RF Building Blocks: Mixers and PLLs

Tip

Exam Focus

High-yield items: the mixer as a multiplier producing $\cos(\omega_{RF}\pm\omega_{LO})t$, with $f_{IF}=\left\lvert f_{RF}-f_{LO}\right\rvert$ and the image on the opposite side of the LO; the general spur family $\left\lvert m f_{RF}\pm n f_{LO}\right\rvert$; passive vs active mixers (loss/linearity vs gain/integration) and metrics (conversion gain/loss, isolation, NF, $P_{1\text{dB}}$, IP3); the PLL chain phase detector $\to$ loop filter $\to$ VCO with capture range typically smaller than lock range; and the synthesizer relation $f_{ref}=f_{xtal}/R$, $f_{out}=Nf_{ref}=(N/R)f_{xtal}$ plus PLL use in FM demod, carrier/clock recovery and frequency synthesis.

RF Signal-Chain Context

Communication systems partition processing into frequency regions: baseband (message near DC), intermediate frequency (IF) (a fixed band chosen for filtering and gain) and radio frequency (RF) (the transmitted/received carrier band). A frequency converter moves information among these regions: the mixer supplies nonlinear multiplication, the local oscillator (LO) supplies the translation frequency and a filter selects the wanted product.

RF signal chain: a transmitter up-converts baseband/IF to RF before the power amplifier; a receiver amplifies, down-converts to a fixed IF/baseband and detects.

Frequency Conversion

Frequency conversion translates every spectral component by a controlled amount while ideally preserving relative amplitude, phase and modulation. A mixer combines the input with the LO; sum and difference products appear, and a filter selects the wanted band.

Up-conversion (IF$\,\to\,$RF) selects the sum product; down-conversion (RF$\,\to\,$IF) selects the difference product. The same mixer core serves both directions.

Up- and Down-Conversion

Mixing $f_{IF}$ with $f_{LO}$ gives products at $f_{LO}+f_{IF}$ and $\left\lvert f_{LO}-f_{IF}\right\rvert$; a bandpass filter selects the desired RF, commonly $f_{RF}=f_{LO}+f_{IF}$ (transmitter translation, channel allocation, microwave links). The receiver mixes $f_{RF}$ with the LO and selects $f_{IF}=\left\lvert f_{RF}-f_{LO}\right\rvert$ (superheterodyne reception, spectrum analysers, SDR front ends, channelizers).

Caution

Key Formula — Conversion products

$$f_{RF}=f_{LO}+f_{IF},\qquad f_{IF}=\left\lvert f_{RF}-f_{LO}\right\rvert.$$

Low-Side, High-Side Injection and Spectral Inversion

Both LO choices $f_{LO}=f_{RF}-f_{IF}$ (low-side) and $f_{LO}=f_{RF}+f_{IF}$ (high-side) yield the same IF; the choice affects image location, LO leakage, tuning range and filter design. Selecting a difference product can reverse the order of sidebands (spectral inversion) depending on whether the LO is above or below the signal — critical for SSB and complex I/Q signals.

Mixer Principle

An ideal mixer is a multiplier. With $v_{RF}=A_{RF}\cos\omega_{RF}t$ and $v_{LO}=A_{LO}\cos\omega_{LO}t$, $v_o=K\,v_{RF}v_{LO}$, and using $\cos A\cos B=\tfrac12[\cos(A+B)+\cos(A-B)]$:

Caution

Key Formula — Ideal multiplicative mixing

$$v_o=\frac{K A_{RF}A_{LO}}{2} \bigl[\cos(\omega_{RF}+\omega_{LO})t+\cos(\omega_{RF}-\omega_{LO})t\bigr].$$

The mixer creates both products; a following filter selects one.

Mixer frequency conversion: $f_{RF}$ and $f_{LO}$ multiply to give $f_{IF}=\left\lvert f_{RF}-f_{LO}\right\rvert$ (selected) plus the sum, while an image on the opposite side of the LO maps to the same IF.

