Formula sheet

Every formula this tool uses, grouped by circuit: AM basics shared by everything, the JFET modulator, the diode-and-tank modulator, and the envelope demodulator. Each one is derived from the one before it, in the same order as the tool's own "show the math" panels.

AM basics

The AM signal

A carrier A_p cos(omega_p t) whose amplitude follows the message m(t), scaled to stay between -1 and +1. The bracket is the envelope; n, the modulation index, is how far the message may push the amplitude away from A_p.

x_{AM}(t) = A_p\left[1 + n\,m(t)\right]\cos(\omega_p t)

How to use: every circuit in this tool either produces this form (modulators) or undoes it (demodulator).

Spectrum: where the sidebands come from

For a single-tone message m(t) = cos(omega_m t), multiplying out with the product-to-sum identity splits the signal into three sinusoids: the carrier, untouched, and two shifted copies of the message, the sidebands, each of amplitude nA_p/2. The message never appears at its own frequency.

\cos\alpha\,\cos\beta = \dfrac{\cos(\alpha-\beta) + \cos(\alpha+\beta)}{2}

x_{AM}(t) = A_p\cos(\omega_p t) + \dfrac{nA_p}{2}\cos\big((\omega_p-\omega_m)t\big) + \dfrac{nA_p}{2}\cos\big((\omega_p+\omega_m)t\big)

How to use: the signal occupies a band 2 f_m wide around f_p; that is the bandwidth the tank modulator's filter has to pass.

Modulation index from the envelope

The envelope peaks at A_p(1+n) where m = +1 and dips to A_p(1-n) where m = -1; subtracting and adding the two removes A_p.

V_{max} = A_p(1+n),\quad V_{min} = A_p(1-n) \ \Rightarrow\ n = \dfrac{V_{max} - V_{min}}{V_{max} + V_{min}}

How to use: read V_max and V_min off a scope. n > 1 folds the envelope (the carrier flips phase) and an envelope detector then recovers a distorted message; n < 0.7 wastes most of the power on the carrier. Target 0.7 ≤ n ≤ 1.

Power efficiency

The average power of a sinusoid of amplitude A is A²/2. The carrier carries A_p²/2, each sideband (nA_p/2)²/2, and only the sidebands carry the message; the efficiency is the sideband share of the total.

\eta = \dfrac{P_{sidebands}}{P_{carrier} + P_{sidebands}} = \dfrac{2 \times \dfrac{n^2 A_p^2}{8}}{\dfrac{A_p^2}{2} + \dfrac{n^2 A_p^2}{4}} = \dfrac{n^2}{2 + n^2}

How to use: n = 1 gives 33%, n = 0.33 gives about 5%. The carrier is kept anyway because it lets a rectifier plus a low-pass filter recover the envelope.

JFET modulator

Channel conductance (ohmic region, small V_DS)

In the ohmic region the gate controls the width of the channel. The standard drain-current model there is quadratic in V_DS; for a small V_DS the squared term is negligible and what remains is Ohm's law with a conductance that depends on the gate only. V_P is the pinch-off voltage (negative for N-channel), I_DSS the current at V_GS = 0.

I_D = \dfrac{2 I_{DSS}}{V_P^2}\left[(V_{GS}-V_P)\,V_{DS} - \dfrac{V_{DS}^2}{2}\right] \ \approx\ G(V_{GS})\,V_{DS}

G(V_{GS}) = \dfrac{1}{r_{DS}} = \dfrac{2 I_{DSS}}{V_P^2}\,(V_{GS} - V_P)

How to use: valid for V_P ≤ V_GS ≤ 0 and |V_DS| well below 2(V_GS - V_P). G is a straight line: 0 at V_GS = V_P, 2 I_DSS/|V_P| at V_GS = 0. Measure I_DSS and V_P on the actual part; both vary a lot between individual JFETs.

Bias point

Halfway along the gate range is halfway up the conductance line, leaving equal room to swing both ways.

V_C = \dfrac{V_P}{2}, \qquad G(V_C) = \dfrac{I_{DSS}}{|V_P|}, \qquad r_{DS}(V_C) = \dfrac{|V_P|}{I_{DSS}}

How to use: the DC level the conditioning chain must deliver to the gate.

Gain cell

Non-inverting amplifier with the JFET channel as the bottom leg of the feedback divider (drain at the - input, source grounded) and R_b as the top leg. The - input follows the + input, so the channel current equals the R_b current.

\dfrac{x_p}{r_{DS}} = \dfrac{V_{out} - x_p}{R_b} \ \Rightarrow\ V_{out} = x_p\big[1 + R_b\,G(V_{GS})\big]

How to use: the carrier x_p(t) drives the + input, the gate voltage V_GS(t) carries the message.

Message on the gate, modulation index

With V_GS = V_C + x_m(t) and the gate swinging by a fraction s of the |V_P|/2 room, the linear conductance becomes G(V_C)[1 + s m(t)]; x is R_b in units of the channel resistance at bias. The output is then exactly the AM form.

