2025-06-07
Noise is an unwanted random signal that interferes with the desired operation of circuits, especially in sensitive analog applications such as amplifiers, ADCs, and oscillators.
Noise generally manifests itself in three forms: thermal noise, shot noise and flicker or “1/f” noise. Thermal noise arises from thermal agitation of electrons in a conductor.
Thermal noise arises due to the random motion of charge carriers in resistors and other passive components.
\[ \overline{v_n^2} = 4kTR \Delta f \]
Where:
- \(k\): Boltzmann constant
- \(T\): Temperature in Kelvin
- \(R\): Resistance in Ohms
- \(\Delta f\): Bandwidth in Hz
An example of a simple circuit with this thermal noise can be observed below in Figure 1.
Figure 1: Thermal Noise in a simple Resistor circuit
source
Also called flicker noise, 1/f noise is dominant at low frequencies in MOSFETs and other semiconductor devices.
\[ S(f) = \frac{K}{f^\alpha} \]
Typical values: \(\alpha \approx 1\), and \(K\) depends on the process and device.
An example of a simple circuit with this Flicker noise can be observed below in Figure 2.
Figure 2: Flicker Noise in a simple MOSFET circuit
source
thermal_noise <- rnorm(10e3,mean=0,sd=sqrt(4*1.38e-23*1000*300*1e3))
r_plot <- ggplot(data.frame(vn = thermal_noise), aes(x = vn)) +
geom_histogram(bins = 50, fill = "skyblue", color = "black") +
labs(title = "Thermal Noise Voltage Distribution",
x = "Noise Voltage (V)",
y = "Frequency") 
Thermal noise is modeled as white noise with a flat spectral density: \[ S_{\text{thermal}}(f) = 4kTR \]
Flicker noise, or 1/f noise, decreases with frequency: \[ S_{1/f}(f) = \frac{K}{f^\alpha}, \quad \alpha \approx 1 \]
The total noise over a bandwidth \([f_1, f_2]\) is the area under the PSD curve: \[ \overline{v_n^2} = \int_{f_1}^{f_2} S(f) \, df \]
In combined systems, the total PSD becomes: \[ S_{\text{total}}(f) = S_{\text{thermal}} + S_{1/f}(f) \]
This model helps predict the total RMS noise in analog circuits.