Transformer characterization: the low-frequency magnetizing-inductance trap

Working teaching note for the future transformer and Bode 100 laboratory section. This material is separate from the introductory recap. It concerns a transformer with one or a few primary turns, not a device with only one winding. The numerical example is authored; no hardware measurement has been performed.

1. The riddle: is the measured voltage ratio the turns ratio?

A high-current DC/DC transformer can have a very small number of primary turns and a small magnetizing inductance. Apply a sine wave to the primary with the secondary unloaded. At low frequency the secondary voltage can be much smaller than the ideal turns ratio predicts. This is a real circuit effect: the magnetizing reactance becomes small compared with series resistance, leaving little voltage across the magnetic branch.

The consequence is not that low-frequency characterization is universally impossible. A raw transfer ratio can be a poor estimate of turns ratio, and parameter extraction becomes sensitive to small measured voltages and phase errors. The useful measurement method and frequency depend on the parameter being identified.

2. Start with the smallest useful circuit

Let n = Ns/Np. Initially neglect leakage inductance, core loss, capacitance and receiver loading. Rp is the primary winding resistance; Rs is the generator's Thévenin resistance. Vg is the generator's open-circuit Thévenin voltage, Vp is the voltage at the primary terminals, and Vm is the voltage across Lm and the ideal transformer's primary.

 Vg ---- Rs ----+---- Rp ----+---- ideal transformer, ratio 1:n ---- Vsecondary
               |            |
             Vprimary       Lm
               |            |
 primary return+------------+

 Secondary is unloaded; its return is a separate winding terminal.
 Vprimary is measured across the entire primary winding, including Rp.

The magnetizing branch forms a divider with the winding resistance:

Its corner is fp = Rp/(2πLm). At frequencies far below fp the secondary/primary-terminal voltage ratio rises at +20 dB/decade, rather than equalling n. Far above fp it approaches n within this simplified model. The terminal voltage includes the copper drop; measuring Vp does not directly measure Vm.

Referencing the output to the generator instead gives a different ratio:

Measuring Vsec/Vp removes the generator divider from the ratio algebraically. It does not restore signal amplitude or eliminate the winding divider. These are different issues.

For a fixed core and linear magnetic circuit, Lm scales approximately with Np². Reducing primary turns therefore tends to reduce Lm strongly. Rp also changes with conductor length, section and construction; the corner cannot be inferred from turn count alone. With a given external Rs, reducing Lm raises the generator-related corner directly.

3. Authored few-turn example

Use Lm = 2 µH, Rp = 2 mΩ, n = 4 and Rs = 50 Ω. These values illustrate the mechanism; they are not specifications of a commercial transformer. The winding-related corner is 159.2 Hz, while the generator-related corner is 3.979 MHz.

At 1 Hz, the predicted secondary signal is only about 1.005 µV RMS for a 1 V RMS open-circuit generator. The model still has a well-defined transfer ratio, but measuring it accurately requires an adequate noise and systematic-error budget. At 1 kHz the terminal-referenced ratio is already close to four, while the generator-referenced ratio is still only about 0.001. Confusing these ratios would imply a wildly incorrect turns ratio.

For less than 1% magnitude error relative to n in the terminal-referenced ratio, this idealized example requires f > 1.117 kHz. This is a magnitude-only condition, not a phase-accuracy or complete measurement-uncertainty requirement. Leakage, loading and capacitance impose additional constraints on the upper usable frequency.

The curves extending toward 100 MHz show the mathematical model only. They do not claim that this low-frequency equivalent circuit remains accurate there; real capacitances and resonances may intervene before a clean generator-referenced plateau appears.

4. Which parameter becomes difficult to identify?

With an open secondary, negligible core loss and negligible leakage, input impedance is

At very low frequency the impedance is mainly resistive. Lm is then recovered from a small imaginary component, so a small phase error can produce a large relative inductance error. Holding magnitude exact, a small phase error δφ in radians gives

Here the impedance phase at 1 Hz is only 0.360°. An illustrative +0.1° phase error would cause approximately +27.8% inferred-inductance error. At 10 Hz it causes about +2.78%. The 0.1° is a sensitivity example, not a Bode 100 accuracy specification; magnitude error, calibration residuals, fixture resistance and noise must also be considered.

Low frequency can be useful for winding-resistance extraction because inductive reactance is small, while being poorly conditioned for inductance extraction. Conversely, an open-secondary inductance measurement includes primary leakage as well as magnetizing inductance. An open-circuit sweep alone does not separate them. A short-circuit measurement estimates the referred leakage/resistive network, subject to short-fixture impedance and the chosen equivalent-circuit convention.

