KCL, Reactance and Common-Mode Current
KCL, Reactance and Common-Mode Current: What the Coax Is Actually Doing
Why “ELI the ICE man” does not let the outgoing and return currents drift apart—and why a non-zero current sum points to a real third path, not to reactance alone.
A familiar explanation says that a reactive antenna shifts the phase of the two coax currents, prevents them from cancelling and thereby creates common-mode current. It sounds plausible because every ham learns that inductors and capacitors shift voltage and current. But it joins two different phase relationships as though they were the same one.
Reactance changes the phase of current relative to voltage; it does not, by itself, change the equal-and-opposite relationship between the outgoing and return currents of a pure differential mode.
Three Correct Ideas—and One Incorrect Leap
Current cannot simply disappear. Every current path must close through conduction, displacement current or the surrounding electromagnetic structure.
Capacitance and inductance change the phase between terminal voltage and terminal current. They also store and return energy.
If two same-reference conductor currents do not cancel, their net component is physically meaningful and can excite an external field.
The incorrect leap is to conclude that Idea 02 automatically causes Idea 03. To make the conductor currents unequal, the real installation must provide another current path or a mechanism that couples energy into another mode.
What KCL Actually Says
Kirchhoff’s Current Law says that the signed sum of currents at a node is zero. It does not say that every current at the node is identical. A junction may have several currents entering and several leaving; KCL requires the accounting to balance.
Σ ik(t) = 0
For an ideal isolated two-terminal component, with both current arrows referenced into the component:
i1(t) + i2(t) = 0
That second result follows because there are only two external terminals. If current enters through one and does not leave through the other, net charge must accumulate or current must be using an unmodelled third path. In an RF antenna installation that third path may be stray capacitance to earth, the outside of the coax, a mast, station wiring, nearby metalwork or radiation into the surrounding field.
A coax run is not one lumped node. At RF, voltage and current vary with position and time. Currents measured at two different positions can have different phases because of propagation. The comparison relevant to modal cancellation is between the conductor currents at the same transverse plane through the line.
The exact local statement of charge conservation is:
∇·J = −∂ρ/∂t
Using Gauss’s law, Maxwell’s total current density gives:
∇·(J + ∂D/∂t) = 0
The term ∂D/∂t is displacement-current density. In a capacitor, conduction current reaches each plate and displacement current continues through the dielectric. Ordinary lumped KCL is the electrically-small circuit form of this more general conservation law. If a physical component couples capacitively to its enclosure or surroundings, those parasitics are additional RF terminals even when they are absent from the schematic.
ELI the ICE Man Is About Voltage Versus Current
For an ideal inductor and capacitor:
vL(t) = L × di(t)/dt
iC(t) = C × dv(t)/dt
These relations produce the familiar 90° phase relationships in sinusoidal steady state. They do not give one terminal current permission to arrive later than the other terminal current. The instantaneous branch current still enters one terminal and leaves the other.
An inductor stores magnetic energy; a capacitor stores electric energy. Both can alter current magnitude, terminal voltage, phase, resonance and reflection. None of that violates current continuity.
The Spreadsheet Trap: Inserting the Result as an Assumption
Suppose a model defines the two longitudinal conductor currents, both referenced in the same physical direction, as:
I1 = I0∠0°
I2 = −I0∠δ
IΣ = I1 + I2
When δ is not zero, the sum is non-zero. The arithmetic is correct. The problem is physical interpretation: the relative phase offset δ was assigned before the sum was calculated.
A non-zero sum says that the chosen two-conductor subsystem is exchanging current with something else. The model must identify that something else and its impedance. Without it, the spreadsheet has not derived common mode from reactance; it has encoded a missing external path in the assumed phase difference.
If 1 A goes up the centre conductor and only 0.8 A comes back on the intended shield surface at the same instant and position, the remaining 0.2 A must be somewhere. The useful question is not what the subtraction gives. The useful question is: where is the other path?
A Reactive Reflection Is Still Differential Mode
A reactive or mismatched termination produces a reflected wave. Voltage and current then form standing-wave patterns along the feedline. That changes the impedance seen at different measurement planes, but reflection is not automatically mode conversion.
For a uniform line:
V(z) = V+e−γz + V−eγz
IDM(z) = (V+/Z0)e−γz − (V−/Z0)eγz
The minus sign on the reflected current wave is required by its direction of travel. At every transverse plane, the centre conductor carries +IDM(z) and the inside shield surface carries −IDM(z). The standing wave changes both together; it does not require an external mode.
| Phenomenon | What changes | What it does not prove |
|---|---|---|
| Load reactance | The phase and ratio of differential terminal voltage and current. | That the outgoing and intended return currents have become unequal. |
| Impedance mismatch | Reflection coefficient and differential standing-wave pattern. | That current exists on the outside of the coax. |
| Mode conversion | Energy coupled from the intended mode into an external mode. | That reactance alone caused the coupling. |
Coax Has Three Relevant RF Surfaces
Calling coax a “two-conductor line” is correct for its intended TEM mode, but it can hide an important practical detail. When the shield is thick compared with skin depth, its inside and outside surfaces support substantially independent current distributions.
