we have added lumped capacitances Cp and Cs to the primary and secondary windings. Rp, a primary winding resistance has also been added to the model, but its use is problematic as is Cp IIn the case of a one turn, 'clamp-on' circuit, one has to ask what has changed when the current is being measured from when it is not?

Certainly for the one turn primary current case, Rp at most could represent a portion of the resistance of the conductor carrying the current being measured. The same is true for Cp, being at most the change in capacitance for the remainder of the conductor due to the placement of the clamp-on measurement transformer.

Now IF the transformer was embedded into the circuit, and the primary turns greater than unity, Rs and Cp could be measured or calculated. It could be useful to include these items were the circuit used for complex current measurements, as in an oscilloscope current probe.

In any event, the next question becomes how to make a model of the circuit. As a first pass, one could include just the linear elements/parameters indicated by the topology of Figure 3.

Now versions to incorporate a nonlinear magnetization inductance for Lm would require different versions for each way that Lm could be modeled, and there could be several. It would seem appropriate to make a second model with terminals brought out to where an Lm model can be attached. For versatility this could be directly across the primary or secondary of the ideal transformer device. However, this would also accommodate a linear representation of Rcl and Lm, so we will make one model which does all these things. Refer to Figure 4 in the following section.

Best CT model:

Figure 4
Best CT transformer model

Figure 4 shows the model topography as well as the parameters to be passed to the model. The device designations were changed to make them different from the passed parameter values

The magnetization inductance Lm (and Rcl if required) are to be connected between LMp1 and LMp2 (or LMs1 and LMs2) terminals as determined for a 1 turn or 'n' turn magnetization inductance model respectively (assuming that the ideal transformer primary winding has 1 turn). If the primary winding had 3 turns, then a 3-turn Lm would be provided for the primary or a 3*n-turn Lm provided at the secondary.

An advantage of this model is that an inter-winding capacitance could be added from LMp1 to LMs1 (and from LMp2 to LMs2 if applicable). However, there is another little problem with the model as shown. B2SPICETM does not like elements with the same name as that of the parameters being passed, hence the symbol device designations were renamed. When the external connections points for the magnetization were placed, this results in a node naming conflict. To resolve this two zero voltage sources were placed in the circuit, resulting in the circuit of Figure 5 following:

Figure 5
Ctrans part and device model

Figure 5 shows the final model and the symbol chosen for the part. Now we need to test the circuit to be sure it is correct. In Figure 6 which follows, we have a contrived circuit whose values do not necessarily represent anything more than a 1:1 transformer model driven by a 1A, 60 Hz current source.

Figure 6
Ctrans test circuit #1

In Figure 6 we have chosen a 1 ohm load for the basic circuit and the Ctrans device model shown by the U2 symbol.

A frequency sweep of this circuit is shown in Figure 7 following:

Figure 7
Ctrans test circuit #1 graph

Here we see the very low frequency gain starts at zero and increases to unity when the magnetizing inductance becomes effective. After some time a resonance is reached between Lm and Cp, and then it decreases. The responses overlap and only a single curve is reached. The midband gain is almost zero dB.

We could do some more experimentation, and should do so, however the model is believed to be accurate and to keep the article to a reasonable length further testing will be left to the user to verify the model.

A netlist for the model follows:

************************
* B2 Spice Subcircuit
************************
*
* Created by Harvey Morehouse
*
*
* Pin # Pin Name
* LMp1 LMp1
* INp INp
* LMs1 LMs1
* OUTp OUTp
* INn INn
* LMp2 LMp2
* LMs2 LMs2
* OUTn OUTn
.Subckt Ctrans LMp1 INp LMs1 OUTp INn LMp2 LMs2 OUTn

***** subcircuit definitions

************************
* b2 spice subcircuit
************************
* pin # pin name
* n2 n2
* n3 n3
* n1 n1
* n4 n4
.subckt itrans2 n2 n3 n1 n4
***** subcircuit definitions
***** main circuit
r1 n2 n3 1e10
e1 3 n4 n2 n3 1.000000000000e+000
f1 n2 n3 vam1 1.000000000000e+000
r2 6 n1 1e-9
vam1 3 6 0
.ends

***** main circuit
XU1 LMs1 LMs2 LMp1 LMp2 itrans2
Lkp1 5 LMp1 {Lkp}
Lks1 LMs1 6 {Lks}
Rs1 6 OUTp {Rs}
Ri LMp2 LMs2 10Meg
Rp1 INp 5 {Rp}
Cp1 5 INn {Cp}
Cs1 6 OUTn {Cs}
V1 INn LMp2 1.000000000000e+000
V2 OUTn LMs2 1.000000000000e+000

.ends

Conclusions:

It was convenient to prepare a model for a two winding current transformer. Ctrans device was made for this purpose. Besides the use as a current sensing/ monitoring device for a scope or other current reporting/display device, there are other interesting possibilities.

One interesting application for this device could include a circuit which could be powered by a power line that could be used to report the current through the line to a remote reporting device. This would be complicated by the fact that powerline currents can vary from a few amperes to tens of thousands of amperes.

If one could vary the burden based on the current amplitude it could be possible to power the device over a wide range of currents. If one made a battery powered load, then it should be possible to charge the battery from the line, and at the same time, with suitable calibration, be able to report the line current.