Four-Phase Signal Timing — Worked Example with Protected Lefts
An arterial crossing with protected left-turn phases on both streets, a 3% downgrade on the main approach and 72-ft crosswalks. We set the change and clearance intervals, check pedestrian time, pick the cycle with Webster's formula, split the green by critical flow ratio, and check v/c and delay for every phase.
The intersection
| Approach | 85th-percentile speed 40 mi/h, main approach on a 3% downgrade, intersection width 72 ft, vehicle length 20 ft |
| Drivers | Perception–reaction t = 1.0 s, deceleration a = 10 ft/s² |
| Pedestrians | 72-ft crosswalks, 3.5 ft/s walking speed, 7-s walk |
| Phasing | Four phases in sequence, 4 s lost time each |
| Phase | Movement | Critical lane flow v (veh/h) | Saturation flow s (veh/h/ln) |
|---|---|---|---|
| 1 | NB/SB protected left | 180 | 1,750 |
| 2 | NB/SB through | 560 | 1,800 |
| 3 | EB/WB protected left | 140 | 1,750 |
| 4 | EB/WB through | 400 | 1,800 |
Signal timing tool: speed 40, grade −3, W 72, crosswalk 72, and the four phase rows above.
Open the signal timing calculator →Yellow and all-red
Y = 1.0 + 58.7 / (2 × 10 + 2 × 32.2 × (−0.03)) = 1.0 + 58.7/18.07 = 4.25 s (level ground: 3.93 s)
AR = (72 + 20)/58.7 = 1.57 s
Change interval per phase Y + AR = 5.82 s (field timing commonly rounded up to 4.5 + 2.0 s)
Minimum time for the crosswalks
Webster and the HCM minimum
L = 4 × 4 = 16 s
Co = (1.5 × 16 + 5)/(1 − 0.716) = 29/0.284 = 102.2 s → use C = 105 s
Cmin (Xc = 0.90) = 16 × 0.90/(0.90 − 0.716) = 78.3 s
Allocate C − L = 89 s by critical flow ratio
| Phase | y | Effective g (s) | Displayed G (s) | Capacity (veh/h/ln) | v/c | Uniform delay d1 (s/veh) |
|---|---|---|---|---|---|---|
| 1 NB/SB left | 0.103 | 12.8 | 11.0 | 213 | 0.845 | 45.1 |
| 2 NB/SB through | 0.311 | 38.7 | 36.8 | 663 | 0.845 | 30.4 |
| 3 EB/WB left | 0.080 | 9.9 | 8.1 | 166 | 0.845 | 46.8 |
| 4 EB/WB through | 0.222 | 27.6 | 25.8 | 473 | 0.845 | 36.7 |
Xc = ΣY × C/(C − L) = 0.716 × 105/89 = 0.845 — equal for every critical lane group by construction
Volume-weighted uniform delay = Σv·d1/Σv = 36.2 s/veh
Pedestrians and minimum greens
Phase 4 (peds cross the major street): 25.8 + 5.8 = 31.6 s ≥ 27.6 s ✓
Phase 3 left: 8.1 s displayed green — above a 5–7 s minimum, but short; consider a lead-lag or permitted+protected left if queues spill back
Tools used in this example
Signal timing calculator, traffic flow and PHF, and Stopbar to simulate the result. Guide: How to time a traffic signal.
Frequently asked questions
How does grade affect the yellow interval?
The ITE formula divides approach speed by 2a + 2gG. On a downgrade G is negative, so the denominator shrinks and yellow gets longer: at 40 mi/h it rises from 3.93 s on level ground to 4.25 s on a 3% downgrade.
How many phases count toward lost time?
Every phase in the critical path of the ring. With four sequential phases and 4 s of lost time each, L = 16 s. Lost time is why adding a protected left-turn phase is expensive: it raises Webster's cycle by about 1.5 × 4 / (1 − ΣY) seconds even before its own green.
Do pedestrians need time in every phase?
Only in the phases that run concurrently with a crosswalk, usually the through phases. Here pedestrians cross with phases 2 and 4, which provide 42.6 s and 31.6 s of green plus change interval against the 27.6 s needed.
What level of service does the delay correspond to?
HCM signalized intersection LOS thresholds on control delay are 10, 20, 35, 55 and 80 s/veh for A through E. The volume-weighted uniform delay here is about 36 s, so LOS D before random and oversaturation delay are added.