Real Nonlinear Mixer

A practical device has a nonlinear law $v_o=a_1v+a_2v^2+a_3v^3+\cdots$ with $v=v_{RF}+v_{LO}$; cross-products create the spur family:

Caution

Key Formula — Mixer spur family

$$f_{out}=\left\lvert m f_{RF}\pm n f_{LO}\right\rvert,\qquad m,n\in\mathbb{Z}_{\ge 0},\quad (m,n)\ne(0,0).$$

The wanted conversion is a low-order product; the rest are spurs needing frequency planning and filtering. The RF port carries the signal band, the LO port a strong switching/pump waveform and the IF port the translated band; a passive reciprocal mixer can often convert in either direction.

Passive and Active Mixers

Mixer classification: passive diode/FET topologies (single-ended, single-balanced, double-balanced/ring) versus active transistor mixers such as the Gilbert cell. Passive NF is approximately conversion loss plus excess noise; active mixers can provide gain but add device noise and may have a lower high-frequency limit.

Passive mixers use diodes or MOSFETs as LO-driven switches (single diode, diode ring, FET commutating). They need no DC core power, usually show conversion loss, offer good linearity and large-signal handling, require high LO drive, and (double-balanced) suppress LO/RF feedthrough. For a passive mixer at the standard noise temperature, noise figure in dB is approximately its conversion loss plus practical excess noise, subject to the stated SSB/DSB convention. Active mixers (e.g. Gilbert cell) use biased transconductance plus switching: they can provide conversion gain, accept lower LO drive and integrate well with LNA/IF stages, but consume DC power, add device noise and often have lower linearity/dynamic range; transistor speed and parasitics can also limit their highest useful frequency range.

Feature	Passive mixer	Active mixer
DC power	None / very low in core	Required (bias)
Conversion	Loss	Gain or lower loss possible
LO drive	Usually higher	Often lower
Linearity	Often higher	Often lower
Integration	Baluns awkward	Excellent in ICs
Noise	Loss directly hurts cascaded NF	Device noise, but gain helps

Passive versus active mixer trade-offs.

Balance types: unbalanced (simplest, large feedthrough); single-balanced (symmetry suppresses one feedthrough family); double-balanced (ideally suppresses both RF and LO feedthrough and even-order products). No practical balance is perfect due to device/transformer mismatch.

Mixer Performance and Nonidealities

Caution

Key Formula — Conversion gain and loss

$$G_c=P_{IF}(dBm)-P_{RF}(dBm),\qquad L_c=P_{RF}-P_{IF}=-G_c\ dB.$$

These equations use a power-conversion convention. A quoted mixer result must state whether it is power or voltage conversion and, for power, whether available or transducer gain/loss is meant. It must also give the defining test conditions: RF/IF frequencies and terminations, LO frequency and drive, RF input level, and relevant bandwidth/temperature for noise measurements.

Port isolation is the attenuation of unwanted leakage between ports (LO$\to$RF limits LO radiation from the antenna; LO$\to$IF limits LO feedthrough; RF$\to$IF measures direct leakage). Higher isolation in dB is better.

• Noise figure: $NF=10\log_{10}(SNR_{in}/SNR_{out})$; image-band noise can convert to IF, so single- vs double-sideband NF must be stated.

• $1$-dB compression point ($P_{1\text{dB}}$): input/output power where actual conversion gain falls $1\,dB$ below the small-signal extrapolation — onset of large-signal compression, not an operating point.

• Third-order intercept (IIP3/OIP3): two tones $f_1,f_2$ create in-band $2f_1-f_2$ and $2f_2-f_1$; extrapolated fundamental and IM3 lines meet at the intercept. Higher IP3 $=$ better linearity, but it is extrapolated.

• Spurious/image: any input with $\left\lvert m f_{RF}\pm n f_{LO}\right\rvert=f_{IF}$ maps to IF; the ordinary image is the $m=n=1$ term on the opposite side of the LO. Preselection and multiple conversion reduce responses.

Note

Key Point — Mixer nonidealities

• LO phase noise causes reciprocal mixing of strong adjacent interferers.

• DC offset / LO self-mixing dominates direct-conversion receivers.

• Even-order distortion and I/Q imbalance create unwanted images.

• Impedance mismatch shifts conversion gain, noise and spur performance.

Phase-Locked Loop (PLL)

A PLL is a negative-feedback system that adjusts a voltage-controlled oscillator (VCO) so its output phase/frequency follows a reference. It comprises a phase detector, loop filter and VCO, with optional feedback division.

