Thermal noise, which folks also call Johnson-Nyquist noise, shows up in every electrical conductor. It’s the result of random thermal agitation of charge carriers—yeah, electrons—even when there’s no current flowing at all.
This noise puts a hard limit on how sensitive any electronic system can be, no matter how clever the design or fancy the materials.
You can’t blame this noise on circuit flaws. It just comes down to the basic physics of temperature and resistance. The Johnson-Nyquist theory lays out how noise power links up with temperature, resistance, and bandwidth. That gives engineers a roadmap for predicting and handling it.
Since this is “white” noise, its power spreads out evenly across frequencies. That’s why it matters so much in things like radio receivers and precision measurement systems.
Understanding thermal noise helps explain why cooling some devices makes them work better. It also shows why bandwidth control is important, and how thermal noise stacks up against other types like shot noise.
With some effort, you can measure, predict, and minimize its effects, turning what seems like a hard limit into something you can actually work with.
Fundamentals of Thermal Noise
Thermal noise pops up from the random motion of electric charge in conductors, thanks to temperature. You’ll find it in all resistive materials. It sets a floor for signal detection and depends on both resistance and absolute temperature.
You can’t avoid this noise if you’re using passive components. It follows some pretty strict physical rules.
Definition and Physical Origins
Thermal noise—or Johnson-Nyquist noise—is electrical noise that comes from the random movement of charge carriers inside a conductor. It shows up even when you don’t apply any voltage. Complete darkness? Circuit not doing anything? Still there.
Absolute temperature makes this noise stronger, and you’ll find it in any resistive element: metal, semiconductor, or electrolyte. The noise has a nearly constant power spectral density across frequency, so it’s “white” over most frequencies you care about.
You can model the voltage noise using a Thévenin equivalent—just a noiseless resistor in series with a Gaussian noise voltage source. The root mean square (RMS) noise voltage over a bandwidth ( B ) looks like this:
[
V_{n} = \sqrt{4 k T R B}
]
Here, ( k ) is the Boltzmann constant, ( T ) is temperature in kelvins, ( R ) is resistance, and ( B ) is bandwidth.
Thermal Agitation and Charge Carriers
When the temperature isn’t absolute zero, electrons and other charge carriers in a conductor keep moving around because of thermal agitation. They bump into atoms, impurities, and each other, which randomly changes their speeds and directions.
This movement creates small voltage and current fluctuations. These aren’t from defects or outside interference—they’re just the result of particles having energy.
As temperature goes up, agitation gets more intense. Higher temperature means the charge carriers have more kinetic energy, so noise power goes up too.
That’s why engineers cool down sensitive devices, like radio telescope receivers. Cooling cuts down thermal noise and boosts measurement accuracy.
Thermal Equilibrium and Dynamic Equilibrium
Thermal noise shows up when a conductor sits in thermal equilibrium—no net energy flows in or out. The tiny energy swaps between charge carriers and the lattice balance out as time passes.
This balance is a kind of dynamic equilibrium. Particles keep moving and trading energy, but the system’s average energy stays steady.
The equipartition theorem says each degree of freedom in the system holds an average energy of ( kT/2 ). In electrical terms, this energy shows up as voltage or current fluctuations across resistive elements, even if you don’t apply a signal.
Johnson-Nyquist Theory
When charge carriers in a conductor move around because of thermal agitation, they create measurable electrical noise—even if you don’t apply a voltage. People call this Johnson-Nyquist noise.
You can explain and predict it using statistical mechanics and basic thermodynamic ideas. The theory connects temperature, resistance, and bandwidth to the random voltage or current fluctuations you see in resistors.
Historical Background: John B. Johnson and Harry Nyquist
Physicist John B. Johnson spotted small, random voltage fluctuations across resistors in thermal equilibrium. He noticed these fluctuations happened without any outside bias and depended on temperature and resistance.
At the same lab, Harry Nyquist figured out the theory behind it. He used thermodynamics and electromagnetic theory to show that this noise comes from electrons jiggling around because of heat.
Nyquist’s work proved the effect didn’t depend on the material or shape. It’s just a fact of life for conductors at any temperature above zero. This insight linked thermal energy to electrical noise, which helped engineers calculate noise limits for circuits.
Nyquist’s Theorem and Statistical Mechanics
Nyquist’s theorem says a resistor at temperature T puts out noise power that’s proportional to kT for every unit of bandwidth, with k as the Boltzmann constant (1.380649×10⁻²³ J/K).
This comes from the equipartition theorem in statistical mechanics, which gives kT/2 of average energy to each degree of freedom. In electrical circuits, each frequency mode in a resistor acts like a harmonic oscillator with this energy.
