The first transistor the signal ever meets
Rung 1 walked the whole receiver chain end to end: antenna, then a string of amplifiers, mixers, filters and converters that march the signal from radio frequency down to bits. Now we zoom all the way in on the very first link — because in a receiver, the first link is not just the start of the chain, it quietly sets the limit on everything the chain can ever do. That first active block is the low-noise amplifier (LNA), and it lives a strange double life: it must be loud and gentle at the same time.
Loud, because the incoming signal is desperately weak. A 4G or GPS signal landing on a handset antenna might be −100 dBm or fainter — a tenth of a picowatt, a hundred-billionth of the power your phone's screen burns. Everything downstream (the mixer, the analog-to-digital converter) is comparatively deaf; if the LNA doesn't lift the signal up by twenty-odd decibels first, those later blocks will never make sense of it. Gentle, because the LNA must not corrupt the one thing that actually matters: the ratio of signal to noise.
Where the noise comes from: thermal and flicker
Why can't we build a perfectly silent amplifier? Because noise is not a defect to be engineered away — it is woven into the physics of charge itself. The dominant source is thermal noise (Johnson–Nyquist noise): the electrons inside any resistive material are in ceaseless thermal motion, jostling like dust motes in sunlight, and that random jostling appears across the terminals as a tiny fluctuating voltage. It exists in every resistor, every wire, and crucially inside the conducting channel of every transistor. The available noise power is P = kTB — Boltzmann's constant times absolute temperature times bandwidth — which at room temperature (290 K) sets a floor of about −174 dBm per hertz. No clever circuit gets below it.
In a MOSFET, the channel acts like a noisy resistor, so its thermal (or 'drain') noise scales with the device's transconductance. Designers fold it into a single figure called the transistor's minimum noise figure, NF_min — the best the device can do even with a perfect source impedance. But there is a second villain that matters at low frequencies: flicker noise, also written 1/f noise. It comes from charge carriers being trapped and released at imperfections in the gate-oxide interface, and its power rises as frequency *falls* — doubling roughly every octave down. At a few gigahertz, flicker noise is usually buried below thermal noise and the LNA designer barely worries about it.
Noise figure: the headline metric
We need a single number that answers: how much does this block dirty the signal? That number is noise factor F — the ratio of the signal-to-noise ratio at the input to the SNR at the output, measured under a standard 290 K source. Quote it in decibels and you have the noise figure (NF = 10·log₁₀ F). A perfect, noiseless amplifier amplifies signal and noise equally and adds nothing of its own: F = 1, NF = 0 dB. A real stage with NF = 3 dB has cut your SNR in half — it injected as much new noise as the source itself delivered. An LNA's headline spec is precisely this number, and the best CMOS LNAs at a few GHz reach NF below 1 dB.
Noise factor and noise figure (290 K reference source):
SNR_in
F = --------- , NF = 10*log10(F) [dB]
SNR_out
Get a feel for the scale:
F = 1.00 -> NF = 0.0 dB (ideal, impossible)
F = 1.12 -> NF = 0.5 dB (a great CMOS LNA at 2 GHz)
F = 1.26 -> NF = 1.0 dB (excellent)
F = 2.00 -> NF = 3.0 dB (SNR halved)
For a passive, lossy part (cable, filter, switch):
NF = its insertion loss. A 1.5 dB pre-select
filter in FRONT of the LNA spends 1.5 dB of NF
before the signal even arrives.Two facts to carry forward. First, a purely passive lossy component has a noise figure equal to its loss: a 1.5 dB filter doesn't merely throw away 1.5 dB of signal, it adds 1.5 dB of NF to the whole chain. Second — and this is the key that unlocks everything — noise figure is *not* a fixed property of the transistor alone. It depends on what source impedance the transistor sees. Feed the device its one special 'optimum' source impedance and it hits NF_min; feed it anything else and the noise figure climbs. That dependence is the entire reason LNA design is an art, as we're about to see.
Friis's formula: why stage one is king
A receiver is a cascade: LNA, then mixer, then filters, then baseband amplifiers, each with its own gain Gₙ and noise factor Fₙ. In 1944 Harald Friis worked out exactly how the noise stacks up across such a chain, and the answer is one of the most consequential formulas in electronics. Its punchline is brutal and beautiful: the *first* stage's noise lands on the total with full weight, while every later stage's extra noise is divided down by all the gain that came before it.
Friis cascade formula (use LINEAR F and G, not dB):
F2 - 1 F3 - 1 F4 - 1
Ftot = F1 + ------ + --------- + ------------- + ...
G1 G1*G2 G1*G2*G3
What the algebra is telling you:
* F1 enters with FULL weight -> the LNA dominates
* F2 is shrunk by G1
* F3 is shrunk by G1*G2 (often already negligible)
If the LNA has G1 = 100 (20 dB) and F1 = 1.26 (1 dB),
a mixer with F2 = 10 (NF 10 dB) contributes only
(10-1)/100 = 0.09 to Ftot. The LNA's gain made the
noisy mixer almost invisible.Read it as a story. The LNA's own noise, F₁, is unavoidable — it sits on the bill at full price. But the mixer behind it, however noisy, gets its excess noise divided by the LNA's gain G₁. Crank that gain to 100× and a hideous 10 dB mixer barely dents the total. This is the deep reason RF receivers are built the way they are, and it's the design rule that flows straight from Friis: the LNA must combine a low noise figure with substantial gain. Low NF alone isn't enough; without gain, the LNA fails to shield the chain behind it. High gain alone isn't enough; a noisy first stage poisons everything no matter how loud. You need both, in the very first block.
