Sunday, July 18, 2010

Impedance matching and transit

This morning, the DHT takes a minor break from discussion of the Lake Oswego transit project, to emphasize a fundamental principle of transit design (one that certainly has relevance to Lake Oswego, and beyond).  Yours truly hold an MS in Electrical and Computer Engineering from Oregon State University, which implies that I know something about electricity and electronics.  (That said, I've spent the past fifteen since graduation doing software, which implies that I've probably forgotten most of it).  However, one important principle of electrical engineering, that of impedance matching, has significant relevance by analogy to transit.

Impedance is defined by Wikipedia as a quantity which "describes a measure of opposition to alternating current (AC)."  While I have some quibbles with that particular phrasing, they're off topic here on the DHT--the definition given is good enough for the purposes of this post.  Impedance can be broken down into two fundamental components: resistance (opposition to non-changing, or direct, current--always a positive number), and reactance (opposition to changes in current, which can be positive, negative, or zero).  Electrical circuits can be modified by a source (typically a voltage source, such as a battery or power supply), and a load (the device that does whatever work or performs whatever function is desired).  Both the source and the load have an impedance associated with them, and the behavior of the circuit depends on the impedance of both components.

Why impedance matters

In many applications, it is important that the impedance of the source and load be matched--a condition which occurs when the resistance of source and load are the same, and the reactance of source and load sums to zero.  Impedance matching is required for maximum transfer of power from source to load, and for minimizing "reflections" of the signal (which can disrupt communications).   The latter phenomenon is too complicated to do justice to in a post for laypersons; but the former can be be explained easily enough:  The power consumed in a circuit element is the product of the voltage drop across it, and the current which flows through it:  P = VI (I is used for current in the field of electronics--C is used for capacitance).   The current through a circuit element can be computed by dividing voltage over impedance: I = V/Z (Z = impedance; don't ask me why).   The total voltage of the source is distributed across the source and load impedances, V = VS + VL; where VS = VZS/(ZS + ZL) and VL = VZL/(ZS + ZL).  Thus, doing a bit of math produces the result that the power dissipated by the load is ZL (V/(ZS+ZL))²  Given that the source impedance ZS is often fixed, a bit of calculus shows that the load power is maximized when the load impedance ZL is set to match it. 

One problem with impedance matching is that while load power is maximized; so is source power, such circuits are only 50% efficient.  Efficiency can be computed by dividing load power over total power, under the assumption that only load power is used for doing useful work. To maximize efficiency, one deigns so that the load impedance as high as possible.  This limits the amount of power which can be delivered to the load, but in many cases that doesn't matter--if you have a 60W light bulb, it doesn't matter that the household circuit may be capable of delivering 2kW--you only need 60W to make the thing work.  Maximum efficiency is achieved by making the load impedance as high as possible relative to the source impedance.   This technique, impedance bridging, is generally only used in situations where reflections aren't an issue. 

On the other hand, if the load impedance is lower than the source impedance, to much power is wasted in the source or the interconnecting wires.  This is generally not a useful case, and in extreme cases is known as a "short circuit".

What of the source impedance itself? The source impedance of a power supply is in many ways, a figure of merit--lower is better.  A power supply with a lower source impedance will be able to deliver a higher maximum power for a given voltage level than a supply with a higher impedance; and will waste less power no matter what the load.  Unsurprisingly, designing power supplies with very low impedances is difficult and expensive. 

(EEs may note that I'm glossing over power supplies modeled as current sources--and they would be correct.  I'm glossing over lots of things...) 

What does all of this have to do with transit?

Good question. 

When designing a transit line, there are several important factors to consider:
  • The nature of the route (including the vehicles): 
  • The nature of the stops (distance between, amenities, platform height, etc)
A transit vehicle or system does two important things:  It stops to pick up or drop off passengers, and it moves to the next stop.   For both functions, minimization of time is required.  It's relative easy to come up with first-order figures of merit for both operations:

First, consider a transit vehicle which is deadheading on its route.  Here, the vehicle traverses the route, running under normal conditions and at normal speed--stopping at traffic lights, stop signs, and for obstacles in the road (other traffic, pedestrians, debris) as necessary--but does not stop to pick up passengers.   We could measure its velocity over the route, under various conditions, to come up with a number we'll call the deadheading velocity, VD.  Higher is better, obviously--and various things can contribute to a higher deadheading velocity:  higher-performance vehicles, a dedicated guideway (no interacting with other traffic), priority at intersections, etc.

