Amplitude Modulation: Modulation Index, Bandwidth and Power
Amplitude modulation (AM) is the oldest broadcast technique and still the best starting point for learning modulation. The amplitude of a high-frequency carrier is varied in step with the message, and a very simple receiver, a diode with a capacitor, can recover the audio. Medium-wave broadcasting in India still uses AM, and the same mathematics appears in ASK, QAM and many lab experiments.
In this article you will derive the AM spectrum, measure modulation index from an oscilloscope, and calculate how much transmitted power actually carries information.
The AM Equation and Its Spectrum
Take a carrier of amplitude Ac and frequency fc, and a single-tone message of amplitude Am and frequency fm. Standard AM, also called double-sideband full carrier (DSB-FC), is:
s(t) = Ac [1 + m cos(2 pi fm t)] cos(2 pi fc t)
m = Am / Ac (modulation index, 0 <= m <= 1)
Expanding with cos A cos B = 1/2[cos(A-B) + cos(A+B)]:
s(t) = Ac cos(2 pi fc t) carrier
+ (m Ac / 2) cos(2 pi (fc - fm) t) lower sideband (LSB)
+ (m Ac / 2) cos(2 pi (fc + fm) t) upper sideband (USB)So a single tone produces three spectral lines: the carrier at fc and two sidebands at fc minus fm and fc plus fm, each with amplitude m Ac / 2. For a real audio signal with highest frequency fm(max), the sidebands become bands, and the occupied bandwidth is:
BW = 2 x fm(max) Example: audio limited to 4.5 kHz -> BW = 9 kHz (Medium-wave channels in India are spaced 9 kHz apart.)
Modulation Index from an Oscilloscope
On an oscilloscope the AM wave looks like a carrier whose envelope follows the message. Measure the maximum peak-to-peak height of the envelope and the minimum peak-to-peak height. Then:
m = (Vmax - Vmin) / (Vmax + Vmin) Example: Vmax = 15 V p-p, Vmin = 5 V p-p m = (15 - 5) / (15 + 5) = 10 / 20 = 0.5 (50 % modulation)
| Condition | Modulation index | What you see |
|---|---|---|
| Under-modulation | m < 1 | Envelope never touches zero; clean recovery with envelope detector |
| Full modulation | m = 1 | Envelope just touches zero at the troughs |
| Over-modulation | m > 1 | Envelope folds over; carrier switches off for part of the cycle; distortion and splatter into adjacent channels |
When several tones modulate the carrier at once, the effective index is the root of the sum of squares:
m_t = sqrt(m1^2 + m2^2 + m3^2 + ...) Example: m1 = 0.3, m2 = 0.4 -> m_t = sqrt(0.09 + 0.16) = 0.5
Power in an AM Wave
Into a load resistance R, the carrier power and each sideband power follow from the amplitudes above:
Pc = Ac^2 / (2R) P_LSB = P_USB = (m^2 / 4) x Pc Pt = Pc + P_LSB + P_USB = Pc (1 + m^2 / 2) Current form: It = Ic x sqrt(1 + m^2 / 2) Transmission efficiency (power in sidebands / total): eta = m^2 / (2 + m^2)
Worked Example 1: Transmitter Power
A 1000 W carrier is modulated to a depth of 80 percent. Find the total power, the power in each sideband and the efficiency.
Pt = 1000 x (1 + 0.8^2 / 2) = 1000 x (1 + 0.32) = 1320 W P_SB = 1320 - 1000 = 320 W total, so 160 W in each sideband eta = 0.64 / (2 + 0.64) = 0.242 -> 24.2 %
Worked Example 2: Finding m from Antenna Current
The antenna current is 8 A unmodulated and rises to 8.93 A when modulated. Find m.
(It / Ic)^2 = 1 + m^2 / 2 (8.93 / 8)^2 = 1.246 m^2 / 2 = 0.246 -> m^2 = 0.492 -> m = 0.70
Even at m = 1 the efficiency is only 1/3, so two thirds of the power goes into the carrier, which carries no information. This waste motivates the variants below.
DSB-SC, SSB and VSB
| Type | What is transmitted | Bandwidth | Receiver |
|---|---|---|---|
| DSB-FC (standard AM) | Carrier + both sidebands | 2 fm | Simple envelope detector |
| DSB-SC | Both sidebands, carrier suppressed | 2 fm | Coherent (synchronous) detector |
| SSB-SC | One sideband only | fm | Coherent detector or product detector with BFO |
| VSB | One sideband plus a vestige of the other | Slightly more than fm | Used in analog TV video |
For SSB at the same message power, all transmitted power goes into one sideband and the bandwidth is halved, which is why amateur radio and HF point-to-point links prefer it. The cost is a more complex, frequency-accurate receiver.
Envelope Detection and Its RC Time Constant
The classic AM detector is a diode followed by a parallel RC. The capacitor charges to each carrier peak and discharges slowly through R between peaks. The time constant must be long compared with the carrier period, but short enough to follow the fastest drop in the envelope:
1 / fc << RC < sqrt(1 - m^2) / (2 pi fm m) Example: fc = 1 MHz, fm = 5 kHz, m = 0.5 Upper limit = sqrt(0.75) / (2 pi x 5000 x 0.5) = 0.866 / 15708 = 55 us Lower limit: 1 us Choose RC around 10-20 us, e.g. 10 k and 1.5 nF = 15 us
If RC is too large the output cannot follow the falling envelope and you get diagonal clipping; if too small, carrier ripple remains in the audio.
Common Mistakes
- Using peak amplitudes in Vmax and Vmin but mixing peak and peak-to-peak values. Use the same measure for both.
- Writing sideband power as m^2 Pc / 2 for each sideband. That is the total of both; each is m^2 Pc / 4.
- Forgetting that bandwidth depends on the highest message frequency, not on the carrier frequency.
- Over-modulating in a lab trainer and blaming the detector for the distortion.
Summary
Standard AM puts a carrier and two sidebands on air, occupies twice the highest message frequency, and has total power Pc(1 + m^2/2). The modulation index can be read straight from an oscilloscope using (Vmax - Vmin)/(Vmax + Vmin), and must stay at or below 1 to avoid distortion. Efficiency never exceeds 33 percent, which is why DSB-SC and SSB exist, and why FM, covered next, takes a different approach altogether.