Noise is always a problem with sensors and other sensitive circuits. Some noise comes from digital circuits, power supplies, or RF interference. Thermal agitation is another source of noise in all components with resistance. This random noise has no pattern. In audio it sounds like hiss, and in cameras it looks like speckles or old analog-TV snow. Averaging can help reduce the amount of noise in a sensor reading, and the Fourier Transform can pull weak signals out from high levels of noise. This post discusses the causes of random noise, and how to reduce it in analog circuits.

### Quantify Noise with RMS Voltage

The Root Mean Square (RMS) voltage describes the amplitude of noise. In the graph showing a 1V RMS waveform, the peak-to-peak variation is about 6 volts. Some instruments just multiply the RMS value by 6 to find the peak-to-peak voltage of a noise waveform. Oscilloscopes with phosphor, either analog or implemented digitally, do a better job of displaying noise.

### Noise is Free!

Every resistor creates its own voltage noise source. If it is convenient, the Norton equivalent current source can be analyzed instead. The noise waveform has a random Gaussian Distribution with an RMS value that increases with the square root of the resistance. The amount of noise measured also increases with the square root of circuit bandwidth. This leads to the unusual square-root-of-Hertz units. For example, if I could measure the resistor noise waveform with a 400MHz oscilloscope, it would have twice as much noise as would be measured with a 100MHz oscilloscope.

One way to reduce noise in a system is to use averaging. To cut the noise in half, the number of samples in the average needs to be doubled. This gets slow, and taken to an extreme, the signal is long gone before enough averages can be taken.

Thermal noise depends only on resistance, temperature, and bandwidth. There are no other variables. Capacitors and inductors only play a role by changing the bandwidth of the circuit. See the short, interesting paper by Nyquist (1928), “Thermal Agitation of Electric Charge in Conductors” for the details.

The example shows how the formula for noise voltage in a resistor works out for the front-end of my oscilloscope. With nothing attached to the scope, it reads the noise on its 100KΩ input. My 100MHz analog scope has a noise bandwidth of about 150MHz, which results in a measured noise of about 0.5mV RMS. My scope shows this as a glowing stripe that is just over 1.5mV wide.

### Calculating Noise Bandwidth

Noise bandwidth is different from the 3dB bandwidth.. This graph shows a one-pole RC low-pass filter. The brown trace is the normal voltage output, with the 3dB frequency normalized to 1.0. The noise bandwidth corresponds to the bandwidth of an ideal brick-wall low-pass filter that results in the same RMS output noise level. This is found by integrating the area of the square of the filter’s voltage response. In the case of a single-pole RC filter, the math works out so that the ratio of the noise bandwidth to the 3dB bandwidth is π / 2. One way to visualize the noise bandwidth is to imagine the brick wall filter dividing the square of the gain curve into two equal areas. This requires plotting the gain squared on a linear scale. In the example graph where dB are used for the vertical scale, the areas do not match.

The noise bandwidth of a filter is always greater than the 3dB bandwidth, and is almost always less than the noise bandwidth of an ideal 1-pole RC filter. This means that noise bandwidth is typically somewhere between 1.05 and 1.57 times the 3dB bandwidth.

### Analyzing Circuits with Noise Sources

Noise sources add in series in a different way. Instead of adding voltages as a simple sum, they add as the square root of the sum of the squares. It would take 64 1.5V noise sources in series to add up to 12V of noise. It is a good thing that batteries don’t rely on random noise!

Simulators can analyze noise voltage. To understand the simulator output, first analyze this simple voltage divider circuit. Then use the circuit simulator to make sure that it gives the expected answer.

This circuit can be solved by superposition. That is, for each voltage source, short-circuit all the other voltage sources, and then add the resulting voltages. Here are three circuits, each with one voltage source:

After doing the math, and combining the noise voltages, the solution should equal the expected result, which is the Thévenin equivalent circuit:

### Using LTspice to Simulate Noise

When making noise calculations, there is almost always a signal to compare with the noise. The LTspice circuit simulator insists on having an AC voltage source, and a completed AC simulation, before it will calculate noise voltage. LTspice is restricted to linear simulation of noise. Nonlinear noise simulation is sometimes important, and requires a different simulator.

This inverting amplifier uses an operational amplifier (opamp) to provide a gain of -10 for the input signal VIN. Use the simulator to find the effects of choosing different opamps and resistor values. The different curves show the output voltage noise and also the contribution of each resistor to the output noise.

LTspice can integrate the noise at the output V(VOUT).

Display the RMS value of the noise by control-clicking the names of the outputs on the plots. While 85μV seems like a small voltage, the peak-to-peak voltage will be about 6 times this amount, or about 0.5mV.

### Conclusion

While there are many aspects to low-noise design such as using good power planes, having clean power supplies, and using good shielding techniques, sometimes the limits to circuit performance are inherent in the physics of the devices. It’s always best to take extra care when choosing low-noise components, and use simulation and measurements to verify the design.