The simultaneous noise-and-power match
Now the central puzzle of LNA design. The antenna and the filter ahead of the LNA present a 50 Ω source — a universal convention so RF blocks plug together like LEGO. Two things want to happen at that 50 Ω input, and they are in tension. First, you want a power match: the LNA's input impedance should look like 50 Ω so the antenna's signal transfers in cleanly instead of bouncing back (good return loss). Second, you want a noise match: the transistor needs to see its special optimum source impedance, Z_opt, to reach NF_min. The cruel fact is that for a bare transistor, Z_opt is *not* 50 Ω, and it's *not* even the conjugate of the input impedance. Chase the best noise and you ruin the match; chase the match and you spoil the noise.
For decades this looked like a forced compromise — you split the difference and accepted a noise figure above NF_min. Then came an elegant escape: a topology where, through a careful choice of an inductor, the input *power* match at 50 Ω lands almost exactly on top of Z_opt. The transistor gets the source impedance it wants for minimum noise *and* the antenna sees a clean 50 Ω, simultaneously. This is called a simultaneous noise-and-power match, and the circuit that achieves it is the canonical LNA we'll now build.
Worked example: the inductively-degenerated cascode CMOS LNA
Here is the most-built LNA on Earth, found in countless Wi-Fi, Bluetooth, GPS and cellular front-ends. It's two stacked transistors with three inductors, and every component has a precise, beautiful reason to be there. Let's read it from the bottom up.
VDD
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+--+--+
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(Ld) load (tank / resistor)
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+--+--+
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out o----o RF output
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||-+ M2 (common-gate CASCODE)
Vb o---|| gate at AC ground
||-+
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+----- node X
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Lg ||-+ M1 (common-source, the gain device)
o--/\/\--||
RFin g ||-+
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(Ls) source-degeneration inductor
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GND
M1 : sets transconductance gm and the noise
Ls : creates a REAL 50 ohm input resistance (gm*Ls/Cgs)
Lg : tunes out Cgs so the input is purely resistive at f0
M2 : cascode -> high gain, isolation, no Miller
Ld : resonates the output, delivers the gainStart with M1, the bottom transistor in a common-source configuration. This is the workhorse: bias it with enough current and its transconductance gm turns the input voltage into a signal current. M1 is also the dominant noise source, so its size and bias current are chosen to push NF_min as low as possible at the band of interest. Now the clever part — the inductor Ls in M1's source, the 'degeneration' inductor. By feedback, Ls makes the LNA's input impedance contain a *real, resistive* part equal to gm·Ls/Cgs, where Cgs is M1's gate-source capacitance. Pick Ls so this works out to 50 Ω, and you've conjured a perfect resistive input match — without using a physical resistor, which would have dumped its own thermal noise straight onto the weakest signal in the system.
The gate inductor Lg finishes the job. M1's Cgs is capacitive, which would leave the input reactive; Lg resonates that capacitance away at the operating frequency f₀, so right at the band the input looks purely like 50 Ω of clean resistance. Lg, Cgs and Ls together form a series resonance, and that resonance is also why the LNA gain peaks sharply at f₀ — a narrowband, tuned amplifier, exactly what a radio channel wants. On a chip, both Lg and Ls are usually on-chip inductors: flat metal spirals, a few hundred microns across, sculpted into the top thick-metal layers. Their finite quality factor Q (typically 8–15 in CMOS) is one of the real-world limits on how low the noise figure can actually go.
Finally, the top transistor M2 in common-gate, stacked on M1: the cascode. Why stack a second device? Three payoffs. It boosts gain by raising the output impedance, so the LNA delivers more amplification from the same M1. It provides reverse isolation, shielding the delicate input match from whatever the output is doing. And — subtly but vitally — it kills the Miller effect: in a lone common-source stage, the gate-drain capacitance gets multiplied by the gain and crushes both bandwidth and the input match. The cascode pins M1's drain at a near-constant voltage, so that capacitance never gets amplified. The output inductor Ld then resonates with the load capacitance at f₀ to form a tuned tank, converting M1's signal current back into a large output voltage. Two transistors, three inductors, one beautifully quiet amplifier.
- Choose M1's width and bias current to minimise NF_min at the target frequency — a power/noise trade often guided by 'constant-current' or 'power-constrained noise optimisation'.
- Pick the source inductor Ls so that gm·Ls/Cgs ≈ 50 Ω, synthesising the real input resistance with no noisy resistor.
- Pick the gate inductor Lg so that Lg, Cgs and Ls resonate at f₀, leaving a purely resistive 50 Ω input right in-band.
- Stack the cascode M2 for gain, isolation, and Miller suppression, biased so M1 stays in saturation.
- Resonate the output with Ld (and load capacitance) at f₀ to form the tuned load that realises the gain.
- Verify in simulation: NF, S11 (return loss), S21 (gain), stability, and linearity — then iterate, since the inductors' finite Q nudges every number.