# The 4 times rule for matching transformer impedance

Started by January 17, 2015
```Hi all,
I have always heard and used the 4 times rule for the impedance of a
transformer winding.
Example;
To match 50 ohms to 400 ohms, the primary inductance should be 200 ohms
at the lowest frequency of interest. 4 x 50 = 200 ohms XL.
Tonight I'm searching for a reference for the 4 times.
I found this;

The math is more than I can handle, but his conclusion is;

properties of broadband RF transformers taking into account the finite
inductive reactance of their primary and secondary windings. We have
shown that the &acirc;&euro;&oelig;four times the impedance&acirc;&euro;&#157; design guideline for the
inductive reactance of the transformer windings yields an impedance
transformation that departs from its &acirc;&euro;&oelig;ideal&acirc;&euro;&#157; behavior by no more than 3%,
and introduces a phase shift of no more than 14 degrees."

Now some may quibble with the 3% and 14 degrees as being picked out
of the air (me) but something needs to be picked
and those numbers seems reasonable.

Also, they reference Broadband RF transformers,

Can I assume that scales to audio transformers?

Mikek

```
```On 18.1.15 04:46, amdx wrote:
> Hi all,
>   I have always heard and used the 4 times rule for the impedance of a
> transformer winding.
> Example;
> To match 50 ohms to 400 ohms, the primary inductance should be 200 ohms
> at the lowest frequency of interest. 4 x 50 = 200 ohms XL.
> Tonight I'm searching for a reference for the 4 times.
> I found this;
>
>
>
>
>   The math is more than I can handle, but his conclusion is;
>
> properties of broadband RF transformers taking into account the finite
> inductive reactance of their primary and secondary windings. We have
> shown that the &acirc;&euro;&oelig;four times the impedance&acirc;&euro;&#157; design guideline for the
> inductive reactance of the transformer windings yields an impedance
> transformation that departs from its &acirc;&euro;&oelig;ideal&acirc;&euro;&#157; behavior by no more than 3%,
> and introduces a phase shift of no more than 14 degrees."
>
>    Now some may quibble with the 3% and 14 degrees as being picked out
> of the air (me) but something needs to be picked
> and those numbers seems reasonable.
>
>   Also, they reference Broadband RF transformers,
>
> Can I assume that scales to audio transformers?
>
>                                Mikek

The model applies to the low end of the frequency range. The high end is
controlled by other phenomena. In a broadband transformer, the high end
is effected by the length of the transmission line of the windings
compared to the wavelength and associated phase shift.

For audio transformers, the formula is an usable approximation for the
low end, although the leakage inductances of the windings will affect
the results, adding attenuation and phase shift earlier when going
toward the low end of the frequency range.

--

Tauno Voipio

```
```On 1/18/2015 3:42 AM, Tauno Voipio wrote:
> On 18.1.15 04:46, amdx wrote:
>> Hi all,
>>   I have always heard and used the 4 times rule for the impedance of a
>> transformer winding.
>> Example;
>> To match 50 ohms to 400 ohms, the primary inductance should be 200 ohms
>> at the lowest frequency of interest. 4 x 50 = 200 ohms XL.
>> Tonight I'm searching for a reference for the 4 times.
>> I found this;
>>
>>
>>
>>
>>
>>   The math is more than I can handle, but his conclusion is;
>>
>> properties of broadband RF transformers taking into account the finite
>> inductive reactance of their primary and secondary windings. We have
>> shown that the &acirc;&euro;&oelig;four times the impedance&acirc;&euro;&#157; design guideline for the
>> inductive reactance of the transformer windings yields an impedance
>> transformation that departs from its &acirc;&euro;&oelig;ideal&acirc;&euro;&#157; behavior by no more than 3%,
>> and introduces a phase shift of no more than 14 degrees."
>>
>>    Now some may quibble with the 3% and 14 degrees as being picked out
>> of the air (me) but something needs to be picked
>> and those numbers seems reasonable.
>>
>>   Also, they reference Broadband RF transformers,
>>
>> Can I assume that scales to audio transformers?
>>
>>                                Mikek
>
>
> The model applies to the low end of the frequency range.

Yes, as I understand it, low end frequency response is the first to
degrade as the inductance of the winding is decreased.

> The high end is
> controlled by other phenomena. In a broadband transformer, the high end
> is effected by the length of the transmission line of the windings
> compared to the wavelength and associated phase shift.
>
I understand there is an effect increasing response of the high end
by the interwinding capacitance.

> For audio transformers, the formula is an usable approximation for the
> low end, although the leakage inductances of the windings will affect
> the results, adding attenuation and phase shift earlier when going
> toward the low end of the frequency range.
>
My end thought/goal is to match a very high Q crystal radio tank
circuit at resonance. The R at resonance could be over 1 Megaohm.
So, I'm trying to find out, do I need a primary impedance of 4 times
1 megaohm?
Then I would add a secondary winding with taps to reflect back the 2000
The best transformers in the crystal radio world seem to be an
autoformer with 359 Henries and a UTC 0-15 listed as a 15k to 1Meg
transformer. I find no inductance data on the UTC transformer.
I want to know Is UTC matching 1 Megaohm at 100Hz using the 4 times
rule? That would mean they have an inductance over 2000 Henries.
The UTC 0-15 transformer is on Catalog Page 16 here,
http://www.junkbox.com/electronics/utc_transformer_catalog_1963.pdf
I tried ringing it with 470pf and 1290pf, I get a peak but with a Q
less than 1 and L of 2550 Henries and 2000 Henries, at 145 and 100 Hz.
So I don't know if that is real. The DC resistance is "posted" as 5000
ohms. So Q will be low. I'm not going to measure it because I'm afraid
of magnetizing my core.
Back to the bench, I'll ring the 15k winding and try extrapolating
from that data.

Thanks, Mikek

