**AMPLITUDE MODULATION AND PEP
Bacon, WA3WDR - July 4, 2002
**

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 the 375W carrier output example, this would mean 375 * 4 = 1500W PEP.

In fact, the average overall output power level increases to 1.5 times carrier power level with 100% sinusoidally modulated AM (a carrier modulated 100% by a sine wave). This is where the extra power goes that the modulator provides. In the 375W carrier, 100% sinusoidally modulated example, that would produce 375 * 1.5 = 562.5W RMS total.

375W carrier, 562.5W RMS total with modulation, 1500W PEP - 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 changes 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 carrier, 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 asymmetrical 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 sinusoidal
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 sidebands, 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 percentages; 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, 1950s up to mid-1960s.

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

The Mechanics 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 experimenting
with DSB with and without carrier. Costas is also famous for the Costas-loop,
used in Phase Locked Loops to this day.

Bacon, WA3WDR