



Here the Oscilloscope Y1 amplifier is presented with a voltage that is in-phase with the primary current, and this is also used to trigger the timebase. Note that the cable leading to the Y2 amplifier is properly terminated, and so the input to the cable is a good approximation to a pure 50Ω resistance. Hence, with the switch open, a 12 turn secondary winding of 9.7μH inductance is loaded with 100Ω. The Y2 voltage measurement is effectively made at a tap half-way down the load resistance, but since this network is a resistive potential divider, the relative phase recorded is the same as the phase of the voltage across the transformer secondary. With the generator (a radio transmitter) operating at 1.6MHz, the oscilloscope timebase, gain, shift and trigger-level controls were manipulated so that one cycle of the waveform was 10cm long. The phase difference between the primary current and the secondary voltage was then found to be 13±0.5mm on this scale, with the secondary voltage leading (inductive). Hence the phase difference in degrees was:

(13±0.5/100)×360° = +46.8±1.8°

The calculated reactance of the full 12 turn secondary winding at 1.6MHz is:

2π × 1.6×106 × 9.7×10-6 = 97.5Ω

and with a load of 100Ω, the calculated phase angle is:

Arctan(100/97.5) = 45.7°

This is perfectly in agreement with the measured value (which is why we don't need to bother taking the error in the inductance measurement into account).

Finally, operating the switch to join the centre taps made no observable difference to the phase measurement or the output voltage. Henc e equation ( x.1 ) is correct.



We have at this stage, resolved the paradox, but we have yet to explain it. With that in mind, a further experiment was conducted with the test jig rewired as per the circuit below:







Now, one half of the secondary winding is 'completely isolated' from the other (or not, as the case may be). With the switch open, the phase difference between the primary current and the secondary voltage at 1.6MHz was found to be:

(18±0.5/100)×360° = +64.8±1.8°

The calculated reactance of the 6 turn (2.425μH) winding at 1.6MHz is:

2π × 1.6×106 × 2.425×10-6 = 24.4Ω

and with a load of 50Ω, the calculated phase angle is:

Arctan(50/24.4) = 64°

Now the interesting part: when the switch was closed, the phase angle changed back to 46.8±1.8° exactly as in the previous experiment. The phase angle for 9.7μH in parallel with 100Ω is exactly the same as the phase angle for 4.85μH in parallel with 50Ω. Hence, Closing the switch has the effect of doubling the inductance of the secondary winding connected to the oscilloscope.

An explanation is called for, and it is this: reactance is indicative of energy storage. Apart from parasitics, which only have an effect at higher frequencies, when a secondary winding is disconnected, its inductance ceases to exist. When no current flows in a winding, no energy can be stored in the inductance of that winding. When the switch is closed however, a current flows in the floating winding and an energy storage mechanism is activated. The two secondary windings moreover, regardless of any DC path between them, are intimately connected by the closed magnetic circuit of the toroidal core. Because a current transformer controls its own input voltage according to how it is loaded, closing the switch alters the flux density in the core. In effect, by transfer of energy between the two secondary windings; when the switch is closed, an extra energy storage mechanism becomes available to the winding connected to the oscilloscope.

If we call the separate inductances of the two secondary windings L a and L b , the total inductance of the windings is:

L i = L a + L b + M ab + M ba

where the M ab and M ba are the mutual inductances. If there is no leakage inductance (as is true to a 98% good approximation for high-permeability toroidal cores) then:

L a = L b = M ab = M ba

which is why the inductance of the two windings in series is 4 times larger than that of a single winding (when the windings are connected start to end), and is also why the inductance of a coil is proportional to N². Hence, when both windings are loaded, the effective inductance of a single winding is not L i /4, but L a +M ba = L i /2, where M ba represents the auxiliary inductance provided by the other winding. Hence, the correct model for the split-secondary current transformer, regardless of whether there is a DC connection between the secondaries, is as shown below:







V i /2 = V [ (R i /2) // (jX Li /2) ] / ( N Z A )