Ballast Resistance Measurement Theory and Practice

Ballast Resistance Measurement Theory and Practice Stuart Landau, PE, MIRSE Signal and Train Control Engineer CH2M 119 Cherry Hill Road, Suite 300 Parsippany, NJ 07054 Stuart.Landau@ch2m.com 4,120 words ABSTRACT Ballast and ties provide a leakage path between the running rails for track circuit current. This leakage affects the adjustment and operation of the track circuit. Excessive leakage may also affect the choice of track circuit equipment or require ballast remediation. Measuring the rail-to-rail, or ballast, resistance is important for proper setup and operation of track circuits and predicting their performance. Various methods of measuring ballast resistance are available, ranging from quick and easy with simple calculations to more involved measurements with more mathematics. Open-circuit measurements taken at the feed end of the track circuit with all other rail connections removed are simple to perform but also include the effects of series rail impedance. To compensate for rail impedance, open-circuit measurements may need to be performed at other locations along the track circuit; or both open- and short-circuit measurements can be taken at the feed end and one set of measurements compensated for the other with additional mathematics. Measurement methods are derived from first principles, including transmission-line modelling and the laws of Kirchhoff and Ohm. A track circuit simulation tool compares results from the different measurement techniques at various ballast conditions and highlights the tradeoffs between accuracy and practicality. Actual measurements are presented, including multiple measurements of the same track circuits over a period of time, and measurements showing the effects of insulated tie plates as one way to remediate poor ballast conditions. INTRODUCTION Track circuits have been in use for well over a century. By definition, track circuits use the running rails as part of the electrical path. A voltage applied across the rails at one end of an unoccupied track section will energize a track relay (or other device) at the opposite end of the track section. But the rails are not perfect conductors and the structure between them is not a perfect insulator. Each rail has impedance and the rails have a leakage path between them through the tie plates, ties, and ballast. The leakage between the rails, although flowing through many different components of the track structure, is usually considered as flowing through the ballast. The impedance of the ballast at frequencies where capacitance is negligible is usually referred to as ballast resistance. This leakage, which is distributed throughout the entire length of the track circuit, affects the amount of current available at the relay end to energize the relay and so affects the adjustment of the source voltage and other components of the track circuit. It is also useful to know ballast resistance to predict shunting sensitivity; and track circuit equipment specifications typically state a minimum ballast resistance, possibly at a maximum length, for proper operation. NOMENCLATURE Hat ( ) denotes an estimated value; prime ( ) denotes with respect to distance. Rail impedance values are for the track (both rails). estimated ballast resistance (Ω); estimated ballast resistance normalized for 1,000 ft (Ω kft); not compensated for rail impedance true ballast resistance (Ω); true ballast resistance normalized for 1,000 ft (Ω kft) estimated rail impedance (Ω); estimated rail impedance normalized for 1,000 ft (Ω/kft); not compensated for ballast resistance AREMA 2015 1

true rail impedance (Ω); true rail impedance normalized for 1,000 ft (Ω/kft) rail-to-rail voltage at distance x from relay end under operating conditions (V) rail current at distance x from relay end under operating conditions (A) rail-to-rail voltage at feed end with relay end open (V) rail-to-rail voltage at distance from relay end with relay end open (V) rail-to-rail voltage at the given fraction of total length from feed end with relay end open (V) rail current at feed end with relay end open (A) open-circuit impedance measured at feed end, (Ω) rail-to-rail voltage at feed end with relay end shorted (V) rail current at feed end with relay end shorted (A) rail current at distance from relay end with relay end shorted (A) short-circuit impedance measured at feed end, (Ω) length of track section (ft) distance from relay end (ft) fraction of length from feed end to relay end; 1 ; 0 at feed end, 1 at relay end length of transmission-line (track) element (ft), elemental rail impedance (Ω/ft), total rail impedance (Ω), elemental ballast admittance (S/ft), total ballast admittance (S) ballast conductance (S) propagation constant (ft -1 ) MEASUREMENT METHODS Only rail-to-rail leakage through the ballast is of interest, so the effects of any other leakage path must be excluded. In all methods, track relay coils, impedance bonds, negative returns, and any other connections must be either removed from the rails or otherwise isolated (such as pulling track circuit fuses). The only remaining connection to the rails is a source of energy with which to make voltage and current measurements. This source of energy can be the one already in place to feed the track circuit. However, where ballast resistance is high, a low feed voltage may cause the current leaking between the rails to be too low to measure reliably. In this case, feed transformer taps may need to be adjusted before taking measurements or a separate source of higher voltage may need to be applied. Figures 1 and 2 show the typical track circuit under test and the same track circuit with isolated rails, respectively. AREMA 2015 2

