PLLs, Mixers & Analog Communication Circuits

Analog circuits that move and lock frequencies. Learn phase-locked loops (phase detector, charge pump, loop filter, VCO, divider), lock range and loop bandwidth, integer-N and fractional-N synthesis, phase noise and reference spurs, clock recovery; mixers and frequency translation, the superheterodyne receiver and image frequency, AM modulation and envelope detection, FM and Carson's rule, PLL FM demodulation, LNAs and impedance matching basics, AGC and RSSI — with Python computing synthesizer outputs, loop bandwidth, AM/FM bandwidths and superhet image frequencies.

Radios, clocks, serial links and motor drives all need frequencies that are precise, tunable or locked to a reference. The PLL does that by forcing a VCO to track a reference in phase — a feedback loop where the controlled quantity is phase.

Mixers shift signals in frequency, letting receivers do most filtering and gain at a fixed intermediate frequency. Together, PLLs and mixers are the backbone of analog communication.

An analogy

A PLL is like a runner pacing a metronome: the runner (VCO) speeds up or slows down whenever their steps drift ahead or behind the beat (phase detector), until they run exactly in step. A divider lets them take N steps per beat — a new, precise tempo.

PLL building blocks

Each block and its role:

Loop dynamics and noise

Inside the loop bandwidth, output phase noise follows the reference (×N, i.e. +20 log N dB); outside it follows the VCO. Choose bandwidth to minimise integrated jitter, typically ≤ f_ref/10 to suppress reference spurs and ensure stable sampled-loop behaviour. Wider bandwidth locks faster; narrower filters more reference noise and spurs. Phase margin 45–65° gives well-damped settling.

Receivers and modulation

The superheterodyne mixes the incoming RF with a tunable LO to a fixed IF where selective filters and AGC amplifiers work best; the image frequency (2·IF away) must be removed before the mixer. Zero-IF and low-IF receivers use I/Q mixers and digital processing. AM is demodulated with envelope detectors (diode + RC) or synchronous detectors; FM with discriminators, quadrature detectors or PLLs.

RF front-end essentials

LNAs set receiver noise figure (Friis: the first stage dominates) and must be matched (often 50 Ω) using L/C networks or common-gate inputs. Mixers are specified by conversion gain, noise figure, IP3 and LO–RF isolation. Clock-and-data-recovery PLLs, frequency multipliers, and PLL-based motor speed control reuse the same loop ideas. Use S-parameters, Smith charts and EM-aware layout at GHz frequencies.

Formulas

Worked example: Synthesizer

Given: f_ref 10 MHz, N = 400

Find: f_out.

Answer: 4.0 GHz

Worked example: FM bandwidth

Given: Δf 5 kHz, f_m 3 kHz

Find: Carson bandwidth.

Answer: 16 kHz

Worked example: Image

Given: RF 100 MHz, IF 10.7 MHz, high-side LO

Find: Image frequency.

Answer: 121.4 MHz

Charge-pump PLL synthesizer

  f_ref ─► [Phase/frequency detector] ─► [Charge pump I_CP] ─► [Loop filter R, C1, C2] ─► [VCO] ─┬─► f_out
            ▲                                                                                    │
            └──────────────────────────── [÷ N divider] ◄───────────────────────────────────────┘
  Locked: f_out/N = f_ref, phases aligned

The phase-frequency detector compares the reference with the divided VCO output and drives the charge pump; the loop filter converts current pulses into a smooth control voltage; the VCO frequency changes until the divided output matches the reference in phase. The loop filter sets bandwidth, settling, stability and spur suppression.

Synthesizer, loop bandwidth, AM/FM and superhet image

import math
# PLL synthesizer: output frequency, loop bandwidth estimate; AM/FM bandwidths and superhet image
fref, N = 1e6, 2400
print(f"integer-N: fout = N*fref = {N*fref/1e9:.3f} GHz, channel step {fref/1e6:.0f} MHz")
Kvco, Icp = 50e6, 1e-3                            # Hz/V, A
R = 2.2e3
fc = Icp * R * Kvco / N                           # |T| = 1 where the loop-filter R dominates (Kvco in Hz/V)
print(f"type-II loop bandwidth ~ Icp*R*Kvco/N = {fc/1e3:.1f} kHz (should be < fref/10 = {fref/10/1e3:.0f} kHz)")
fm, m = 5e3, 0.8
print(f"AM: 5 kHz tone, m = 0.8 -> bandwidth {2*fm/1e3:.0f} kHz, sideband power {m**2/2/(1+m**2/2)*100:.1f}% of total")
dev = 75e3
print(f"FM broadcast: deviation 75 kHz, 15 kHz audio -> Carson bandwidth {2*(dev+15e3)/1e3:.0f} kHz")
frf, fif = 100e6, 10.7e6
flo = frf + fif
print(f"superhet: LO {flo/1e6:.1f} MHz, image at {(flo+fif)/1e6:.1f} MHz (2*IF = {2*fif/1e6:.1f} MHz away)")
# integer-N: fout = N*fref = 2.400 GHz, channel step 1 MHz
# type-II loop bandwidth ~ Icp*R*Kvco/N = 45.8 kHz (should be < fref/10 = 100 kHz)
# AM: 5 kHz tone, m = 0.8 -> bandwidth 10 kHz, sideband power 24.2% of total
# FM broadcast: deviation 75 kHz, 15 kHz audio -> Carson bandwidth 180 kHz
# superhet: LO 110.7 MHz, image at 121.4 MHz (2*IF = 21.4 MHz away)

A 1 MHz reference and N = 2400 give 2.4 GHz with 1 MHz channel steps. With 1 mA, 2.2 kΩ and 50 MHz/V the loop bandwidth is ~46 kHz — below the f_ref/10 guideline. AM with m = 0.8 puts only 24% of the power in the information-bearing sidebands. FM broadcast needs 180 kHz per Carson's rule. A 10.7 MHz IF places the image 21.4 MHz away from the wanted station, so the RF front-end filter must reject it.

Common mistakes

Applications

Key takeaways

Going further