If you model a resistor as part of a matched load, Nyquist showed that the resistor and the circuit share the noise power evenly. This idea works for both voltage and current noise, and it holds for ideal resistors over a wide frequency range—except at super high frequencies where quantum effects sneak in.
Key Equations and Derivations
For an ideal resistor R, the mean-square noise voltage over bandwidth Δf is:
V²ₙ = 4 k T R Δf
- Vₙ = RMS noise voltage
- k = Boltzmann constant
- T = absolute temperature (K)
- R = resistance (Ω)
- Δf = bandwidth (Hz)
The noise power available from the resistor is:
Pₙ = k T Δf
When the load is matched, resistance doesn’t affect this power. Engineers usually write it in dBm, with −174 dBm/Hz being the thermal noise floor at room temperature.
These equations are the foundation for figuring out noise performance in amplifiers, receivers, and measurement systems.
Noise Voltage and Noise Current
You can describe thermal noise in resistors using voltage or current. Both depend on resistance, absolute temperature, and bandwidth. The math is pretty clear, and it helps you predict noise in real circuits.
Root Mean Square (RMS) Noise Voltage
The RMS noise voltage for an ideal resistor is:
[
V_\text{n} = \sqrt{4 k_\text{B} T R \Delta f}
]
Where:
- ( k_\text{B} ) = Boltzmann constant (1.380649×10⁻²³ J/K)
- ( T ) = absolute temperature (K)
- ( R ) = resistance (Ω)
- ( \Delta f ) = bandwidth (Hz)
This formula tells you noise voltage goes up with temperature, resistance, and bandwidth.
Say you have a 10 kΩ resistor at 300 K over a 1 kHz bandwidth. You’ll get about 0.4 μV RMS of thermal noise.
The noise follows a Gaussian distribution and doesn’t care about any DC voltage you might apply. It’s “white” for most practical frequencies, so its power spreads out evenly across the spectrum.
RMS Noise Current in Resistors
You can get the RMS noise current from the RMS noise voltage using Ohm’s law:
[
I_\text{n} = \frac{V_\text{n}}{R} = \sqrt{\frac{4 k_\text{B} T \Delta f}{R}}
]
So, as resistance goes up, noise voltage rises but noise current drops.
Take a 1 kΩ resistor at 300 K over 1 MHz. The RMS noise current works out to about 4 nA.
When you model a resistor’s thermal noise, you can use:
- A series voltage source with a noiseless resistor (Thévenin form)
- A parallel current source with a noiseless resistor (Norton form)
Both models work fine—it just depends on what’s easier for your analysis.
Power Spectral Density
Power spectral density (PSD) shows how noise power spreads out over frequency. For thermal noise in a resistor, the PSD stays flat:
[
\text{PSD} = 4 k_\text{B} T R \quad (\text{V}^2/\text{Hz})
]
Take the square root of PSD to get the noise voltage density, usually in nV/√Hz.
At 300 K, a 1 kΩ resistor has a noise voltage density of about 4.0 nV/√Hz.
Because the PSD is constant, doubling the bandwidth doubles the total noise power. That’s why filtering bandwidth is a go-to trick for reducing thermal noise in sensitive circuits.
Bandwidth and Temperature Dependence
Thermal noise in resistors depends directly on the frequency range and the component’s temperature. The relationship is simple and lets you estimate noise levels pretty accurately.
Role of Bandwidth in Noise Power
Noise power from a resistor climbs in direct proportion to the measurement bandwidth. If you use a wider bandwidth, you catch more of those random voltage or current wiggles, so total noise energy goes up.
For an ideal resistor, the noise is “white” and the power spectral density doesn’t change with frequency. You can calculate total noise power as:
P = k × T × B
Where:
- P = noise power (watts)
- k = Boltzmann’s constant (1.380649 × 10⁻²³ J/K)
- T = absolute temperature (kelvin)
- B = bandwidth (hertz)
If you want to cut noise in sensitive circuits, just use filters to shrink the bandwidth.
Impact of Absolute Temperature
Thermal noise amplitude rises as absolute temperature goes up. Higher temperatures push charge carriers around with more energy, so the noise gets stronger.
It doesn’t matter what the material or geometry is—if it’s a conductor in thermal equilibrium, this rule applies.
It’s a linear relationship: double the temperature, and you double the noise power. That’s why cooling, like using cryogenic temps in radio astronomy receivers, can really cut down noise and boost signal-to-noise ratio.
This temperature effect comes straight from statistical mechanics. It doesn’t depend on any applied voltage or current. It’s just the unavoidable energy you get in resistive materials at any temperature above zero.