impedance bond resistor/ reactor Figure 1 Double-rail track circuit with impedance bonds. Figure 2 Feeding isolated rails with bonds and relay disconnected. I. Open-Circuit Measurements at Feed End Once the rails are isolated with only a voltage source connected, an obvious way to proceed is to measure the voltage applied and the resulting current flowing into a rail at the feed end as shown in Figure 3. Since both rails are isolated, any current drawn from the source must be leaking through the ballast. The estimate of the total ballast resistance for the section is then Ω (1) and the ballast resistance normalized for 1,000 feet is 1,000 Ω kft (2) Although voltage and current can be complex for alternating current, only magnitudes need to be measured if only magnitude is desired as a result. The complex calculation involves only division, so the magnitude and phase are calculated separately and the phase can be disregarded. (If and are complex, then ; phase angles are subtracted separately.) Thus, only a common multi-meter is required. AREMA 2015 3

V A Figure 3 Feeding open rails. Measured impedance is typically dominated by ballast. A problem with this simple method is that the calculated resistance will also include the effects of series rail impedance. In track circuits with good (high) ballast resistance, the rail impedance will be very small compared to ballast resistance and the result will be close to actual ballast resistance. But it is often the case that measurements are being made because of a track circuit problem, and this problem may be due to low ballast resistance. In these cases, rail impedance becomes more significant and the calculated value for ballast resistance will be too high due to the added series impedance being measured [1]. II. Open- and Short-Circuit Measurements at Feed End This method takes measurements with the relay end of the track both open and shorted. Before demonstrating how to use both sets of measurements to find true ballast resistance, a digression is made to transmission-line theory. Railway track as described in the introduction, with distributed series (rail) impedance and shunt (ballast) admittance, behaves as a transmission line. In the early twentieth century, transmission-line theory was applied to track circuit analysis [2]. Those results were used to allow both ballast and rail resistances to be measured using direct current while compensating one for the other [1]. This is extended to alternating current for an exact method to measure ballast resistance and rail impedance. Figure 4 Track (transmission-line) element showing elemental rail impedance (resistance and inductance) and elemental ballast admittance (conductance and capacitance). Figure 4 shows one typical element of a transmission line with elemental series (rail) impedance consisting of resistance and inductance and elemental shunt (ballast) admittance consisting of conductance and capacitance. These elements are uniformly distributed throughout the entire track circuit, which is shown in Figure 5 with one AREMA 2015 4

such element. Apply Kirchhoff s voltage law around this element and Kirchhoff s current law at the element s upper-right node: (3) (4) 0 0 0 Figure 5 One transmission-line element of length Δx within a track circuit of length, fed from the left. Divide each side by Δ and take the limit as Δ approaches 0. (Physically, this converts the lumped components of the elements to a perfect distribution of infinite elements, each with zero length.) The left side becomes the definition of derivative of the voltage and current: (5) Differentiate once more with respect to : (6) (8) Substitute the expressions for the first derivative from (5) and (6): (9) (10) Let. Then (11) (12) These are the Telegraph Equations describing voltage and current along a transmission line with propagation constant. They are second-order linear differential equations with constant coefficients of the form 0 (13) where 1, 0, and. Its characteristic equation, 0, has real roots, making the general solution of (7) AREMA 2015 5