G(V_C + x_m) = G(V_C)\big[1 + s\,m(t)\big], \qquad x = \dfrac{R_b}{r_{DS}(V_C)}

V_{out}(t) = x_p(t)\,(1+x)\left[1 + \dfrac{x\,s}{1+x}\,m(t)\right] \ \Rightarrow\ K_0 = 1 + x, \qquad n = s\,\dfrac{x}{1+x}

How to use: n can never reach s (the swing fraction is its ceiling); a bigger R_b gives both more gain and deeper modulation. For a target n, invert:

x = \dfrac{n}{s - n}, \qquad R_b = r_{DS}(V_C)\,x

Signal-conditioning chain

Four small stages turn a small bipolar source into V_C + x_m(t) at the gate. Gain stage: same non-inverting amplifier with a fixed bottom resistor. High-pass: a divider between 1/(sC) and R, corner a decade below the lowest message frequency (0.04 dB loss there). Divider: series current times R_bottom. Summer: currents into a virtual ground, unity weights with equal resistors; its minus sign turns the positive tap into the negative V_C and merely inverts the message.

\dfrac{V_{in}}{R_{bottom}} = \dfrac{V_{out} - V_{in}}{R_{top}} \ \Rightarrow\ \text{gain} = 1+\dfrac{R_{top}}{R_{bottom}} = \dfrac{V_{swing}}{V_{source}}

H_{HPF}(s) = \dfrac{R}{R + 1/(sC)} = \dfrac{sRC}{1 + sRC}, \qquad f_c = \dfrac{1}{2\pi R C} = \dfrac{f_{m,min}}{10}

V_{tap} = V_{cc}\,\dfrac{R_{bottom}}{R_{top}+R_{bottom}} = |V_C| \ \Rightarrow\ R_{top} = R_{bottom}\,\dfrac{V_{cc} - |V_C|}{|V_C|}

\dfrac{V_{ac}}{R} + \dfrac{V_{tap}}{R} = -\dfrac{V_{out}}{R} \ \Rightarrow\ V_{out} = -(V_{ac} + V_{tap}) = V_C - V_{ac}

How to use: gain from the source amplitude and the wanted swing; C from f_c with R fixed; R_top of the divider from |V_C| with R_bottom fixed; every resistor rounded to E24 and the actual values recomputed.

Diode + resonant-tank modulator

Nonlinear mixing (Taylor expansion)

A diode's smooth, strongly curved current-voltage law can be approximated around the bias point by a polynomial. The linear term only scales; the squared term creates new frequencies. Feeding it the sum of carrier and message and squaring, with cos² x = (1 + cos 2x)/2 and the product-to-sum identity, sorts the result into DC, harmonics and the wanted sidebands.

i = I_S\left(e^{v/(\eta V_T)} - 1\right) \approx I_0 + a\,v + b\,v^2 + \cdots, \qquad v = A_p\cos(\omega_p t) + A_m\cos(\omega_m t)

b\,v^2 = \underbrace{\tfrac{b}{2}(A_p^2 + A_m^2)}_{\text{DC}} + \underbrace{\tfrac{b}{2}A_p^2\cos(2\omega_p t) + \tfrac{b}{2}A_m^2\cos(2\omega_m t)}_{\text{harmonics}} + \underbrace{bA_pA_m\big[\cos((\omega_p-\omega_m)t) + \cos((\omega_p+\omega_m)t)\big]}_{\text{sidebands}}

How to use: the diode current holds DC, omega_m, omega_p, 2 omega_m, 2 omega_p and omega_p ± omega_m all at once; the tank keeps only omega_p and its two sidebands. The depth of modulation, n = 2bA_m/a, depends on the diode's curvature and is set on the bench, not designed.

Resonant tank (parallel RLC)

Parallel branches add as admittances. The imaginary part vanishes where the capacitor and the inductor cancel: that is the resonance, where the tank is just R. The impedance falls to R/sqrt(2) where the imaginary part equals 1/R; those two frequencies are 1/(RC) apart, which defines the bandwidth and Q.

Y = \dfrac{1}{R} + j\left(\omega C - \dfrac{1}{\omega L}\right) \ \Rightarrow\ \omega_0 = \dfrac{1}{\sqrt{LC}}, \qquad f_0 = \dfrac{1}{2\pi\sqrt{LC}}

\omega C - \dfrac{1}{\omega L} = \pm\dfrac{1}{R} \ \Rightarrow\ \Delta\omega = \dfrac{1}{RC}, \qquad Q = \omega_0 R C = R\sqrt{\dfrac{C}{L}}, \qquad BW = \dfrac{f_0}{Q}

How to use: BW = 2 × margin × f_m,max (margin ≥ 1) so both sidebands pass; Q = f_p / BW; pick a practical L, then C = 1/(omega_0² L) and R = Q / (omega_0 C), each rounded to a preferred value, then recompute the actual f_0, Q and BW.

Bias margin

The polynomial only describes a conducting diode: the summed voltage must stay above the forward threshold even when the carrier and the message peak together.

V_{DC} \geq A_p + A_m + V_f + \text{margin}

How to use: size the DC-bias input of the summer to at least this value.

