By mains hum we mean AC power hum, which is 50 Hz in Europe, and 60 Hz in the US. Mains hum is a constant problem in neuroscience. Praveen Taneja of Brandeis University claimed that mains hum was caused primarily by capacitive coupling into our analog inputs. We disagreed, saying it was primarily inductive. In order to understand better how to avoid mains hum, we had best understand how it arises, so we performed a few simple experiments to determine its origins.
We start by assuming that the source of mains hum in any circuit can be modeled by a mains-hum voltage source, V, in series with a source impedance, Z. We are going to measure mains hum with an oscilloscope. We define our ground potential as the ground terminal of the oscilloscope probe. The voltage V is with respect to ground. We place the tip of our probe upon the metal frame beneath a wood-topped table. We do not connect the probe ground clip to the frame. We see mains hum of about 1 V. Now we touch the ground tip to the frame. The mains hum disappears completely.
When we touch the ground clip to the table, we are connecting Z directly to ground with a 0-Ω resistor, and all of V is dropped across Z. None is left for our probe to measure. When we do not touch the ground clip to the table, V is connected to Z in series with the impedance of our probe. It is a 10-MΩ probe. With this 10 MΩ loading our mains hum, we are still left with around 1 V.
These two measurment alone do not allow us to estimate the amplitude of V. But if we connect the ground clip through a variety of resistors, we in each case load V with a new resistance, and we get to see how Z shares V with each resistance. In the following table we show how the probe voltage, Vp, varies with with the load resistance, R. Note that R is the 10 MΩ probe impedance in parallel with the resistance we place deliberately in series with the ground clip.
| R (MΩ) | Vp (mV AC) |
|---|---|
| 10 | 1100 |
| 5 | 900 |
| 2 | 400 |
| 0.1 | 20 |
| 0.01 | <10 |
From these measurements, we conclude that Z is of order 10 MΩ. It may be capacitive or inductive, but we doubt it is purely resistive.
We now performed the same experiment on my tongue. In this case, we observe a slight DC voltage as well, which we include in the table.
| R (MΩ) | Vp (mV AC) | Vp (mV DC) |
|---|---|---|
| 10 | 50 | -500 |
| 5 | 50 | -500 |
| 2 | 40 | -250 |
| 0.1 | 10 | -100 |
| 0.01 | <10 | <10 |
From these measurements, we conclude that Z for mains hum on my tongue is of order 1 MΩ.
We see that objects have mains hum. There are two ways they can pick it up. Either they act as transformers in conjuction with mains-carrying power cables, or they act as capacitor plates with mains-carrying power cables. If it's the former, then a loop of wire, which has very little capacitance, will pick up mains hum. So we made a loop of wire 15 cm in diameter. When loaded with our oscilloscope probe, the 15-cm looop picked up 4 mV of mains hum right next to a power supply, but ≶1 mV elsewhere. We made a 3-m diameter loop of wire. It picked up no more than the 15-cm loop.
We conclude that the mains hum on bodies in our lab is due entirely to these bodies acting as one plate of a capacitor, with the mains power lines as the other plate. One way to estimate the capacitive connection between an object and the power lines is to assume that the the body is in free space, surrounded by a large sphere at voltage V. In that case, the connection between the body and V is the free-space capacitance of the body. The free-space capacitance of a spherical conducting body is 4πεrA, where ε is the capacitance of free space and rA is the body's radius.
If we model my body as a sphere of radius 0.5 m, we arrive at a capacitance of 60 pF. The impedance of 60 pF at 60 Hz is 50 MΩ. We observe Z to be only 1 MΩ. On the other hand, the mains amplitude is 160 V (110 V rms), but we observe V to be only 50 mV in my body.
Our spherical model is inaccurate in obvious ways. First of all, the power cables contain various phases of mains voltage. Second, there are some cables far closer to my body than others. Third, there are many other bodies in the lab, each of which picks up mains hum by the same capacitive coupling, and transmits it to my body. By Thevenin's Equivalence Theorum, all of these can be reduce to a single voltage source in series with a single capacitor. The voltage will be much smaller than 110 V, but the capacitance will be much larger than the solitary free-space capacitance of the object.
So this is our understanding of mains hum: it originates in power cables and propagates capacitively (which is to say: by electrostatic action) through the air to all objects in the room, and these objects in turn propagate the mains hum among one another. We expect to observe mains hum source impedances anywhere up to 10 MΩ, and voltages up to 1 V.
Suppose we want to measure the voltage between two points in an animal, and we want to do so without interference from mains hum. If possible, we must connect one of these points through a low impedance to ground. This eliminates the mains hum at that point, and greatly reduces it at points nearby. The other point we can connect to a high-impedance input. We now measure the potential generated by local voltage sources between these two points, and we load these local voltage sources only with our high impedance.
If, on the other hand, we connect both points to the high-impedance inputs of a differential amplifier, we will have up to 1 V of mains hum on both lines. Even with the very best of twisted and shielded cables, we will still end up with 1% of that 1V, which is 10 mV as a differential signal on our input.
In short: there is a time and a place for high-impedance, differential amplifiers, known as instrumentation amplifiers. Whenever you use an instrumentation amplifier, you must to add a third low-impedance connection from the body of your signal source to the ground potential of the amplifier, or else all benefit of your differential inputs will be lost.