Noise Power Calculations
Using P = k × T × B, you can estimate noise power for whatever settings you have. For room temperature (~300 K) and a 1 MHz bandwidth:
P = (1.380649 × 10⁻²³) × (300) × (1 × 10⁶)
P ≈ 4.14 × 10⁻¹⁵ W
People often write this as dBm:
Noise power (dBm) = −174 dBm/Hz + 10 × log₁₀(B)
| Bandwidth | Noise Power at 300 K (dBm) |
|---|---|
| 1 Hz | −174 |
| 1 kHz | −144 |
| 1 MHz | −114 |
These numbers help engineers design systems to hit specific noise targets.
Comparison With Other Types of Electronic Noise
Electronic noise takes a bunch of forms, each with its own cause and impact on circuits. Some come from the way charge carriers behave, others from outside interference or imperfect components.
Knowing the differences helps you pick the right design strategies and understand your limits.
Shot Noise Versus Thermal Noise
Shot noise happens because electric charge is discrete. You see it when current crosses a potential barrier, like in diodes or transistors. Each electron or hole arrives at random, making the current fluctuate.
Thermal noise, on the other hand, comes from charge carriers moving randomly due to temperature. It’s there even if no current flows, as long as the component has resistance and isn’t at absolute zero.
A big difference is dependence on current:
- Shot noise gets bigger as average current goes up.
- Thermal noise only depends on temperature and resistance, not current.
Both have flat frequency spectra, but shot noise is usually smaller in plain resistors. In active devices, shot noise can take over, though.
White Noise Characteristics
Both thermal noise and shot noise fall into the category of white noise. Their power spectral density stays constant across a broad range of frequencies.
You’ll find that noise power per unit bandwidth doesn’t really change with frequency, at least within practical limits.
White noise usually shows:
- A flat spectrum over the range you care about
- Random phase and amplitude shifts
- Often, a Gaussian distribution for those instant values
In electronics, white noise sets a baseline level of interference. You can’t filter it out within the flat region. Most designers measure it as noise voltage or current per square root of bandwidth, like nV/√Hz.
Noise in Electronic Circuits
Electronic circuits deal with a mix of noise sources, like thermal noise, shot noise, flicker noise, and whatever the environment throws in. The main culprit changes with the component and how you’re running it.
Resistors mostly generate thermal noise. Semiconductor devices can have both thermal and shot noise.
Low-frequency circuits often face flicker noise. High-frequency systems usually bump into white noise as a limiting factor.
Engineers try to cut noise by picking low-noise parts, keeping resistance down, and watching the temperature. Shielding and solid grounding help block out external noise that could raise the electronic noise floor.
Applications and Practical Implications
Thermal noise limits how sensitive a measurement system gets, how much info a communication link can handle, and how clean an amplified signal stays. Its effect really comes down to temperature, resistance, and the system’s bandwidth.
Measurement Techniques and Instrumentation
If you want to measure thermal noise accurately, you’ll need equipment with a noise floor lower than your signal. Johnson noise thermometers actually use voltage fluctuations across a resistor to measure temperature without ever touching it.
High-precision gear usually measures noise within a set bandwidth, and filters keep out unwanted frequencies. That way, you can figure out noise power using:
[
P_\text{noise} = kTB
]
Here, k stands for the Boltzmann constant, T is absolute temperature, and B is bandwidth.
People use spectrum analyzers and low-noise amps for these measurements. Careful calibration makes sure the instrument’s own noise doesn’t cover up the real thermal noise from the device you’re testing. Shielded boxes and stable reference resistors help boost accuracy.
Limitations in Signal Processing and Communications
Thermal noise draws a hard line for the smallest signal you can detect or send. In digital communications, it drags down the signal-to-noise ratio (SNR), which means you can’t push data rates past a certain point, thanks to the Shannon, Hartley theorem.
Take a receiver with a set bandwidth. If you double that bandwidth, total noise power climbs, even if signal power doesn’t budge. That can mess with bit error rates unless you crank up the signal.
In analog setups, like audio or radio receivers, thermal noise just shows up as a steady background hiss. This noise doesn’t care about your signal and you can’t filter it out without losing some of what you actually want to hear.
Noise Minimization Strategies
Engineers usually tackle thermal noise by tweaking temperature, bandwidth, and resistance. When you lower the operating temperature, you cut down on electron agitation, and that drops the noise power. Folks working with cryogenic detectors or radio astronomy receivers really rely on this trick.
Limiting bandwidth helps a lot too. If you keep the system focused on just the frequencies you need, you can reduce total noise power without messing with your actual signal.
Circuit design plays a big role here. Choosing low-noise components, matching impedance, and using differential signaling all help boost SNR. Sometimes, honestly, just cranking up the signal power is the best option if you can’t get the noise any lower.