Carries one side of the intended coaxial differential mode.
Carries the equal-and-opposite differential return current.
Can carry an external current whose return path involves the wider station and environment.
Icentre = IDM
Ishield,inside = −IDM
Ishield,total = −IDM + Ioutside
For a probe around the complete coax:
Iprobe = Icentre + Ishield,total = Ioutside
The probe is therefore not inventing a current by arithmetic. It responds to the magnetic field produced by real net longitudinal current through its aperture. The differential currents cancel in the probe; the outside-surface current does not.
For two same-direction conductor references, modal currents are often defined as:
IDM = (I1 − I2)/2
ICM = (I1 + I2)/2
With that definition, a whole-cable probe reads the net current I1 + I2 = 2ICM. In practical amateur work, that probe reading is itself often called “the common-mode current.” Both usages exist. Declaring the convention prevents an avoidable factor-of-two argument.
Common mode is indeed a mathematical decomposition rather than a new species of electron. Differential mode is also a decomposition. Both are useful because they correspond to different field patterns, impedances, return paths and radiation behaviour.
Where the External Current Really Comes From
A two-terminal impedance value such as 50 + j0 Ω or 35 − j120 Ω is a port description. It does not describe every coupling path in the installed antenna system.
Mode conversion can be produced by:
- unequal antenna-arm impedances or unequal capacitance to the surroundings;
- feeding a balanced radiator from an unbalanced line without sufficient isolation;
- the coax leaving the feedpoint asymmetrically or running close to one element;
- coupling to a mast, tower, roof, gutter, building wiring, ground or operator;
- a matching network, connector or enclosure that does not preserve the intended symmetry;
- an end-fed or off-centre-fed system whose external return path is not deliberately controlled;
- finite shield transfer impedance, slots or imperfect connections at sufficiently high frequency.
Tuning can change any of the voltages and current distributions that drive these couplings. Common-mode current may therefore rise or fall after tuning. That is a real effect—but it shows that tuning changed the complete electromagnetic system, not that reactance is itself the missing return path.
Match and Balance Are Separate Axes
SWR answers a differential-mode impedance question at a stated reference plane. A whole-cable current measurement answers an external-current question at a stated position. Neither measurement can replace the other.
| Low common-mode current | High common-mode current | |
|---|---|---|
| Low SWR |
Matched and well controlled. The line sees a good differential match and the feedline is not strongly participating. |
Matched but unbalanced. The reassuring SWR can hide substantial outside-shield current. |
| High SWR |
Mismatched but balanced. There are differential reflections, but little evidence of external feedline current. |
Two separate problems. Mismatch and external-mode excitation both require investigation. |
Why a Choke Does Not “Burn the Reflected Power”
A common-mode choke presents impedance to net current through its aperture. Equal-and-opposite differential currents ideally produce cancelling ampere-turns in the core, so the wanted coaxial mode sees little added impedance.
ZCM = RCM + jXCM
Ploss ≈ ICM,rms2 × RCM
The resistive part dissipates real power as heat. The reactive part stores and returns energy during each RF cycle. Real ferrites have both components, and some choke designs intentionally use a substantial resistive part to damp the external-mode current over a useful bandwidth.
Current itself is not “burned.” Energy is dissipated in the resistive part of the choke, conductors and dielectric. The relevant current in the loss equation is the current that remains after the choke is installed, because the choke impedance changes that current.
Dimensional analysis is a useful guardrail. Reactance is measured in ohms, current in amperes and insertion loss as a power ratio or in decibels. An expression that subtracts current from reactance cannot define differential insertion loss.
Differential insertion loss is measured under differential excitation—normally as a terminated two-port power ratio or calibrated S21. Common-mode impedance is measured under common-mode excitation. They are different tests.
If the shield is floated while only the centre conductor is excited, the return current must travel through the fixture, instrument case, earth or surrounding capacitance. That arrangement creates net enclosed current and is therefore not a clean differential-mode test. The test jig has become part of the common-mode circuit.
Can a Choke Change the Measured Antenna Impedance?
Yes—if the outside of the feedline was participating in the antenna. In that case, the coax exterior is an unintended additional radiator or return conductor. Adding a choke changes its boundary condition, changes the current distribution of the complete structure and can change the driving-point impedance measured by an analyser.
