Debunking
Transformer Performance Myths
by
Steven M. Sandler and Danny
Chow,
AEi Systems, Los Angeles, California.
In
this article we intend to dispel three common myths regarding
transformers, and more generally, magnetic coupling.

Myth
1: The turns ratio of a transformer is the and is the reciprocal of

Myth
2: The leakage inductance of a transformer is based on the coupling
from one winding to another

Myth
3: The inductance of a transformer is
To
provide the proof that these myths are false, three transformers were
constructed. Each transformer included only two windings, with 15
turns on each winding.
A
series of measurements were made for each of the three transformers.
All of the measurements were made using one piece of test equipment,
OMICRON Lab’s Bode100 portable vector network analyzer.^{1} The measurements for each transformer were:

The
turns ratio, measured in both directions, as and

The
inductance of each winding

The
inductance of each winding with the second winding shorted.
This
series represents many more measurements than are actually necessary
to assess transformer performance
as it relates to the three myths. However, we wanted to show three
different methods of assessment, all yielding similar results.
Same
Turns Ratio, Different Winding Configurations
The
picture in Figure 1 shows the three wound transformers. One
transformer was wound bifilar, with two wires, covering the entire
core (360 degree coverage). A second transformer was wound with
windings on opposite sides of the core, each in a sector that is
approximately 30 degrees. The final transformer was wound with one
winding covering the entire core (360 degrees) and the second winding
on top of the first, and covering a sector that is approximately 30
degrees of the core.
The
Bode 100 was calibrated to a BNC jack and the transformer windings
were soldered directly to the BNC, as shown in Figure 2. This figure
shows the inductance measurement being performed.
Figure
1. The
three transformer samples.
Figure
2. Test
setup showing the inductance measurement.
Measuring
Turns Ratios
However,
before taking readings on the inductance values of each of the
transformer windings, the equipment in Figure 2 was used to measure
each transformer’s turn’s ratio in both directions. The
first measurement showed the ratio as the secondary voltage divided
by the primary voltage (Vsec/Vpri), while the second measurement
recorded the primary voltage divided by the secondary voltage
(Vpri/Vsec). In Figures 3 and 4 these measurements are shown for the
bifilar transformer sample.

Frequency 
Trace1 
Trace2 
Cursor
1 
707.798
kHz 
0.167
dB 

Cursor
2 
72.663
kHz 
0.114
dB 

Delta
C2C1 
635.135
kHz 
0.053
dB 

Figure
3. Turns
ratio of the bifilar transformer, measured as gain = Vsec/Vpri.

Frequency 
Trace1 
Trace2 
Cursor
1 
707.798
kHz 
0.294
dB 

Cursor
2 
72.663
kHz 
0.338
dB 

Delta
C2C1 
635.135
kHz 
0.044
dB 

Figure
4. Turns
ratio of the bifilar transformer, measured
as gain = Vpri/Vsec.
The
result in Figure 3 yielded an average measurement of 0.140 dB for a
turns ratio of approximately 0.985. Figure 4, which shows Vpri/Vsec,
is slightly different, with an average measurement of 0.316 dB or a
ratio of 0.964. This is slightly different, but still close to the
forward measurement. So, the bifilarwound transformer sample has a
turns ratio of 0.985 in one direction and 0.964 in the other
direction. Similar measurements were performed on the other two
transformer samples with the results presented later in the article.
Measuring
Inductance
The
next measurement was a measurement of the primary inductance with the
secondary open.

Frequency 
Trace1 
Trace2 
Cursor
1 
3.987418
MHz 
62.477
dB 

Cursor
2 
50.184
kHz 
14.671
dB 

Delta
C2C1 
3.937234
MHz 
47.805
dB 

Figure
5. Measurement
of winding 1 (primary) inductance with open secondary.
The
impedance of 14.671 dB at a frequency of 50.184 kHz was 5.4 ohms
reactive, which corresponds to an inductance of 17.17 µH.
Similarly, the measurement of winding 2 is shown in Figure 6.

Frequency 
Trace1 
Trace2 
Cursor
1 
3.987418
MHz 
62.509
dB 

Cursor
2 
50.184
kHz 
14.679
dB 

Delta
C2C1 
3.937234
MHz 
47.830
dB 

Figure
6. Measurement
of winding 2 (secondary) inductance with open primary.
The
impedance of 14.679 dB at a frequency of 50.184 kHz is 5.42 ohms
reactive, which corresponds to an inductance of 17.19 µH.
These two inductance measurements were also made with one winding
shorted, in order to directly measure the leakage inductance. This
method assumes that the shorted winding effectively shorts the core.
Figure 7 shows the result of the first of these measurements,
measuring the inductance of winding 1 with winding 2 shorted.

