AM Modulator / Demodulator Design

An AM modulator needs a gain that follows the message. The trick used here: a JFET operated with a small drain-source voltage behaves like a resistor whose value is set by its gate voltage, so it can be dropped in wherever a resistor sets an amplifier's gain.

Why a resistor: in the ohmic (triode) region, where V_DS is small, the JFET channel is a conducting path whose width the gate controls. A more negative gate narrows it, until at V_GS = V_P (the pinch-off voltage, negative for an N-channel part) it closes completely. The standard model for the drain current there is:

I_DSS is the current at V_GS = 0 with the channel fully open. For a small V_DS the squared term is negligible next to the first one, and what is left is Ohm's law, I = G V, with a conductance that depends on the gate only:

That conductance is a straight line in V_GS: zero at V_GS = V_P (channel closed, infinite resistance) and 2 I_DSS/|V_P| at V_GS = 0 (channel wide open). Biasing the gate exactly halfway, at V_C = V_P/2, lands exactly halfway up that line, which leaves the same room to swing in both directions:

One condition to respect: dropping the squared term needs |V_DS| well below 2(V_GS - V_P). In this circuit V_DS is the carrier voltage at the op-amp's inverting input, and the tightest moment is the most negative gate swing, V_GS = -3.80 V, where 2(V_GS - V_P) = 0.40 V. Keeping the carrier amplitude several times smaller than that keeps the JFET a clean resistor; a larger carrier bends the gain within each carrier cycle and distorts the output.

The gain cell is a non-inverting amplifier with the JFET channel in place of the bottom resistor: the carrier x_p(t) drives the + input, the channel goes from the - input to ground, and R_b feeds the output back to the - input. With negative feedback an ideal op-amp keeps its two inputs at the same voltage and draws no input current, so the - input sits at x_p(t), and the current through the channel must also flow through R_b:

So the gain is 1 + R_b G, and G is set by the gate. Now put the message on the gate on top of the bias, V_GS(t) = V_C + x_m(t). Because G is a straight line in V_GS, adding x_m to the gate adds a proportional amount to G. Writing the gate swing as a fraction s of the room between V_C and V_P (that room is |V_P|/2), x_m(t) = s (|V_P|/2) m(t) with m between -1 and +1:

Substitute that into the gain and call x = R_b G(V_C) = R_b / r_DS(V_C), the feedback resistor measured in units of the channel resistance at bias:

That is exactly the AM form A_p[1 + n m(t)] cos(ω_p t) from the basics, with the carrier amplitude multiplied by a constant gain and the message riding on it:

Two things to read off. n can never reach s, because x/(1+x) is always below 1: the swing fraction is the ceiling of the modulation index. And a bigger R_b raises both the gain and the depth of modulation, which is what a scope shows when R_b is changed on the same JFET. To hit a target n, invert the relation:

Over one message cycle the gate swings by ±- around V_C and the gain moves between 2.700 and 33.300: the carrier comes out 12.33 times larger at the crest of the message than in its trough, which is the ratio (1+n)/(1-n) of the envelope.

The gate needs V_C + x_m(t): a DC level of -2.00 V with the message swinging ±- on top of it. A signal generator delivers something else, typically ±1.00 V centered on 0 V. Three small stages bridge the gap: amplify to the right swing, strip any DC the amplifier may add, then add back exactly the DC wanted, taken from a resistor divider off the supply.

1. Gain stage. The same non-inverting amplifier as the gain cell, with a fixed resistor instead of the JFET. The - input sits at V_in (ideal op-amp with feedback), and the current through R_bottom equals the current through R_top:

2. DC-blocking high-pass. A capacitor in series, then a resistor to ground, is a voltage divider between the capacitor's impedance 1/(sC) and R. At DC the capacitor is an open circuit and nothing gets through; well above the corner it is a short and everything does:

Putting f_c a decade below the lowest message frequency (100 Hz) costs the message almost nothing: at f = 10 f_c the loss is 10 log(1 + 0.01) = 0.04 dB and the phase shift under 6 degrees. R is fixed at 100 kΩ, C is solved from f_c and rounded to a stock value:

3. Bias divider. Two resistors in series across the supply carry the same current, so the tap voltage is that current times the bottom resistor. The tap has to sit at |V_C|; with R_bottom fixed, R_top follows:

4. Summer. An inverting amplifier with two inputs through equal resistors R. The - input is a virtual ground (0 V), so each input pushes a current V/R into that node, the op-amp pulls the sum back out through the feedback resistor, and with all three resistors equal the weights are exactly 1:

The minus sign is useful, not a nuisance: it turns the positive tap +2.00 V into the negative V_C = -2.00 V an N-channel gate needs, and it merely inverts the message, a 180 degree phase shift that changes nothing in the AM envelope. The gate now sits at V_C + x_m(t), which is what the gain cell was derived for.

A message such as audio lives at low frequencies. Sent as it is, it cannot be radiated (the antenna would need to be kilometres long) and two senders could never share the air. Modulation moves the message up to a carrier frequency of the sender's choosing. Amplitude modulation is the simplest way to do it: let the message set the size, the amplitude, of a fast carrier wave.

Write the carrier as A_p cos(ω_p t) and the message as m(t), scaled so that it stays between -1 and +1. Amplitude modulation multiplies the carrier by a factor that follows the message:

The bracket is the envelope, the slowly varying outline of the fast oscillation, and n is the modulation index: how far the message is allowed to push the amplitude away from A_p. Nothing in the bracket is invented; it reads "amplitude A_p, plus a bit proportional to the message".

What frequencies does that contain? Take the simplest message, a single tone m(t) = cos(ω_m t), and multiply out. The product of two cosines is where new frequencies appear:

So an AM signal is three sinusoids: the carrier itself, untouched, plus two copies of the message shifted to either side of it, the lower and upper sidebands, each of amplitude nA_p/2. The message never appears at its own frequency ω_m; only the sidebands carry it, and the whole signal fits in a band 2 f_m wide around the carrier.

On a scope the envelope is read directly: its highest value is A_p(1+n), where m = +1, and its lowest is A_p(1-n), where m = -1. Subtracting and adding the two removes A_p:

n cannot usefully exceed 1: past that the bracket goes negative during part of the cycle, the carrier flips phase there and the envelope folds back on itself. An envelope detector only sees the size of the signal, never its sign, so it recovers a distorted message. Below about 0.7 the sidebands become small next to the carrier and most of the transmitted power is wasted, as the next lines show.

The average power of a sinusoid of amplitude A (into 1 ohm) is A²/2. Applied to the three lines above: the carrier carries A_p²/2, each sideband (nA_p/2)²/2, and only the two sidebands carry the message. The fraction of the total power that is actually message is therefore:

Even at n = 1 only a third of the power is message. The carrier is kept anyway because it is what lets a receiver recover the envelope with nothing more than a rectifier and a low-pass filter (see the Demodulator mode).