How to measure flux in transformer in LTspice?

Question

I wish to check the flux through a transformer core in my LTspice simulation. I have set up a push-pull converter simulation using a transformer with a turns ratio of 1 : 1.1, a supply voltage of 12 V and two FETs pulling either side low for almost 1 µs in turn. There is a small dead time of 50 ns, where neither of the FETs are conducting.

In order to determine the flux in the core, I know of two tentative approaches:

• Determine the product of inductance \$L\$ and current \$I\$ for each winding and sum them: $$\Phi_1(t)=\sum_i^\text{windings}{L_i\cdot I_i(t)}$$
• Measure the volt-second integral normalized by the turns ratio \$N\$ of the windings and sum them: $$\Phi_2(t)=\sum_i^\text{windings}\int_0^t{\frac{V_i(\tau)}{N_i} \text d\tau}$$

So, I set up two bv sources in LTspice to calculate each of the two expression:

I know that the absolute turns count is missing in the second expression, only the turns ratio is respected. So it should be wrong by a constant scale factor. However, what I see is that both expressions yield qualitatively different results, which are plotted below.

Question

Which of these two expressions (if any) is correct and why is the other one (or both) wrong?

Answer 1

The flux in the transformer core can be assumed as the integral (over time) of the voltage at the inner winding, divided by the number of its turns. So the expression for the voltage should taking only one of the windings.

Answer 2

None of them is correct.

flux1 looks more reasonable because flux2 uses unswitched windings as well.

Remember the transformer model:

^{simulate this circuit – Schematic created using CircuitLab}

The current you are measuring in the simulation is \$i_P\$ which includes energy-related currents \$i_{E1}\$ and \$i_{E2}\$, and the magnetisation current \$i_M\$ therefore \$i_P = i_M + i_{E1}\$.

Magnetisations caused by energy-related currents cancel each other therefore all that is left is the magnetisation caused by \$i_M\$.

So you should use either the L-I product or the Volt-seconds product for the primary winding only.

$$ v_p = L_p \ \frac{di_M}{dt} = N_p \ A_e \ \frac{dB}{dt} $$

So

$$ B=\frac{1}{N_p \ A_e}\int v_p \ dt = \frac{L_p}{N_p \ A_e}\int di_M $$

Answer 3

Which of these two expressions (if any) is correct and why is the other one (or both) wrong?

Both are wrong...

The flux in a transformer core is due to the magnetization current supplied by the primary-side voltage. It is exclusively load independent (for most reasonable analytical situations) so, measuring secondary currents and voltages (that result in \$\Phi_2\$) are not inapplicable to measuring core flux density.

In other words, the flux in a transformer core is due to the applied voltage at the driven winding (what we normally call the primary winding) and, you might as well (for your experiment) use a VCVS from the primary voltage driving an inductor of value 110 μH and look at the waveform.

Alternatively, unload the output (C3 and R3 disconnected) and use the \$\Phi_1\$ measurement. Please remember to discuss any changes to your question (that you think are needed) as a result of any answers given.