PLL core: the phase detector (PD) compares reference and feedback phase; the loop filter averages the error; the VCO output is fed back (optionally $\div N$) to close the loop.

Block	Function
Phase detector / PFD	Compare reference and feedback phase; produce error
Charge pump (digital PLL)	Convert PFD pulses to controlled current
Loop filter	Average error, reject detector products, set loop dynamics
VCO	Produce frequency controlled by loop voltage
Divider ($\div N$, optional)	Scale output before comparison for synthesis

PLL building blocks.

Caution

Key Formula — Linearised PLL relations

$$v_d=K_d(\phi_i-\phi_f),\qquad f_o=f_{free}+K_{VCO}\,v_c,$$

where $K_d$ is detector gain (V/rad) and $K_{VCO}$ is VCO sensitivity (Hz/V).

Operating States

1. Free-running: without useful error, the VCO runs near $f_0$.

2. Acquisition/capture: average correction pulls the VCO toward the input.

3. Locked: average frequencies are equal (or follow the divider ratio). Phase error is bounded and constant for a static offset.

4. Loss of lock: input change, noise or modulation exceeds tracking/hold capability.

Capture and Lock Ranges

Typical PLL relation: the capture range (acquire from unlocked) is smaller than the lock/hold range (remain locked). Exact definitions and limits depend on detector, loop topology and test method.

The capture range is where an unlocked PLL can acquire lock; the lock/hold/tracking range is where an already-locked loop stays locked as the input changes slowly.

Note

Law — Capture vs lock

The capture range is typically smaller than the lock range: an unlocked detector can produce fast alternating error that the low-pass loop averages away, while a locked loop can hold via DC feedback across a wider tuning range. The exact definitions and relation are topology- and test-method-dependent, so numerical limits require the detector, loop filter and VCO model.

Loop Bandwidth Trade-Off

Wider loop bandwidth tracks faster modulation and acquires quickly but passes more noise/spurs; narrower bandwidth filters noise but acquires/tracks slowly. Loop damping sets overshoot, settling and stability. A first-order loop has no independent filter pole; second-order loops (most common) trade bandwidth against damping.

PLL Applications

PLL applications span demodulation, synchronization, synthesis, tracking and multiplication. For AFC, filtered control voltage corrects slow local-oscillator drift.

FM/FSK Demodulation

The VCO control voltage tracks instantaneous input frequency, so the loop-filter output is the recovered message:

Caution

Key Formula — PLL FM demodulator output

$$v_c(t)\approx\frac{f_i(t)-f_0}{K_{VCO}}.$$

For FSK, the control voltage settles to two levels (mark/space); a threshold recovers bits if both tones lie in the tracking range and the symbol duration is long enough for the loop to settle sufficiently before the decision instant.

PLL as an FM demodulator: the recovered message is taken as the loop-filter control voltage that forces the VCO to follow the input.

Carrier and Clock Recovery

Suppressed-carrier DSB/PSK needs a coherent reference: a Costas loop uses I/Q products to form a data-insensitive phase error; a squaring loop makes a carrier multiple then divides; a pilot-aided loop locks to a transmitted pilot. BPSK’s $180^\circ$ ambiguity is handled by differential coding or training. Clock recovery uses a timing-error detector plus loop filter and controlled oscillator to place sampling instants at symbol centres (a timing PLL, even when digital).

Frequency Synthesis and Multiplication

With a reference divider, define $f_{ref}=f_{PFD}=f_{xtal}/R$. Inserting $\div N$ in feedback gives, in lock, $f_{out}/N=f_{ref}$, so $f_{out}=N f_{ref}$; changing integer $N$ selects channels spaced by $f_{ref}$. A feedback divider multiplies by $N$; a divider after the VCO yields coherent lower frequencies. Frequency multiplication scales carrier frequency, frequency deviation and phase deviation by the multiplication factor. Reference-derived phase noise rises ideally by $20\log_{10}N$ in dB, and practical PLL output also contains loop, divider and VCO noise.

Integer-$N$ frequency synthesizer: reference $\div R$, PFD, loop filter, VCO and $\div N$ feedback give $f_{out}=Nf_{ref}=(N/R)f_{xtal}$.

