[AMRadio] Power Levels

[Bob Bruhns]

I discovered one more error -

In the equation for PEP with 200% modulation, I showed 3^3
(three cubed) when I meant 3^2 (3 squared).  The section
should have read:

With 200% modulation,
M = 2,
M + 1 = 2 + 1 = 3,
3^2 = 9,
so PEP= 9C.

Sorry for the repeat, but for the record, here is my whole
statement, with technical corrections, one more reference,
and a few spelling and punctuation corrections.

  Bacon, WA3WDR
-------------------------------

Modulation does interesting things in AM.  500W input, 375W
output is roughly true of the CW carrier, but then we apply
AM to the carrier.  With 100% symmetrical modulation, this
varies the carrier amplitude from its normal level up to
twice carrier voltage on positive peaks, and down to zero on
negative peaks.

With a resistive load (which we have), doubling the voltage
results in doubling the current.  But when both the voltage
and current are doubled, the power is multiplied by four.
Hence with 100% symmetrical modulation, and likewise with
any AM modulation waveform producing 100% positive
modulation, PEP = 4 * carrier.   .

In fact, the average overall output power level increases to
1.5 times carrier power level with 100% sinusoidal AM.  This
is where the extra power goes that the modulator provides.

It's an interesting subject.  Sometimes it doesn't seem to
make sense, but it really does.

Another fine point: the carrier itself is a sine wave.  When
we modulate it, it is slightly distorted from a sine wave
because it is changing in amplitude.  But these change are
usually very slow compared to the carrier frequency, so we
ignore them.  The carrier is essentially a sine wave.  This
is what the FCC is talking about when they say that the peak
RMS power of the RF envelope shall not exceed 1500 watts,
averaged over a carrier cycle.  This was to avoid ambiguity
about the meaning of peak level.  It could have meant the
peak voltage that the carrier sine wave reached, which would
be 1.414 times the RMS; in this 1500W PEP example, the power
at that instant would be 3000 watts.  But that's not what
they meant; they meant the peak RMS level on a one-cycle RF
timescale.

---------------------------

(Someone thought that an AM signal amplified through a 100W
PEP amplifier would have 33.3W carier, 33.3W upper sideband,
and 33.3W lower sideband.)

Classical AM has a carrier, an upper sideband and a lower
sideband.  However, the power is not distributed 1:1:1
(carrier/usb/lsb) unless supermodulation is used.

I refer to classical or classic AM to distinguish it from
compatible modes such as asymmetircal sideband used in
analog television broadcasts, and similar techniques
sometimes used in AM broadcasting to avoid adjacent channel
interference.

If you had a classic 100% modulated AM signal on 1000 KHz,
modulated by a 1 KHz sine wave, it would have a carrier at
1000 KHz, an upper sideband at 1001 KHz and a lower sideband
at 999 KHz.  The following would be true:

1) The energy in the upper sideband would be equal to the
energy in the lower sideband.

2) The peak RMS sideband energy in both sidebands combined,
averaged over one RF carrier cycle, would be equal to the
carrier level.  (This way, they cancel to zero at the
negative peak.)

3) The RMS energy in each sideband would be 1/4 of the
carrier level.

#3 seems strange, after #2, but it is the case.  In this
example, the upper sideband is a CW signal at 1001 KHz and
the lower sideband is a CW signal at 999 KHz.  Take the
carrier away, and you get a 2 KHz beat note.

Remember, we said this was a 100% modulated classic AM
signal.  In this example, if the carrier level is 1.0V RMS,
the upper sideband level will be 0.5V RMS, and the lower
sideband level will be 0.5V RMS.

The sidebands are on different frequencies, so one is
cycling faster than the other, and it is alternately in
phase, and out  of phase, with the other sideband.  As the
upper and lower sideband add and subtract from each other
(that 2 KHz beat note), the voltage adds to 1.0V RMS when
the two sideband frequencies are in phase, then drop to 0
when the two sideband frequencies are out of phase, then add
to 1.0V RMS when the two sideband frequencies are in phase
again, and drop to zero when the two sideband frequencies
are out of phase again, and so on.

