Motivations
"Common-mode noise" is a fundamental concept in electronics. Yet, because it appears in a lot of related but different contexts - amplifiers and signaling, 120 V or 230 V power supply, or EMI/EMC - just to name a few, it has many seemingly contradictory definitions. This situation creates so much confusion among beginners.
For example, EMC engineers may tell you: "Your system's radiation is coming from the 5-volt DC cable due to common-mode noise." And you don't understand what they're talking about.
So you look it up and found the definition: "A differential-mode voltage exists between the inverting and non-inverting input of a differential amplifier, a common-mode voltage exists between the inputs and ground." You are confused. Amplifier? What amplifier? So you keep looking.
Then you stamped upon another definition: The noise between Live and Neutral conductor in the AC mains is differential-mode noise, and it can be suppressed by a Class-X capacitor, the noise between Live/Neural and Earth conductor is common-mode noise, and it's suppressed by a Class-Y capacitor or a common-mode choke. You are even more confused. AC mains? Neural? But it's a 5-volt power supply. Earth? But it's battery powered.
Finally, you found yet another definition before giving up: "If we have two conductors, the current that flows on both conductors in the same direction is the common-mode current. The current that flows on both conductors in the opposite direction is the differential-mode current." At this point, you finally gave up. I have +5 VDC and GND here. How can a current flows in the same direction? Does it even make sense?
In my opinion, none of the existing answer is satisfactory because they don't explain this problem. My answer is an attempt to clarify this confusion in the context of EMI/EMC.
A Current Definition of Common-Mode Noise
When we are speaking of analog or digital signaling with a differential amplifier, the voltage definition is often used - it's the noise across a signal and a common ground reference. However, when we are talking about it in a more general sense, it's often unclear where the "common ground" is. Thus, it's more useful to use a current definition of common-mode noise.
To put it in plain but sloppy language, in many practical applications, a common-mode current is often the unwanted current that interferes with the circuitry, while a differential-mode current is often the useful current necessary for the operation of the circuitry (but there can be differential-mode noise too). Ideally, the common-mode current in cables should be zero, in other words, it should not even exist.
To understand what I meant, consider the previous 5-volt DC power supply example. We have an electronic device powered by a 5-volt DC supply. The power is delivered by a DC cable with two wires: + 5 VDC and GND. We can represent this system in a simplified schematic.
Kirchhoff's Current Law tells us, \$ \sum_{k=1}^n {I}_k = 0 \$. In other words, there should be an equal and opposite current on +VDC and GND. The current flows into the load must equal to the current flows out of the load. For example, if Ivdc = 500 mA, Ignd = -500 mA. The net current in the cable as a whole should be zero.
However, if we perform this experiment with a real switched-mode power supply, and a real circuit board with several digital logic chips on it, you'll see something different: Ivdc may be 500.0025 mA, while Ignd may be -499.9975 mA. To our surprise, at the ground wire, the 5 μA of current is missing, and there's a net current of 0.005 mA (5 μA) flowing into the load. This 5 μA current, is the common-mode current.
Mathematically, we can say that the current in the cables is a superposition of two different currents. There's a differential-mode current that flows toward the opposite directions, and a 2.5 μA current on both VDC and GND wires that flows toward the same direction to the load.
Thus, we have our first definition of current-mode current: If we have two conductors, the current that flows on both conductors in the same direction is the common-mode current.
But KCL tells us that "what goes in, must come out", and the current must return to the source. How can the current travels on both wires on the same direction? What's going on? It's easy to see the answer once we realize that (1) we have AC current, not DC current. Because both the switched-mode power supply and the digital chips on the circuit board work by switching rapidly, the frequency components of the current can be as high as 100 MHz. (2) Parasitic capacitance is everywhere in a real circuit, and at 100 MHz, the impedance of a 5 pF capacitor is just 318j Ω.
Thus, the 5 μA common-mode current simply jumps to a nearby conductor via the parasitic capacitance and returns to the source via an alternative path. The exact path taken by the current is often ill-defined and somewhat unpredictable. Often it's the metal enclosure. For systems without a metal enclosure, It can be whatever conductive object that happens to be here nearby: a metal table, a file cabinet, or literally the ground.
Source: Electromagnetic Compatibility Engineering by Henry Ott, fair use.
Due to the uncontrolled nature of the common-mode current, it usually travels in a large loop, and this loop radiates electromagnetic interference. And actually, the common-mode current is not merely flowing in a conductor according to basic circuit theory, it's the result of electromagnetic field. A wire on over a plane creates a monopole antenna that produces radiated electromagnetic waves.
A special scenario is when the system is a 120 V or 230 V mains power supply. In this case, we indeed have a well-defined reference. The AC mains have three conductors: Live, Neutral, Earth, and the enclosure is grounded to Earth, a current that otherwise should return via Neutral often returns via Earth instead. Thus, this explains another definition of common-mode noise: The noise between Live and Neutral conductor in the AC mains is differential-mode noise, the noise between Live/Neural and Earth conductor is common-mode noise.
Measuring Common-Mode Noise
Common-mode noise current cannot be measured by a multimeter, because they are created by RF current at hundreds of megahertz, with a magnitude of a few microamps. Instead, in EMI/EMC pre-compliance testing, it's measured by a spectrum analyzer with an RF current probe. And the result is a noise voltage, not a current. When the probe is characterized with a transfer impedance curve, the voltage can be converted to current.
The RF current probe senses the current via the magnetic field. If we put the current probe across an individual wire, the current on that wire is measured. If we put the current probe across the entire cable, e.g. including both power and ground, the net magnetic flux is zero inside the probe, and we can measure the leakage, or common-mode noise current.
Here's an example of a commercial RF current probe.
Source: Review: Tekbox TBCP1 RF current probe, EDN, fair use.
If one doesn't have a spectrum analyzer, a low-cost solution is using a SDR dongle. A $10 RTL-SDR is enough to show the frequency peaks of these EMI sources. Another option is amplifying the current probe with an RF Low-Noise Amplifier (LNA), then analyzing the result on a digital oscilloscope via FFT is also an option. Both methods are unreliable and unsuitable for quantitative measurements, but good enough to be an educational demonstration.
The path taken by the common-mode current is often ill-defined, and can be whatever conductive object that happens to be here nearby: a metal table, a file cabinet, or literally the ground. Thus, to make a reliable and repeatable measurement in conductive EMI compliance testing, solid metal plates are grounded to Earth and placed on the horizontal and vertical directions near the device-under-test, so that it creates a predictable path for common-mode current.
Source: Electromagnetic Compatibility Engineering by Henry Ott, fair use.
For a simple pre-compliance testing at the workbench, Ott suggested that one can use a DIY lab cart in the following arrangement to approximate the test environment.
Source: Electromagnetic Compatibility Engineering by Henry Ott, fair use.