Phase difference of voltages at one side of a transformer

Question

Consider the circuit in picture

^{simulate this circuit – Schematic created using CircuitLab}

Suppose that I measure VA and VB with respect to GND using an oscilloscope.

Will VA and VB be in phase or out of phase with respect to each other?

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Edit : I tried to make measurement on a real circuit with the values of parameters as in picture (V1 is 220V). I measured VA and VB with an oscilloscope and I got

VA=6 V

VB=17.5 V

I used the MATH function on oscilloscope to get the difference VA-VB

VA-VB= 20 V

I did not measure directly the phase shift between VA and VB signals, but, from the amplitude I can deduce that pase shift d was such that

sqrt(VA^2+VB^2-2*VAVBcos(d))=20

Solving I find d around 106°.

Is this normal? I expected a 180° shift while I get a value which is much lower

Answer 1

If the transformers shown are ideal, VA will be 180 degrees out of phase with VB. If not ideal then VA and VB will tend to be 180 degrees out of phase plus or minus a small error.

Answer 2

In generic terms, for a given voltage difference between the \$a\$ and \$b\$ points, there will be an infinite number of possibilities for the voltages \$v_a(t)\$ and \$v_b(t)\$to assume, when measured relative to a reference point. The exact setup will be given by the conditions of the circuit. For example, in the case of AC (sinusoidal), the \$v_{ab}(t)\$ voltage can be given as the difference:

$$ V_{ab}\cos\omega t = V_a\cos(\omega t+ \theta_a) -V_b\cos(\omega t+ \theta_b) $$

including \$\theta_a\$ and \$\theta_b\$ phase shifts. Assuming that \$V_a\$ and \$\theta_a\$ are known and developing the right side, leads to:

$$ \theta_b = atan2(V_a\sin\theta_a,V_a\cos\theta_a-V_{ab})$$ $$ V_b = \sqrt{ V_a^2-2V_aV_{ab}\cos\theta_a+V_{ab}^2} $$

Considering TR1 turn ratio as 11, I simulate your circuit (obtaining results consistent with the above expressions) for two values of R1:

The phase difference is 111.6 deg (\$\theta_a\$=64.19 deg and (\$\theta_b\$=175.78 deg). Changing R1 to 10R:

Here the phase difference is 108 deg (\$\theta_a\$=54.69 deg and (\$\theta_b\$=162.69 deg). That is, many possibilities for phase difference other than 180 deg.