(14) where C 1 and C 2 are constants to be determined. Using the identities cosh sinh (15) coshsinh (16) equation (14) can be rewritten cosh sinh cosh sinh cosh sinh (17) The first derivative of this solution is equal to equation (5): sinh cosh (18) Divide by Zʹ to find the solution for current: sinh cosh (19) To solve for the constants, consider voltage and current at the relay end. Here, 0, so cosh cosh 0 1, and sinhsinh00. Substitute these into equations (17) and (19): 0 (20) 0 0 0 0 Substitute (20) and (22) back into (17) and (19) and expand γ: (21) (22) 0 cosh 0 sinh (23) 0 sinh 0 cosh (24) In track circuit analysis it is often more useful to know the voltage and current at the boundaries (feed and relay ends) rather than at arbitrary locations. Evaluate these equations at the feed end by letting. Then set the units of length of and to so that impedance and admittance per unit length become total impedance and admittance for the track circuit length. The expressions involving values with respect to length become (25) (26) At low and power frequencies ballast capacitance is usually ignored, so ballast admittance can be replaced by ballast conductance. Then (23) and (24) become 0 cosh 0 sinh (27) 0 cosh 0 sinh (28) AREMA 2015 6

These equations appear in [2], [4], and many other sources. They define feed-end voltage and current in terms of relay-end voltage and current and track characteristics and allow analysis starting with relay parameters and resulting in required feed voltage. They are the starting point in [1], which will now be discussed. As in method I, open-circuit resistance is an approximation of ballast resistance. With the rails isolated and the relay end of the track open as in Figure 3, there is no current at the relay end, so 0 0. Replacing ballast conductance G with the reciprocal of ballast resistance, 1/B, equations (27) and (28) become, for open-circuit voltage and current, 0 cosh (29) 0 sinh (30) Approximate ballast resistance is then tanh (31) This is an estimated ballast resistance, as opposed to true ballast resistance, since it includes the effects of series rail impedance. V A Figure 6 Feeding shorted rails. Measured impedance is typically dominated by rails. Now consider the short circuit condition which gives an approximation of rail impedance. With the rails isolated and the relay end of the track shorted as in Figure 6, there is no voltage at the relay end, so 0 0 and (27) and (28) become, for short-circuit voltage and current, 0 sinh (32) 0 cosh (33) Approximate rail impedance is then tanh (34) This is an estimated rail impedance, as opposed to true rail impedance, since it includes the effects of shunt ballast resistance. AREMA 2015 7

Dividing (34) by (31) and taking the square root yields equation (35). Rewriting and and combining with (31) and (34) yields (36) and (37) which relate measured and true values of ballast resistance and rail impedance: tanh (35) tanh tanh (36) tanh tanh (37) Combining equations (35) to (37) and normalizing yields tanh 1,000 Ω kft (38) tanh 1,000 Ω/kft (39) Now true ballast resistance and rail impedance can be calculated as functions of measured ballast resistance and rail impedance, with all measurements taken at the feed end. This is an accurate method, but for alternating current both magnitude and phase must be carried through all calculations. So proper application requires more sophisticated instruments to measure phase angles as opposed to common magnitude-measuring test equipment such as voltmeters and ammeters. With direct current, phase angles are zero and this method can be applied easily. As already noted, this method yields rail impedance in addition to ballast resistance. Since rail impedance can be known beforehand (its characteristics are fixed and predictable), calculating it from measurements provides a check of the test setup and technique. If the result is as expected, there is more confidence in the ballast resistance measurement for which there may be no prior expectation. Rail impedance that is higher than expected may indicate bad rail bonding or other problems. It is important to note that, unlike most applications of transmission-line theory, the wavelengths used in power- and audio-frequency track circuits are much longer than the length of the track circuit. This means that there is no concern for standing waves, reflections, etc. so track circuit adjustment does not require the use of Smith charts. III. Open-Circuit Measurements at Some Distance from Feed End To simplify the calculations and still get a true measure of ballast resistance, consider that leakage between the rails causes the rail-to-rail voltage to decrease from the feed end to the relay end. This implies that there is some distance from the feed end at which the open-circuit rail-to-rail voltage has decreased enough such that the ballast resistance calculated by equation (40), with voltage in the numerator, has decreased enough to compensate for the series rail impedance. (True ballast resistance is lower than measured since the measurement includes the series effects of rail impedance. Decreasing the voltage in the numerator decreases the calculated ballast resistance.) The general form of this approach, whose measurements are shown in Figure 7, is (40) AREMA 2015 8