When we add the carrier, of course we get carrier level when
the sidebands cancel each other to zero, but we get either
2X carrier level when the sidebands add to 1.0V RMS, if the
combination at that point is in phase with the carrier, or
we get 0 level when the sidebands add to 1.0V RMS, if the
combination at that point is out of phase with the carrier.
In this example, the sidebands produce peaks of 1.0V RMS
that are alternately in phase and out of phase with the
carrier.  This results in 100% positive and negative
modulation.

The interesting thing is the power levels involved.  Since
two 1.0V signals (the carrier, and the combination of the
two sidebands), of equal power levels, are adding to produce
a 2.0V RMS positive peak, wouldn't you think the peak power
could only be 2X carrier level?  But it doesn't work that
way.  The voltage is doubled, and the peak power is
multiplied by 4X.

Likewise, if there is 1/4C power in each sideband, then
wouldn't that add up to 0.5X power total, and wouldn't that
only modulate the carrier partially?  No, because that
analysis is static, that is it does not allow for the 2 KHz
beat, it just takes the average.

In fact, the total signal energy increases exactly in that
way, to 1.5X carrier power with 100% sinusoidal modulation,
if you take the composite signal RMS over a full cycle of
modulation, or several full cycles of modulation, or over a
long period of time.  But the instantaneous power varies
during this time, because of the beat note, and because the
combined sidebands are alternately in and out of phase with
the carrier.  The average power levels are squeezed into the
varying AM envelope, and this is how two sidebands of only
1/4C each add up with a carrier of 1C to produce positive
peaks of 4X carrier (RMS taken over one carrier cycle), and
average power of 1.5C (RMS taken over one modulation cycle,
or an integer number of modulation cycles, or over a long
period of time).

So if I understand the example (classical AM at 100W PEP
output, with sinusioidal modulation?), you would have 25W
RMS carrier output, 6.25W RMS in each sideband, and 100W RMS
PEP.  (Remember that RMS in the context of PEP is taken over
one RF carrier cycle.)

33-1/3W carrier with 33-1/3 W in each sideband would add up
like this: 1.0C RMS carrier + 1.0C RMS USB + 1.0C RMS LSB =
3.0C peak = 3 x 3 = 9X carrier level on peaks, so you would
have 33-1/3 x 9 = 300W PEP.  Modulation percentage would be
1.0C +/- 2.0C SB, or 200% modulation.

Note that compared to 100% modulation of a 33-1/3W carrier,
which would require only 16-2/3 W total from both sideands,
or 8-1/3W from each sideband, 200% modulation requires
33-1/3W per sideband, so you need 4X the modulator power for
a 2X increase in modulation.  That's why we like high
modulation perentages; the sideband energy increases
quickly.

An equation for PEP of an AM signal.: PEP = ((M+1)^2)C,
where M is the modulation factor.  100% modulation would be
M = 1, 200% would be M = 2, etc.  So with 100% modulation,
M = 1,
M + 1 = 2,
2^2 = 4 (2^2 means 2 squared),
so PEP = 4C.

With 200% modulation,
M = 2,
M + 1 = 2 + 1 = 3,
3^2 = 9,
so PEP= 9C.

A few good references:

The older ARRL Handbook issues, maybe 1981 and back

Radio Engineers Handbook, by Frederick Terman, McGraw-Hill,
1943

The Mechaics of Modulation" by Paul R. Huntsinger, October
1931 QST, with a missing figure or to appearing in
Corrections, QST, November 1931 (page 34).

"Lop Sided Speech and Modulation" by George Grammer,
February, 1940 QST

"A Course in Radio Fundamentals" by George Grammer (Part 6 -
Modulation, QST November, 1942)

"New Sideband Handbook" by Don Stoner, Cowan Publishing
Corp., 1958, 1959, 1960, 1962, 1964, 1966.

Also look over anything you can find by John. R. Costas,
W2CRR.  (I don't know whether he is still living, so his
callsign may be reissued by now.)  Also look for Norgaard.
These guys were pioneers in sideband, and they were fooling
with DSB with and without carrier.  Costas is also famous
for the Costas-loop, used in Phase Locked Loops to this day.

Bacon, WA3WDR