```
```On Sat, 17 Jan 2015 20:46:23 -0600, amdx wrote:

> Hi all,
>   I have always heard and used the 4 times rule for the impedance of a
> transformer winding.
> Example;
> To match 50 ohms to 400 ohms, the primary inductance should be 200 ohms
> at the lowest frequency of interest. 4 x 50 = 200 ohms XL. Tonight I'm
> searching for a reference for the 4 times. I found this;
>
>
>
>   The math is more than I can handle, but his conclusion is;
>
> properties of broadband RF transformers taking into account the finite
> inductive reactance of their primary and secondary windings. We have
> shown that the &acirc;&euro;&oelig;four times the impedance&acirc;&euro;&#157; design guideline for the
> inductive reactance of the transformer windings yields an impedance
> transformation that departs from its &acirc;&euro;&oelig;ideal&acirc;&euro;&#157; behavior by no more than
> 3%, and introduces a phase shift of no more than 14 degrees."
>
>    Now some may quibble with the 3% and 14 degrees as being picked out
> of the air (me) but something needs to be picked and those numbers seems
> reasonable.
>
>   Also, they reference Broadband RF transformers,
>
> Can I assume that scales to audio transformers?
>
>                                Mikek

You are going to _buy_ your audio transformers, right? So let the tranny
designer worry about the inductance! Impedance transforms as the square of
the turns ratio, in your example, a ratio of 2.8 to 1.
```
```On 1/18/2015 10:52 AM, amdx wrote:
> On 1/18/2015 3:42 AM, Tauno Voipio wrote:
>> On 18.1.15 04:46, amdx wrote:
>>> Hi all,
>>>   I have always heard and used the 4 times rule for the impedance of a
>>> transformer winding.
>>> Example;
>>> To match 50 ohms to 400 ohms, the primary inductance should be 200 ohms
>>> at the lowest frequency of interest. 4 x 50 = 200 ohms XL.
>>> Tonight I'm searching for a reference for the 4 times.
>>> I found this;
>>>
>>>
>>>
>>>
>>>
>>>
>>>   The math is more than I can handle, but his conclusion is;
>>>
>>> properties of broadband RF transformers taking into account the finite
>>> inductive reactance of their primary and secondary windings. We have
>>> shown that the &acirc;&euro;&oelig;four times the impedance&acirc;&euro;&#157; design guideline for the
>>> inductive reactance of the transformer windings yields an impedance
>>> transformation that departs from its &acirc;&euro;&oelig;ideal&acirc;&euro;&#157; behavior by no more than
>>> 3%,
>>> and introduces a phase shift of no more than 14 degrees."
>>>
>>>    Now some may quibble with the 3% and 14 degrees as being picked out
>>> of the air (me) but something needs to be picked
>>> and those numbers seems reasonable.
>>>
>>>   Also, they reference Broadband RF transformers,
>>>
>>> Can I assume that scales to audio transformers?
>>>
>>>                                Mikek
>>
>>
>> The model applies to the low end of the frequency range.
>
> Yes, as I understand it, low end frequency response is the first to
> degrade as the inductance of the winding is decreased.
>
>
>> The high end is
>> controlled by other phenomena. In a broadband transformer, the high end
>> is effected by the length of the transmission line of the windings
>> compared to the wavelength and associated phase shift.
>>
>    I understand there is an effect increasing response of the high end
> by the interwinding capacitance.
>
>
>> For audio transformers, the formula is an usable approximation for the
>> low end, although the leakage inductances of the windings will affect
>> the results, adding attenuation and phase shift earlier when going
>> toward the low end of the frequency range.
>>
>    My end thought/goal is to match a very high Q crystal radio tank
> circuit at resonance. The R at resonance could be over 1 Megaohm.
>   So, I'm trying to find out, do I need a primary impedance of 4 times
> 1 megaohm?
> Then I would add a secondary winding with taps to reflect back the 2000
>   The best transformers in the crystal radio world seem to be an
> autoformer with 359 Henries and a UTC 0-15 listed as a 15k to 1Meg
> transformer. I find no inductance data on the UTC transformer.
> I want to know Is UTC matching 1 Megaohm at 100Hz using the 4 times
> rule? That would mean they have an inductance over 2000 Henries.
> I don't Know about that!
> The UTC 0-15 transformer is on Catalog Page 16 here,
> http://www.junkbox.com/electronics/utc_transformer_catalog_1963.pdf
>   I tried ringing it with 470pf and 1290pf, I get a peak but with a Q
> less than 1 and L of 2550 Henries and 2000 Henries, at 145 and 100 Hz.
> So I don't know if that is real. The DC resistance is "posted" as 5000
> ohms. So Q will be low. I'm not going to measure it because I'm afraid
> of magnetizing my core.
>   Back to the bench, I'll ring the 15k winding and try extrapolating
> from that data.
>
>
>                           Thanks, Mikek
>
I rang the 15K winding and that data is poor.
with 1290pf resonance is 700 Hz, Q=0.83 Calculates to 40 Henries.
with 470pf resonance is 800Hz, Q-0.81 Calculates to 84 Henries.
Nothing seems right. The self capacitance may be messing with
me.
Thoughts,
Thanks, Mikek
```
```"amdx" <nojunk@knology.net> wrote in message
news:m9f6p8\$6gu\$1@dont-email.me...
>   Now some may quibble with the 3% and 14 degrees as being picked out of
> the air (me) but something needs to be picked
> and those numbers seems reasonable.

It's just Magic Numbers.  Why not -3dB and 45 degrees?  Why not -1dB?  Why
not -20dB?

For wideband amps where you might be wanting to chain a few together, -1
or -2dB is a good compromise because you'll end up with -3dB sooner or
later, thus not ending up with an unexpectedly high LF limit.  If you need
extra pulse flatness, you can crank down the phase margin arbitrarily -- but
don't cry if you run out of cores to do it!

Tim

--
Seven Transistor Labs
Electrical Engineering Consultation
Website: http://seventransistorlabs.com