1 V V A Figure 7 Feeding open rails and measuring at feed end and at some distance from feed end. To find an exact location at which to take the voltage measurement, equate the measurement in (40) with the exact ballast resistance of (36), noting that the measurements of are taken at the feed end ( ): tanh tanh (41) Removing the common factor of I o and recalling from (25) that, tanh From equation (23), under open-circuit conditions where 0, and at, Combine (43) and (45): Combine (42) and (46): (42) 0 cosh (43) 0 cosh (44) 0 cosh cosh cosh tanh sinh cosh cosh (45) (46) (47) cosh (48) 1 sinh cosh (49) The fraction of the total distance from the relay end is 1 sinh cosh (50) AREMA 2015 9

Finally, the fraction of the distance from the feed end is 1 1 1 sinh cosh 1 sinh cosh (51) The exact distance at which to take the open-circuit voltage measurement is a function of propagation constant, meaning it is a function of ballast resistance and rail impedance. So in order to find the distance which will compensate for series rail impedance when calculating ballast resistance, we must have a priori knowledge of the quantities to be measured. This difficulty is avoided here with a software tool [3] that makes it possible to take simulated measurements while having control over track characteristics (see next section). Table 1 shows the locations at which measurements should be made at various track circuit lengths and ballast resistances. (Equation 51 and calculations in the next section were evaluated with a script written in the Python programming language; Python includes support for hyperbolic trigonometric functions of complex variables.) (ft) (Ωkft) (ft) 0.2 116 0.423 200 2 116 0.423 20 116 0.423 0.2 289 0.422 500 2 289 0.423 20 289 0.423 0.2 579 0.421 1,000 2 578 0.423 20 577 0.423 Table 1 Locations for measuring to get true ballast resistance. ( = 0.1 60 Ω/kft track) An approximate distance used in established procedures is 0.4 [4]. This location, 40% of the distance from the feed end to the relay end, appears to be a reasonable approximation under a wide range of conditions and is also easy to locate in the field without a calculator. So,. Ω kft (52) 1,000 As with method I, although equation (52) involves complex division, phase need not be considered (or measured) when only the magnitude is of interest. The same methodology can be applied to find a location at which to measure short-circuit rail current in order to calculate true rail impedance. That is, there is some distance from the feed end at which the short-circuit rail current has decreased enough such that the rail impedance calculated by equation (53), with current in the denominator, has increased enough to compensate for the shunt ballast resistance. (True rail impedance is higher than measured since the measurement includes the shunting effects of ballast resistance. Decreasing the current in the denominator increases the calculated rail impedance.) The general form for this approach is (53) But measuring rail current at any location other than the ends of the track section is more difficult than measuring rail-to-rail voltage. So the following manipulations may be made such that rail current does not need to be measured except at the feed end. Proceeding as from (41) through (44) but for rail impedance, we have tanh tanh From equation (24), under short-circuit conditions where 0, tanh (54) (55) AREMA 2015 10

Combining (43), (44), (56), and (57), Now (53) can be rewritten as 0 cosh (56) 0 cosh (57) (58) (59) 1 Ω/kft (61) This is what appears in [4]. If is already normalized, then no further normalization is necessary for. (60) SIMULATED COMPARISON A track circuit analysis software tool [3] is used to compare the results of the various methods. The subject track circuit is 2,000 feet long with a 25-Hz source, a rail impedance of 0.1 60 Ω/kft of track (both rails), and a ballast resistance of 2 Ω kft. The rails are then isolated as in Figure 2 and the relay end opened or shorted as necessary. Voltage and current phase angles are referenced to the source voltage. open-circuit meas. short-circuit meas. results meas. method I II III (V) 1.438 1.438 0.8 1.438 0.4 (V) 1.391 A 1.388 1.388 2.4 1.388 (V) 0.793 36.3 0.793 A 4.107 20.6 4.107 (Ω kft) 2.072 (2) 2.001 0.0 (38) 2.004 (52) (Ω/kft) 0.100 60.2 (39) 0.100 (61) % from true 3.6 0.05 0.0 0.2 % from true 0.0 0.3 0.0 Table 2 Comparison of measurement methods. Track characteristics are = 2,000 ft, = 2 Ω kft, and = 0.1 60 Ω/kft. Numbers in parentheses are equation numbers. The three methods, as summarized from the preceding section, are: I: estimated ; based on measurement of open-circuit voltage and current magnitudes at feed end; calculated with arithmetic operations II: true and ; based on measurement of open- and short-circuit voltage and current, and magnitudes and phases, at feed end; calculated with arithmetic operations, square roots, and hyperbolic trigonometry III: near-true and ; based on measurement of open- and short-circuit voltage and current magnitudes at feed end, and open-circuit voltage magnitude at 40% of the distance from feed end; calculated with arithmetic operations Note that: Method II provides results that are closest to true values but requires phase angles to be measured for alternating current. Methods I and III require only magnitudes to be measured. AREMA 2015 11