Now consider the hypothetical case of a vehicle which traverses its route, stopping at stations and stops in the ordinary manner--but travelling instantaneously from one stop to the next:  when the doors close at one stop, the vehicle and its passengers are teleported instantly to the next needed stop.  We add in a per-stop penalty for to account for acceleration and decelation as well. Assume that each stop is serviced normally--if little old ladies need to fumble through their purse to find exact change, that time is counted.  Were we to sum up all the stop times (averaged over numerous runs), and divide them into the length of the route, we would come up with a number I'll call the service velocity, VS

Now, let's invert the two parameters, VD and VS.  Science doesn't presently have a good term for the inverse of velocity, so I'll coin one:  lethargy, or L (LS, LD).  The lethargy of something is the amount of time it takes to cover a distance, divided by the distance.  Why do we use lethargy?  Because, lethargy is additive.  The total lethargy is simply the service lethargy plus the deadheading lethargy.  (One can invert the total lethargy to get the total velocity, or an approximation thereof--keep in mind, this is a crude model).  If it isn't obvious already; lower is better.

How lethargic is good?

What values of lethargy are good?  Bad?  Typical?  First, lets consider some typical LD values for various modes of transportation:
  • Jet air travel:  < 0.12 (>500MPH)
  • Turbofan air travel, true high speed rail: 0.2 - 0.3 (200-300 MPH)
  • Second-tier high-speed rail : 0.3 - 0.5 (120-200 MPH)
  • Amtrak, rural freeway: 0.7 - 0.9 (65-85 MPH)
  • Urban freeway (no congestion), grade-separated metro: 0.8 - 1.2 (50-75 MPH)
  • Rural highway: 1.1 - 1.3 (45 - 55 MPH)
  • Urban expressway with stoplights, i.e OR224: 1.5-1.7 (35-40 MPH)
  • Median-running light rail:  1.7 - 2.1 (28-33 MPH)
  • Urban boulevard: 1.8 - 2.2 (27-32 MPH)
  • Urban arterial: 2.4 - 3.0 (20-25 MPH)
  • Light rail, traffic in downtown grid: 2.4 - 3.0 (20-25MPH)
  • Residential streets: >3.0 (<20MPH)
Now, LS.
  • Point-to-point auto travel: < 0.1 
  • Point-to-point commuter bus/rail, >10 mile trip:  < 0.5 (assuming 2.5 mins for loading and unloading)
  • Point-to-point air travel: <300 mile trip (40 mins to load, 20 mins to unload plane): >0.2
  • Point-to-point air travel: <300-600 mile trip: 0.1 - 0.2
  • Point-to-point air travel, >600 miles: < 0.1
  • Corridor commuter rail/express bus (such as WES):  0.1 - 0.3
  • True-metro rapid transit, 1-2 mile stop spacing, 30-40 second dwell:  0.33 - 0.5
  • Transit, 0.6-1 mile spacing, 30 second dwell: 0.5 - 0.8
  • Transit, 0.4 - 0.6 mile spacing, 20 second dwell: 0.8 - 1.3
  • Transit, 0.2 - 0.4 mile spacing (1000-2000 feet), 15 second dwell: 0.6 - 1.2
  • Transit, 750 - 1000 feet, 15 second dwell (i.e. Streetcar):    1.2 - 1.6
  • Transit: 500-750 feet, 10 second dwell: 1.1 - 1.5
For a given service type, by adding the corresponding lethargies for the route type and the stop spacing, you get a good estimate of the overall lethargy--and the overall speed.

What lethargies do riders generally expect or encounter?  Values less than 1.5 are generally rare, but here are some "typical" values:
  • Point-to-point commuter rail: 1.5
  • Corridor commuter rail, i.e. Sounder or WES:  1.8-2.2 (WES is 2.4; due to shorter-than-typical stop spacing and tracks not rated for higher-speed operation).
  • Point-to-point highway-running express bus: 1.8-2.0 (i.e. C-Tran 199)
  • Corridor express bus: 2.2 - 2.6 (C-Tran 105)
  • "True" metro: 1.7 - 2.2
  • Dedicated-guideway light rail:  2.0 - 2.8:
  • Dedicated guideway BRT, limited stop, no signal priority:  2.5 - 3.0 (LA Orange Line)
  • Median-running light rail (i.e. yellow line): 2.8 - 3.2
  • Suburban highway bus: 3.0 - 3.5 (the 33 and 35 are both in this range)
  • Rapid streetcar:  2.8 - 3.5
    • Proposed Lake Oswego streetcar, LO-PSU:  ~3.5 
  • Downtown light rail: 3.5 - 4.0
  • Urban boulevard local bus (ie TriMet 9): 3.5-4.0
  • Local bus: 5.0 - 10.0
  • Portland Streetcar downtown:  ~8.0
Excluding express and commuter services, a common cutoff for rapid transit is a lethargy of about 3.0 (an average speed of 20MPH).  High-quality rapid transit can do 2.0 or better; medium-quality, about 2.5.   Frequent stop (non-express) bus service running in mixed traffic on ordinary streets has a hard-time doing better than 3.0, and mixed-traffic streetcar is unsurprisingly, very slow.