```
```"amdx" <nojunk@knology.net> wrote in message
news:m9gobk\$77s\$1@dont-email.me...
>   My end thought/goal is to match a very high Q crystal radio tank circuit
> at resonance. The R at resonance could be over 1 Megaohm.
>  So, I'm trying to find out, do I need a primary impedance of 4 times
> 1 megaohm?
> Then I would add a secondary winding with taps to reflect back the 2000

!!??

Rewind the damned tank.  It's just another transformer!

Better still, if you need impedance transformation in the end, construct the
tank as two coupled resonators, each with different taps suiting their
external connections.  Coupling can be a bridging C, L or proximity (k and
stray C).

Tim

--
Seven Transistor Labs
Electrical Engineering Consultation
Website: http://seventransistorlabs.com

```
```On 1/18/2015 11:24 AM, Tim Williams wrote:
> "amdx" <nojunk@knology.net> wrote in message
> news:m9f6p8\$6gu\$1@dont-email.me...
>>   Now some may quibble with the 3% and 14 degrees as being picked out
>> of the air (me) but something needs to be picked
>> and those numbers seems reasonable.
>
> It's just Magic Numbers.  Why not -3dB and 45 degrees?  Why not -1dB?
> Why not -20dB?
>
> For wideband amps where you might be wanting to chain a few together, -1
> or -2dB is a good compromise because you'll end up with -3dB sooner or
> later, thus not ending up with an unexpectedly high LF limit.  If you
> need extra pulse flatness, you can crank down the phase margin
> arbitrarily -- but don't cry if you run out of cores to do it!
>
> Tim
>
So... you agree the 4 times number is arbitrary, but it does produce
a transformer that is pretty darn good as a rule of thumb to wind
transformers.
This is the first time I ever saw a treatment of the rule.

Mikek
```
```On Saturday, January 17, 2015 at 9:46:34 PM UTC-5, amdx wrote:
<snip dumb question>

Why don't you try to find what the Bible says about it, retard.
```
```On Sun, 18 Jan 2015 09:43:31 -0800, bloggs.fredbloggs.fred wrote:

> On Saturday, January 17, 2015 at 9:46:34 PM UTC-5, amdx wrote: <snip
> dumb question>
>
> Why don't you try to find what the Bible says about it, retard.

If you are going to capitalize "bible" then you should capitalize "the"
as well, retard.

As in The Bible.

But we know that you are full of nothing but babble and pablum.  Your
post here proves that.  Most of your other posts do as well.

Hell, most times you don't even make it past utter retard into pablum
level stupidity.  But that is the Bloggstard for you.
```