In method III, the open-circuit voltage measurement at 42% of the distance instead of 40% (as located by equation 51) is 1.389 V, which results in a better value of 2.001 Ω kft for ballast resistance (0.05 % from true). The software tool used for these simulated measurements is for illustrative purposes under ideal conditions; actual track does not have perfectly distributed characteristics and actual measurements will vary based on testers, equipment, weather, etc. ACTUAL MEASUREMENTS A survey of ballast conditions was performed at a rail transit system in the eastern U.S. Initial measurements used method II at d.c. and later measurements used method III at power line frequencies. Outside surface-running areas were generally in a range of 25 to over 100 Ω kft. Tunnels had the lowest ballast resistance, probably due to accumulation of steel dust, poor drainage, seepage, etc. Some of the lowest values were in river tunnels beneath salty estuary water. Tunnel values were mostly well below 100 Ω kft with the majority of those below 25 Ω kft; many were below 5 Ω kft; and some were as low as 0.2 Ω kft. Portions of the tunnel system were inundated by salt water due to Hurricane Sandy. Some track sections that were measured before the storm were also measured after the storm but before any cleaning or other remediation. In many cases, ballast resistance became worse. Some examples of these changes are from 25 Ω kft to 12 Ω kft, 16 to 8, 10 to 4, and 8 to 4. BALLAST REMEDIATION At a rail transit system in the eastern U.S., it was found that low ballast resistance would not support the planned use of a certain kind of track circuit. The equipment manufacturer specifies a minimum ballast resistance of 5 Ω kft whereas the majority of track circuits in the tunnel portions of the system are below that. A solution that avoided the cleaning or replacing of ballast or other heavy track work was to replace the existing tie plates with insulated tie plates. This did not address the ballast or ties directly (so the title of this section is somewhat misleading); instead, this electrically isolated the rails from the ties and ballast. Before and after measurements (using method III) of track sections that were remediated with insulated tie plates showed an improvement from a range of 1 to 9 Ω kft to a corrected range of 260 to 650 Ω kft. For ease of measurement, the existing feed was used without further adjustment; but where ballast (rail-to-rail) resistance was even higher, the open-circuit leakage current was too low to be measured with standard-issue meters without providing a source of higher voltage. CONCLUSION Various methods were shown for measuring ballast resistance, some of which also measure rail impedance. These are well-established methods that were derived here from first principles. A basic procedure with little calculation provides an estimated ballast resistance (method I). Transmission line theory was the basis for mathematically providing more accurate measures of ballast and rail but at the expense of complicated test equipment, procedures, and calculations (method II). Finally, a modified basic method, where open-circuit voltage is measured at a different location, results in near-true values with simple test equipment and basic arithmetic operations (method III). This provides the best combination of simplicity and accuracy. Using a track circuit simulator tool, it was shown that even the most basic method yields results that are close to true values. REFERENCES [1] Boettcher, Alvin P. D.C. Track Circuit Measurements. Railway Signaling (March 1947): 178-9. [2] Lewis, L. V. Alternating-Current Track Circuit Calculations. The Signal Engineer 4, no. 7 (July 1911): 268-72. [3] Landau, Stuart. Computer-Aided Analysis of Non-Coded Alternating-Current Track Circuits Using a Finite-Element Transmission-Line Model. Proceedings of the American Railway Engineering and Maintenance-of-Way Association Annual Conference, Chicago, IL, September 2014. [4] Association of American Railroads. American Railway Signaling Principles and Practices. Chicago: Association of American Railroads, 1951. Chap. 11, Non-Coded Alternating Current Track Circuits. 28-48. AREMA 2015 12

