Finding Transformer Pi Model Parameters

Given a random transformer, create a decent Spice model… I have to do this rarely enough that I’d better write it down so it’s easy to find. There’s no magic here; it’s all described in ON Semi (nee Motorola) App Note AN-1679/D. See page 4 for the grisly details; I’ve reordered things a bit here.

Go to the basement lab and measure:

  1. Primary & secondary voltages with a sine-wave input: Vp & Vs.
  2. Primary inductance with secondary open: Lps(open)
  3. Primary inductance with secondary shorted: Lps(short)
  4. DC resistance of primary & secondary: Rp & Rs

Then return to the Comfy Chair and calculate:

  1. Turns ratio N = Vp/Vs.
  2. Coupling coefficent k = sqrt(1 – Lps(short)/Lps(open))
  3. Primary leakage inductance LI1 = (1 – k) · Lps(open)
  4. Secondary leakage inductance LI2 = (1 – k) · Lps(open) / N^2
  5. Magnetizing inductance Lm = k · Lps(open)

To wit…

A quick trip to the basement lab produces these numbers for this small high-voltage transformer:

Small HV Transformer
Small HV Transformer
Primary Secondary
Voltage 1.08 27.32
DC resistance 2.03 349
L other open 15.5 mH 9.68 H
L other short 45.0 uH 31.3 mH

You don’t actually need the secondary inductances, but while you have the meter out, you may as well write those down, too. Maybe someday you’ll use the transformer backwards?

And a session with the calculator produces a Spice model:

  1. N = 1.076 / 27.32 = 0.0394
  2. k = sqrt(1 – 45.0 uH / 15.5 mH) = 0.998
  3. LI1 = (1 – 0.998) ·15.5 mH = 22.5 uH
  4. LI2 = (1 – 0.998) ·15.5 mH / 0.0394^2 = 20.0 mH
  5. Lm = 0.998 ·15.5 mH = 15.48 mH

Note: the value of (1 – k) is the small difference of two nearly equal numbers, so you wind up with a bunch of significant figures that might not be all that significant. The values of LI1 and LI2 depend strongly on how many figures you carry in the calculations; if you don’t get the same numbers I did, that’s probably why.

The coupled inductors L1 & L2 form an ideal transformer with a primary inductance L1 chosen so that its reactance is large with respect to anything else. I picked L1 = 1 H here, which is probably excessive.

The coupling coefficient would be 1.0 if that were allowed in the Spice model, but it’s not, so use 0.9999. Notice that this is not the k you find from the real transformer: it’s as close to 1.0 as you can get. [Update: either I was mistaken about 1.0 not being allowed or something’s changed in a recent release; 1.0 works fine now.]

Spice transformer pi model
Spice transformer pi model

The primary inductance and turns ratio determine the secondary inductance according to:

Vp / Vs = N = sqrt(L1 / L2)

So:

L2 = L1 / (N^2) = 1 / 0.0394^2 = 644 H (!)

The models for LI1 and LI2 include the DC resistance, so that’s not visible in the schematic.

And now you can model a high-voltage DC supply…

Memo to Self: It’s G16821 from Electronic Goldmine

  • Primary on pins 2 & 10
  • HV secondary on pin 8 & flying wire
  • Electrostatic shield on pin 3

Note: You can compute the turns ratio either way, as long as you keep your wits about you. With any luck, I’ve done so… but always verify what you read!