If the installation had negligible common-mode current to begin with, an ideal choke should produce little change in the antenna impedance. This gives us a useful interpretation:
When a choke changes the SWR, it does not automatically mean the choke has introduced differential loss. It may mean the original feedline was part of the antenna and has now been partly removed from that role.
A tuner at the transmitter mostly transforms the impedance presented to the transmitter. A feedpoint choke can alter the physical antenna system by suppressing an unintended external conductor. These are not equivalent operations.
A Reproducible Experiment for a Radio Club
The most convincing test changes one physical mechanism at a time. It should also make the measurement setup part of the declared circuit rather than pretending the analyser, radio case and connecting leads do not exist.
Use a well-shielded coaxial fixture and a resistive termination. Map whole-cable current at several feedline positions and record the system noise floor.
Insert shielded two-terminal L or C networks inside the fixture. Confirm that impedance and SWR change while the external geometry remains fixed.
Introduce a known asymmetric external capacitance or conductor, then repeat the current map. This supplies the missing mode-conversion mechanism deliberately.
Record its complex common-mode impedance, differential insertion loss and temperature rise at a stated power and duty cycle.
Keep the feedline route and surroundings fixed. Repeat with and without tuning and choking, one change at a time.
Compare field strength or pattern using the same antenna geometry and the same accepted power—not merely the same transmitter setting.
| Measurement | What it answers | Important limitation |
|---|---|---|
| Complex impedance and SWR | What differential load appears at the calibration plane? | Does not measure common-mode current, pattern or efficiency. |
| Whole-cable current at several positions | Where is net external current present, and how does it change? | One position may coincide with a common-mode standing-wave minimum. |
| Complex choke impedance | How much resistive and reactive impedance opposes the external mode? | The jig, calibration and stray capacitance can dominate the result. |
Differential S21
|
How much wanted-mode power is lost through the choke assembly? | Does not by itself state common-mode suppression in an installation. |
| Choke temperature rise | Is real power being dissipated safely at the stated duty cycle? | Requires thermal equilibrium, known ambient conditions and realistic power. |
| Field strength or pattern | Did the complete radiating system change? | Geometry and accepted power must be controlled. |
Do not let the test jig become the hidden third conductor. Instrument enclosures, USB cables, mains protective earth, a laptop charger and even the operator can complete the external RF path. Isolate or characterise them, keep the geometry fixed and repeat measurements before drawing a causal conclusion.
What the Physics Lets Us Conclude
- KCL does not say that every current at a node is the same; it says their signed sum balances.
- At RF, the continuity equation—including displacement current—is the more general statement.
- Reactance changes current relative to voltage; it does not independently phase-shift the two terminal currents of an ideal isolated two-terminal load.
- A reactive mismatch can produce a differential standing wave with no common-mode current.
- A non-zero whole-cable current requires a real external path and a mechanism that excites it.
- The outside-shield current is physically measurable even though “common mode” is a modal decomposition.
- Matching and balance are separate properties.
50 + j0 Ωdoes not certify a quiet feedline. - A choke adds complex impedance to the external mode. Only its resistive component dissipates average common-mode power.
- If a choke changes the measured impedance, the feedline may have been participating in the antenna.
Mini-FAQ
- Does KCL require all coax currents to be identical? No. It requires signed current continuity at a defined node or boundary. The intended differential currents are equal and opposite because of the mode and two-terminal topology, not because KCL says every branch current is equal.
- Can a purely reactive shielded load create common mode by itself? No. It changes the differential voltage-current phase and reflection coefficient. A separate external path and coupling mechanism are required.
- Can a matched antenna still have common-mode current? Yes. SWR describes differential matching at a stated plane, not balance or outside-shield current.
- Is common-mode current “only a calculated sum”? The mode is a decomposition, but the corresponding net surface current and its electromagnetic field are measurable. A whole-cable probe responds to that net current.
-
Does a choke consume reflected power? Not as a general rule. Differential reflected power remains in the differential mode unless mode conversion occurs. A choke dissipates
I2Rfrom the common-mode current flowing through its resistive impedance. - Why measure at several coax positions? The external mode can form its own standing wave. A single low reading may be a current minimum rather than proof of a quiet feedline.
Technical Foundations
- S. J. Orfanidis, Electromagnetic Waves and Antennas, Chapter 11: Transmission Lines — TEM fields, line voltage and equal-and-opposite conductor currents.
- G. D. Sower, UNM Measurement Note 44: Quad Coaxial Balun — the three-surface coax current model and the independent outside-shield current.
- University of Texas notes on Maxwell’s equations — continuity, displacement current and the capacitor example.
- MIT OpenCourseWare, Electromagnetics and Applications — transmission-line waves, reflections and field-based circuit models.
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