Caution

Key Formula — Synthesizer output frequency

$$f_{ref}=f_{PFD}=\frac{f_{xtal}}{R},\qquad \frac{f_{out}}{N}=f_{ref}\ \Rightarrow\ f_{out}=Nf_{ref}=\frac{N}{R}\,f_{xtal}.$$

The basic integer-$N$ channel step is $f_{ref}$. Fractional-$N$ synthesis modulates the divider sequence so its average ratio is fractional, giving finer channel steps. Divider quantization then creates shaped noise and fractional spurs; the modulation pattern and PLL bandwidth/loop filter must be chosen together to trade settling, in-band noise and spur suppression.

Tracking Filter and AFC

A narrow PLL can act as a tracking filter: it follows the wanted carrier inside its loop bandwidth while rejecting more distant out-of-band noise and interference. In automatic frequency control (AFC), the filtered control/error voltage is fed to the receiver local oscillator so that slow LO drift is corrected and the wanted signal remains centred in the IF passband.

Worked Examples

Note

Example — RF plan — image location (5 marks)

Receive $f_{RF}=100\,MHz$ at $f_{IF}=10.7\,MHz$.
High-side LO: $f_{LO}=100+10.7=110.7\,MHz$, image $=110.7+10.7=121.4\,MHz$.
Low-side LO: $f_{LO}=100-10.7=89.3\,MHz$, image $=89.3-10.7=78.6\,MHz$.
Both give the wanted $10.7\,MHz$ IF; choose by RF filter rejection, tuning, spurs and spectral orientation.
Practice: sketch both plans and mark the image on the opposite LO side.

[!NOTE]

Example — Mixer conversion loss and isolation (5 marks)

A passive mixer has $P_{RF}=-30\,dBm$, $P_{IF}=-36\,dBm$:

$$L_c=P_{RF}-P_{IF}=-30-(-36)=6\,dB.$$

An LO of $+10\,dBm$ leaking at $-50\,dBm$ at the RF port gives LO$\to$RF isolation $=10-(-50)=60\,dB$.
Practice: state test impedances/LO drive with every mixer number.

[!NOTE]

Example — Synthesizer channel plan (5 marks)

With $f_{xtal}=10\,MHz$ and $R=100$: $f_{PFD}=100\,kHz$. Selecting $N=975$ gives $f_{out}=(975/100)\times 10=97.5\,MHz$; $N=976$ gives $97.6\,MHz$, confirming $100\,kHz$ spacing.
Practice: the VCO/lock range must include every selected channel.

[!NOTE]

Example — PLL FM demodulator output (5 marks)

With $K_{VCO}=100\,kHz/V$ and deviation $\Delta f=50\,kHz$:

$$v_{c,\text{peak}}\approx\frac{\Delta f}{K_{VCO}}=\frac{50}{100}=0.5\,V.$$

The loop bandwidth must pass $f_{m(\max)}$ and the deviation while rejecting out-of-band noise.
Practice: relate loop bandwidth to message bandwidth and lock retention.

[!NOTE]

Key Point — Revision — RF Building Blocks

• Mixer $=$ multiplier: products $\cos(\omega_{RF}\pm\omega_{LO})t$; select one with a filter.

• $f_{IF}=\left\lvert f_{RF}-f_{LO}\right\rvert$; image on the opposite LO side maps to the same IF.

• General spur family $\left\lvert m f_{RF}\pm n f_{LO}\right\rvert$; plan frequencies and preselect, with nonnegative $m,n$ not both zero.

• Passive: loss, high linearity, high LO drive; active (Gilbert): gain, bias, easy integration. Double-balanced suppresses RF and LO feedthrough.

• Metrics: $G_c/L_c$, isolation, NF (SSB/DSB), $P_{1\text{dB}}$, IP3.

• PLL chain: phase detector $\to$ loop filter $\to$ VCO $\to$ ($\div N$) feedback; capture range is typically smaller than lock range.

• Synthesizer $f_{ref}=f_{xtal}/R$ and $f_{out}=Nf_{ref}=(N/R)f_{xtal}$; fractional-$N$ gives finer steps with quantization-noise and spur trade-offs.

• PLL uses: FM/FSK demod, carrier/clock recovery, AFC, frequency synthesis/multiplication.