What does this all mean?

The reason I use lethargy, in addition to the fact that it's easy to compute with--is that it's analogous to the electrical property of impedance discussed in the first section.  Furthermore, deadheading lethargy is analogous to source impedance, and service lethargy is analogous to load impedance, in the following ways:
  • Power consumed in the load, and time spent picking up and dropping off passengers, is useful.  Whether a high or a low value is good or bad depends on the application--commuter rail (and point-to-point transport) have very low service lethargies, as the bulk of the time is spent in transit--but don't serve very many people.  Local bus service, on the other hand, has a very high service lethargy--it stops in lots of places--but this limits its effective speed.
  • Deadheading lethargy, like source impedance, is a figure of merit--the lower, the better.  And like source impedance, making it lower costs money.  Time spent in transit is generally time wasted, just as power consumed in the source is wasted energy.  
  • Most importantly, it is useful to match lethargies, just as it is useful in electonics to match impedance.  Building expensive transit infrastructure, but running services with frequent stop spacing on top of it, does not lead to better service, at least as far as this analysis is concerned.  (There may be other reasons--reliability, capacity, sex appeal, whatever, to do so; all of these things are outside the scope of this post).  
  • On the other side of the coin, running limited stop service in mixed traffic is useful--express bus and commuter rail being two examples--but these services generally have low impact.
  • Finally, like impedance, lethargy is logarithmic in its impact.  There's a  bigger difference between lethargies of 2 and 3, then between lethargies of 5 and 7, for instance.  This logarithmic nature makes higher-performance lines difficult and expensive to build; it's far too easy to build too many stops on a line to garner greater political support, sabotaging its technical merit in the process.  (MAX downtown suffers from this problem to some extent--that said, MAX was designed to provide local service through downtown; not to be an efficient means for crosstown trips).


    1. Thanks for the refresher on impedence. It's been a while.

      For comparison, what would L be for the current bus between Lake Oswego and PSU. The fastest weekday run is the 5:03 (LOTC) northbound at 17 minutes and the slowest the 5:19 (4th & Harrison) at 31 minutes. Current estimates for streetcar are for between 30 and 34 minutes depending on alignment decisions.

    2. RA--I thought you might ask. As we both know, best case performance of LO streetcar is not going to be any better than the bus today. I'll do another post on that soon.

      As far as the whole impedance/lethargy thing, I probably should emphasize that this particular bit of analysis is a crude model I cooked up to make a point. While real transit planners and system designers do similar things; their analyses are far more complex, and aren't artificially constructed to resemble electric circuit theory.

      That said, I was attempted to expand the analogy to AC, and model congested corridors as inductors, and transfer stations as capacitors, and service frequency as, well, signal frequency. It works! If there's only one bus or train per day, transfer points block passage; as service frequency increases, transfer points become less and less of a barrier to movement of passengers--and at high frequencies all you would have to do is walk across the platform. Likewise, series capacitors will block DC signals but pass high-frequency ones. Congested corridors have a negative impact on high-frequency service, as bottlenecks form and vehicles bunch up--just like high frequency electrical signals get attenuated when passing through a coil.

      However much "nerd chic" such an analysis would have generated--it was far beyond what was necessary to prove the point I wanted to make. Hence, it is left as an exercise for the reader.

      Now what would be the transit equivalent of an op amp?

    3. "All you would have to do is walk across the platform".

      You've described The Netherlands. They have an absolutely fabulous half-hourly service between everywhere and everywhere, centered not on Amsterdam (it's actually rather peripheral though many direct trains not in this timed transfer network have it as an end point) but Utrecht. It's fabulous. Four trains on four routes (normally Rotterdam/Zwolle/Groningen, Maastricht/Amsterdam, Arnhem/Schiphol, and Hengelo/Den Haag) arrive essentially simultaneously and passengers have about four minutes to scramble among them. Then off they go.

      It happens every fifteen minutes first westbound then eastbound alternately.


      Amsterdam is served by direct trains to and from Rotterdam, Den Haag, Hengelo and Groningen that bypass Utrecht on other routes.


    Keep it clean, please