Railway Track Circuit (3D)

The rails are a leaky transmission line

In a DC track circuit the two running rails carry a small current from a feed (battery, series resistor Rs) at one end to a relay coil Rr at the other. Per kilometre the two rails have a loop resistance r (ohm/km) and the ballast leaks current from one rail to the other with conductance g = 1/Rb (siemens/km, where Rb is the ballast resistance in ohm km: dry 20 or more, soaking wet below 1). This is exactly a transmission line with

gamma = sqrt(r g),   Z0 = sqrt(r / g),   V(x) = V2 cosh(gamma s) + Z0 I2 sinh(gamma s)

where s is the distance to the relay. Starting at the relay with V2 = Rr I2 and propagating back through the line sections gives the voltage and current at every point; the feed condition E = V1 + Rs I1 fixes the scale. Each wheelset that is on the block adds a shunt conductance 1/Rax across the rails at its position - that is how the model treats a train (16 axles for the 4-car train).

Why it is fail-safe, and where the margins are

The relay is a neutral relay with hysteresis: it picks up at I_pu and releases at I_do = ratio x I_pu. The signal shows green only while the relay is picked up. Almost every failure (broken rail, dead battery, wet ballast, a train) lowers the relay current, so the relay drops and the signal goes red: a safe failure. Wet ballast with an empty block is a false occupancy - an annoyance, never a danger. The dangerous direction is the opposite: a train that does not lower the relay current below I_do, for example on a rusty or contaminated rail where each axle only shorts the rails through several ohms. Graph 4 shows the two currents against ballast resistance: the useful window is between the clear current (green) and the best shunted current (red); the thresholds must sit inside that window for every ballast condition.

Numerical checks on this page

(1) The hyperbolic solution is compared with an independent 4000-section ladder network solved by recursion. (2) Power balance: the power delivered by the feed equals the dissipation in the series resistor, the rails, the ballast, the axle shunts and the relay (each integrated from the voltage profile). (3) With ballast leakage switched off the relay current must equal E / (Rs + r L + Rr).

Honest simplifications

Steady-state DC; both rails lumped into one loop resistance; ballast resistance uniform along the block and constant in time except in the rain cycle; the axle contact resistance is the same for all axles; no rail-joint bonding, no traction return current, no relay timing; the signalling rules (overlaps, approach locking) are not modelled. The numeric values are plausible teaching values, not those of a particular railway.

What to try

• Press Rain cycle or load Heavy rain: as the ballast gets wetter the relay current falls below the drop-away current with an empty block - a false occupancy that stops trains safely.
• Load Contaminated rail: 100 ohm per axle, then press Hold train in block and slide the position: the relay stays picked up with a train inside - the dangerous case the design must exclude.
• Raise the feed voltage: more margin in wet weather, but also a higher current when shunted.
• Make the block 3 km long: the voltage at the relay end falls and the margin to the pick-up current shrinks.
• Set the drop-away ratio to 0.95: a tight relay drops sooner and is more sensitive to a poor shunt.
• In graph 3 compare dry (blue) and wet (orange) ballast for the same train position.