LIST OF FIGURES AND TABLES Figure 1 Double-rail track circuit with impedance bonds. Figure 2 Feeding isolated rails with bonds and relay disconnected. Figure 3 Feeding open rails. Figure 4 Track (transmission-line) element. Figure 5 One transmission-line element within a track circuit. Figure 6 Feeding shorted rails. Figure 7 Feeding open rails and measuring at feed end and at some distance from feed end. Table 1 Locations for measuring to get true ballast resistance. Table 2 Comparison of measurement methods. AREMA 2015 13

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AREMA 2015 ANNUAL CONFERENCE Ballast Resistance Mathematical basis of particular methods Derivations in the paper Track circuit simulation tool to verify and compare Actual results What is ballast resistance? Measure of electrical leakage between the rails Leakage path includes tie plates, ties, ballast Call it ballast Models can include shunt capacitance but this is ignored at low frequencies Call it resistance AREMA 2015 ANNUAL CONFERENCE Why measure ballast resistance? Track Circuit Specimen Predict track circuit performance Initial setup and adjustments Track circuit equipment requirements Maximum track circuit length is X kft at Y Ω kft (or Ω km or S/km) Double-rail a.c. with impedance bonds AREMA 2015 ANNUAL CONFERENCE Isolated Track Circuit Open-Circuit Measurements at Feed Eliminate paths other than rail-to-rail leakage (relay coil, impedance bonds, etc.) AREMA 2015 15

AREMA 2015 ANNUAL CONFERENCE Open-Circuit Measurements at Feed Open- and Short-Circuit Measurements AREMA 2015 ANNUAL CONFERENCE Open- and Short-Circuit Measurements Open- and Short-Circuit Basis Transmission-line model Telegrapher s equations Track circuit solution Open- and short-ckt. cases AREMA 2015 ANNUAL CONFERENCE Open- and Short-Circuit Measurements Open-Circuit Measurement at a Distance AREMA 2015 16

AREMA 2015 ANNUAL CONFERENCE Open-Circuit Measurement at a Distance Benefits of Measuring Z Z can be known a priori, so Measurement of Z that is as expected Validates the test setup and technique Provides confidence in measured B Z that measures too high may indicate bad bonding or other problems AREMA 2015 ANNUAL CONFERENCE Comparison of Measurement Methods Track circuit simulator configuration Length = 2000 ft f= 25 Hz B = 2 Ω kft Z = 0.1 60 Ω/kft Comparison of Measurement Methods o.c. @ feed o.c., s.c. @ feed o.c. @ 40% B (Ω kft) 2.072 2.001 0.0 2.004 Z (Ω/kft) 0.100 60.2 0.100 B error 3.6 % 0.05 % 0.2 % Z error 0.0 % 0.0 % Simulation is for illustrative purposes under ideal conditions Actual track does not have uniform characteristics Measurements vary based on env., test eqpt. AREMA 2015 ANNUAL CONFERENCE Actual Measurements Rapid transit system in eastern U.S. Surface 25 to 100+ Ω kft Tunnels 1 to 50 Ω kft (steel dust, seepage, poor drainage) Under- and near-river tunnels As low as 0.2 Ω kft (salty estuary water) Effects of Superstorm Sandy Some track circuits inundated with salt water After measurements are before cleaning or other remediation 25 to 12 Ω kft 16 to 8 10 to 4 8 to 4 AREMA 2015 17

AREMA 2015 ANNUAL CONFERENCE Ballast Remediation Replacement track circuits require minimum ballast resistance of 5 Ω kft Insulating tie-plate pads and hardware Before: 1 to 9 Ω kft After: 250 to 650 Ω kft Ballast not being addressed; rather, the rails are being isolated from the problem Conclusion Some methods are theoretically better than others Considering variability in environment, track, equipment, and testers, all methods seem close enough AREMA 2015 ANNUAL CONFERENCE? AREMA 2015 18