Envelope demodulator

Absolute value keeps the envelope

The envelope is the size of the fast oscillation; the absolute value keeps the size and drops the sign. For n ≤ 1 the bracket is never negative, so it acts on the carrier only.

\left|x_{AM}(t)\right| = A_p\big[1 + n\,m(t)\big]\,\left|\cos(\omega_p t)\right|

Fourier series of the rectified carrier

A rectified cosine is periodic, hence a sum of sinusoids. Full-wave: average 2/pi, first ripple at 2 f_p. Half-wave: average 1/pi and a ripple term at f_p itself, twice as close to the message. Multiplied by the envelope, the constant term is the message and the rest is ripple the low-pass filter removes.

\left|\cos\theta\right| = \dfrac{2}{\pi} + \dfrac{4}{\pi}\left[\dfrac{\cos 2\theta}{3} - \dfrac{\cos 4\theta}{15} + \dfrac{\cos 6\theta}{35} - \cdots\right]

\max(\cos\theta, 0) = \dfrac{1}{\pi} + \dfrac{1}{2}\cos\theta + \dfrac{2}{\pi}\left[\dfrac{\cos 2\theta}{3} - \dfrac{\cos 4\theta}{15} + \cdots\right]

\left|x_{AM}(t)\right| = \underbrace{\dfrac{2A_p}{\pi}\big[1 + n\,m(t)\big]}_{\text{message + DC}} + \underbrace{\dfrac{4A_p}{3\pi}\big[1 + n\,m(t)\big]\cos(2\omega_p t) - \cdots}_{\text{ripple at } 2f_p \text{ and above}}

How to use: the ripple frequency (2 f_p full-wave, f_p half-wave) is the stopband edge of the envelope filter; the message band f_m,max is its passband edge.

Precision full-wave rectifier

Two op-amps, two diodes, R1 = R2 = R3 (any equal value). The diodes sit inside feedback loops, so their forward drop is corrected. Positive input: D2 conducts, D1 is off, no current in R1/R2, both op-amps are followers. Negative input: D1 conducts, D2 is off, U1B is an inverting amplifier fed through R1 with R2 as feedback.

V_{out} = V_{in}\ (V_{in} > 0), \qquad V_{out} = -\dfrac{R_2}{R_1}V_{in} = -V_{in}\ (V_{in} < 0) \ \Rightarrow\ V_{out} = |V_{in}|

How to use: checked against Texas Instruments TIDU030; use it whenever a single diode's ripple and 0.7 V loss are not acceptable.

Envelope low-pass filter

Same design as the Active Filter Design tool, reused directly: fp = f_m,max is the passband edge, fs = the ripple frequency the stopband edge. n is rounded up to the next even integer (every stage is a 2nd-order Sallen-Key, no leftover 1st-order stage). The full derivation of the response, the order formula, the poles and the cutoff factor is on the Active Filter Design formula sheet.

k = \dfrac{f_p}{f_s}, \qquad n \geq \dfrac{\log\!\left[\dfrac{10^{A_{min}/10}-1}{10^{A_{max}/10}-1}\right]}{2\log(1/k)}\ \text{(Butterworth)}, \qquad n \geq \dfrac{\operatorname{acosh}\sqrt{\dfrac{10^{A_{min}/10}-1}{10^{A_{max}/10}-1}}}{\operatorname{acosh}(1/k)}\ \text{(Chebyshev)}

\left|H(j\omega)\right|^2 = \dfrac{1}{1 + \varepsilon^2 \left(\dfrac{\omega}{\omega_p}\right)^{2n}}, \qquad \varepsilon = \sqrt{10^{A_{max}/10} - 1}

\varepsilon^2 \left(\dfrac{\omega_0}{\omega_p}\right)^{2n} = 1 \ \Rightarrow\ \omega_c = \omega_0 = 2\pi f_p\, \varepsilon^{-1/n}\ \text{(Butterworth)}, \qquad \omega_c = 2\pi f_p\ \text{(Chebyshev)}

How to use: Butterworth loses exactly Amax dB at fp only when its pole circle sits at fp times epsilon^(-1/n); that factor is 1 only for Amax = 3.0103 dB. Every stage is scaled by this omega_c.

Sallen-Key low-pass (unity gain)

Two equal resistors R in series to the + input, C_bottom from the + input to ground, C_top from the output back to the R-R junction X; the op-amp is a follower. Two current balances, at X and at the + input, give the transfer function.

\text{at X:}\ \dfrac{V_{in} - V_X}{R} = \dfrac{V_X - V_{out}}{R} + sC_{top}(V_X - V_{out}), \qquad \text{at (+):}\ \dfrac{V_X - V_{out}}{R} = sC_{bottom}V_{out}

H(s) = \dfrac{1}{s^2R^2C_{top}C_{bottom} + 2sRC_{bottom} + 1} \ \Rightarrow\ \omega_n = \dfrac{1}{R\sqrt{C_{top}C_{bottom}}}, \qquad Q = \dfrac{1}{2}\sqrt{\dfrac{C_{top}}{C_{bottom}}}

How to use: pick C_bottom from a preferred series, C_top = 4Q² C_bottom rounded to E12, then solve R from omega_n with the two capacitor values actually used, and round R to E24.