Frequency 
Trace1 
Trace2 
Cursor
1 
19.361728
MHz 
69.040
dB 

Cursor
2 
1.256227
MHz 
16.735
dB 

Delta
C2C1 
18.105501
MHz 
52.305
dB 

Figure
7. Measurement
of winding 1 inductance with winding 2 shorted.
The
measurement of 16.735 dB at 1.256227 MHz results in a leakage
inductance of 870 nH. Similarly in the opposite direction, the
measurement is shown in Figure 8.

Frequency 
Trace1 
Trace2 
Cursor
1 
19.676448
MHz 
67.826
dB 

Cursor
2 
1.256227
MHz 
16.706
dB 

Delta
C2C1 
18.420220
MHz 
51.120
dB 

Figure
8. Measurement
of winding 2 inductance with winding 1 shorted.
The
measurement in Figure 8, 16.706 dB at 1.256227 MHz, results in a
leakage inductance of 867 nH.
The
core used for these three transformers is a 55120A2 MPP core, which
has an A_{L} value of 72mH/1000T^{2}.
The inductance of 15 turns is, therefore, 16.2 µH. The core
is marked +2, which means that the core has an A_{L} value that is 2% above nominal or 73.44 mH/1000T^{2}.
Using this marked inductance reference, the core inductance is 16.52
µH. The equivalent circuit for the transformer is shown in
Figure 9.
Figure
9. Simplified model of the twowinding transformer.
Looking
at the simplified model in Figure 9, we can see that the turns ratio
of the unloaded transformer
(open circuit windings) is:
And
it is also seen from the figure that the measured inductance of any
winding is the sum of the two inductances:
And
lastly, if the core is effectively shorted, the remaining measurement
is the leakage term of the winding.
Using
the first test transformer as an example, the leakage is solved as:
And
using the second measurement the results are computed as:
The
third measurement, which attempted to measure the leakage directly,
resulted in a leakage inductance of 870 nH. Note that this
measurement is greater than the other two. This is due to the
inability to completely short the core, and the inability to create a
zero impedance short across the winding. This is a significant point,
because it shows that this is the least accurate measurement, and yet
it is the most common method specified for the measurement of leakage
inductance. Either of the other two measurements is superior for
accurately determining the leakage inductance.
Without
showing the details of the other measurements, the results are
summarized in Table 1.
Table
1. Summary
of measurements.
Configuration 
Leakage
(µH) 
Lmeasured
(µH) 
Turns
Ratio 

a 
b 
a 
B 
a:b 
b:a 
Bifilar 
0.82 
0.87 
17.17 
17.19 
0.985 
0.964 
30
360deg 
1.04 
4.21 
17.24 
20.41 
0.973 
0.819 
30
30deg 
4.09 
4.06 
20.29 
20.26 
0.730 
0.731 
It
is interesting to note that in the case of the two 30 degree
windings, the measurements are also symmetrical, meaning nearly the
same in both directions, though the turns ratio is no longer 1, but
now is 0.73. This reduction is due to the voltage divider that is
created by the mutual inductance and the leakage inductance, and
exacerbated by the high leakage inductance of the winding technique.
The
asymmetrical winding results in asymmetrical results, meaning that
the inductance of the winding is different in the two measurement
directions, as is the turns ratio, which is nearly 1 in one
direction, and substantially lower in the other direction.
Analyzing
the Findings
We
can make several conclusions from these simple tests.
First,
we have proven that while the A_{L }value
is a significant contributor to the inductance, it is not the only
contributor. It represents the mutual portion of the inductance. The
leakage inductance can also represent a significant portion of the
inductance of a single winding.
Second,
we can conclude that the leakage inductance is not a measure of the
coupling of one winding to another, but of one winding to itself. A
single winding contains a leakage inductance.
Third,
we can conclude that the turns ratio of the transformer is not simply
N1/N2, nor is it the same, as measured in the two possible
directions. Again, the leakage term plays a large role. How large a
role is dependent on the magnitude and location of the individual
leakage terms.
Of
course, we have also learned that the winding strategy plays a large
role as well, and while we have shown these results for two windings,
the implications are the same, though exponentially more complex, as
the number of windings is increased.
Reference:
1.
The Bode 100 is a portable Vector Network Analyzer covering the
frequency range from 1 Hz to 40 MHz. It is available from OMICRON Lab
(www.omicronlab.com).
About the Authors:
Steve Sandler is the
founder and CTO of AEi Systems, LLC.
He is responsible for worst case
circuit analysis of power, RF,
and linear systems as well as the
design of AEi Systems line
of radhard dcdc converters.
Danny Chow is a
Senior Engineering Specialist for AEi Systems, LLC
He is
responsible for
reliability engineering analysis
and bench testing operations.
For further reading on magnetic component design, see the How2Power Design Guide, and search the Design Area category and the Magnetics subcategory.”
