# The quick quiz

Try this quick quiz:

• I have an AM transmitter modulated by a single sine wave and connected to a 50 ohm dummy load with a 50 ohm transmission line.
• I have an oscilloscope connected to a -40dB sampling port on the dummy load.
• The oscilloscope has a 10:1 probe and the Y input sensitivity is set to 20mV/cm.
• The crest of the modulated envelope measures 8cm peak to peak, the valley measures 2cm peak to peak (don't read from the graticule in the picture, use the values stated in this text).

Q1. What is the PEP (ITU pX)?

Q2. What is the carrier power (ITU pZ)?

Q3. What is the modulation index?

Q4. What is the average power ("RMS power" as it is often termed) (you are free to make additional assumptions if necessary, but you must state them as part of your answer);

Q5. When power is calculated as you have done for Q4, is it meaningful for a single channel SSBSC transmitter under voice modulation?

Note that for the the purposes of the exercise, you should assume that all components etc are perfect unless stated otherwise. This means that for instance the oscilloscope is calibrated accurately, sampling port properly terminated, the modulation process is distortion free etc.

## Q1

$$pX=\frac{V_{c_{pp}}^2}{8R} \text{ W}$$ where Vcpp is the peak to peak voltage at the crest of the modulation.

pX=(8*20E-3*10/(10^(-40/20)))^2/(8*50)W

pX=64W

## Q2

$$pZ=\frac{\left [\frac{V_{c_{pp}}+V_{v_{pp}}}{2} \right ]^2}{8R} \text{ W}$$ where Vcpp is the peak to peak voltage at the crest of the modulation and Vvpp is the peak to peak voltage at the valley of the modulation.

pZ=((8+2)/2*20E-3*10/(10^(-40/20)))^2/(8*50)W

pZ=25W

## Q3

For an AM wave

$$m=\sqrt {\frac{pX}{pZ}}-1$$.

m=(64/25)^0.5-1

m=0.6

An alternative calculation that may be applied where pZ may vary with modulation (eg controlled carrier modulation, or SSB transmitters with ALC active in AM mode) is

$$m=\frac{V_{c_{pp}}-V_{v_{pp}}}{V_{c_{pp}}+V_{v_{pp}}}$$.

m=(8-2)/(8+2)=0.6

## Q4

Average power is a means of expressing the (DC) equivalent heating effect of an alternating voltage and current. It requires the averaging of the instantaneous power over a period of time. For a periodic waveform, a period of one complete cycle of the periodic waveform will yield the correct result.

For an AM wave $$pY=(1+0.5m^2)P_c \text{ W}$$ where m is the modulation index and Pc is the carrier power.

Since this sine wave modulated wave repeats with period of the inverse of the modulation frequency, it is necessary to calculate the average power over that period, and the answer is 29.5W. This also complies with ITU pY in this case, but that equivalence is not general.

## Q5

This metric is almost meaningless for an aperiodic waveform such as voice modulation of an SSBSC transmitter, unless you were specifically interested in the heating effect (eg in EMR context). It is a challenge to measure because it requires an instrument that integrates and averages instantaneous power values over a long (perhaps unspecified) period of time.

My previous measurements of the peak to average ratio of an SSB transmitter under normal voice modulation with normal ALC indicate a value of 11dB to 15dB depending on the individual speaker and compression options. So for example, a transmitter with an RMS power (measured over a transmitter on interval or "over") of 10W may be 125W PEP to 316W PEP

# Definitions

The ITU Radio Regulations define the terms Peak Envelope Power, Mean Power and Carrier Power with regard to a radio transmitter. The terms are defined as:

• Peak Envelope Power 'pX' (s1.157) means the average power supplied to the antenna transmission line by a transmitter during one radiofrequency cycle at the crest of the modulation envelope taken under normal operating conditions.
• Mean Power 'pY' (s1.158) means the average power supplied to the antenna transmission line by a transmitter during an interval of time sufficiently long compared with the lowest frequency encountered in the modulation taken under normal operating conditions.
• Carrier Power 'pZ' (s1.159) means the average power supplied to the antenna transmission line by a transmitter during one radio frequency cycle taken under the condition of no modulation.

Note: For use in formulae, the symbol p denotes power expressed in watts and the symbol P denotes power expressed in decibels relative to a reference level.