5. Bode 100 measurement plan for the future laboratory

OMICRON's Transformer modelling application note, V2.0 is the primary practical reference for combining winding-resistance, open/short-circuit, transfer and capacitance measurements into an equivalent circuit. Its worked transformer differs from the few-turn scenario here; neither its test frequencies nor its excitation levels should be copied as universal settings.

The Bode 100 Revision 2 technical data specifies a 50 Ω source and selectable receiver impedances of 1 MΩ or 50 Ω. The high-impedance input also has 40–55 pF capacitance. Its stated typical receiver sensitivity is 1 µV RMS at 10 Hz RBW with zero input attenuation; that is not a guarantee of accurate ratios at a 1 µV DUT signal. Confirm the revision and settings of the instrument actually used.

1. Define the parameter and reference plane. Record the winding under test, open/short/loaded secondary, frequency, excitation and temperature. Label Vg, Vp and Vsec explicitly. A gain measurement and a 50 Ω S21 measurement are different configurations.
2. Measure winding resistance separately. For milliohm windings use a suitable four-wire resistance measurement and inspect connections. A fixture contact resistance comparable with Rp changes the divider substantially.
3. Choose an impedance method for the actual range. Use a calibrated mode and fixture suited to low impedances. The manufacturer's measurement-mode overview identifies shunt-through as a low-impedance method. A one-port reflection measurement near a short can be poorly conditioned because Γ is close to −1; use the method's uncertainty information rather than assuming equal accuracy throughout the range.
4. Measure gain at the DUT terminals. Put the reference channel at the primary terminal plane and the other channel at the secondary plane, with terminations included in the model. Follow the instrument manual for grounding and connections: the two measurement channels must not be assumed to provide independent isolated differential inputs. The bench fixture is a defined unpowered transformer test, not the previous common-mode isolation test.
5. Calibrate at the intended fixture plane. Use the procedure appropriate to the selected measurement mode. Calibration can correct specified fixture/system errors; it cannot remove real winding loss, recover unlimited SNR or make an unsuitable equivalent model valid.
6. Sweep excitation as well as frequency. Repeat at several levels and look for invariance of extracted parameters. For sinusoidal core voltage, Bpk = Vm,RMS/(4.44 f Np Ae). Use the induced core voltage, not blindly the generator setting. In the resistance-dominated limit Vm falls with frequency, so constant generator amplitude does not automatically imply unlimited flux as f tends to zero. If terminal voltage is actively held up, flux and magnetizing current still require explicit checks.
7. Fit a useful frequency window. Seek adequate phase and signal amplitude below the region dominated by parasitic capacitance and resonance. Add leakage, core loss and receiver loading when needed, and compare predicted impedance and gain against independent sweeps. Report residuals and sensitivity, not just a fitted Lm value.

Receiver loading deserves an explicit check. With n = Ns/Np, a secondary load ZL refers to ZL/n² at the primary and appears in parallel with the magnetizing branch. Selecting 50 Ω instead of high impedance can therefore change the DUT response. High impedance is also not an ideal open at every frequency because of receiver and cable capacitance.

Low-impedance fixtures also require attention to return currents. OMICRON's Impedance Measurements application note, V1.1, §2.2.3 explains that internal source/receiver ground connections can produce a shunt-through ground-loop error, especially at low impedance and low frequency. A Kelvin-like connection alone does not guarantee that this systematic error is negligible.

6. Recall exercise and reproducibility

Ask students to predict the two corners, explain why CH2/CH1 can differ from output/generator, and identify whether a failed low-frequency extraction is caused by an incorrect circuit model, poor signal amplitude, phase sensitivity or nonlinear excitation. Then change only Lm, Rp or Rs. Replacing the generator with a lower-impedance source changes the excitation and available SNR; it does not eliminate the internal Rp–Lm divider.

scripts/transformer_low_frequency.py exports figure 73 and transformer-low-frequency-checks.json. It verifies both transfer ratios against an independent two-node solution at 41 frequencies and checks winding active-power balance. The phase-error examples are calculated sensitivity tests. No instrument automation, actual Bode 100 measurement or full nonlinear magnetic simulation is claimed.

This note can later be joined to the existing transformer equivalent-circuit discussion and the common-mode injection example. Recap §11.7 develops the sum/difference basis, power normalization and symmetry that distinguish the modes. Keep differential parameter extraction and common-mode interwinding-capacitance characterization as distinct fixtures with explicit connections.