The simplified speed – time curve for urban and sub-urban service for electrical traction is shown in the Figure.

Quadrilateral Speed – Time Curve

The study of Quadrilateral speed time curve is given below.

Let

α = Acceleration in km per hour per second

β_C = Coasting retardation in km per hour per second

β = Braking retardation in km per hour per second

V₁ = Maximum speed at the end of acceleration ( km per hour )

V₂ = Speed at the end of coasting period ( km per hour )

T = Total time of run ( second )

Acceleration time in second t₁ = V₁ / α …. ( 1 )

Coasting time in second t₂ = V₁ – V₂ / β_C ….. ( 2 )

Braking time in second t₃ = V₂ / β …… ( 3 )

Total distance travelled in ( km )

S = Distance travelled during acceleration ( Area ABC ) +

Distance travelled during coasting ( Area BDEF ) +

Distance travelled during retardation ( DEF )

S = ( ½ V₁t₁ / 3600 ) + { ( V₁ + V₂ ) / 2 × ( t₂ / 3600 ) } + ( ½ V₂t₃ / 3600 )

S = ( V₁t₁ / 7200 ) + { ( V₁t₂ / 7200 ) + V₂t₂ / 7200 ) } + ( V₂t₃ / 7200 )

S = { V₁ ( t₁ + t₂ ) / 7200 } + { V₂ ( t₂ + t₃ ) / 7200 }

As t₁ + t₂ + t₃ = T

S = { V₁ ( T – t₃ ) / 7200 } + { V₂( T – t₁ ) / 7200 }

S = V₁T / 7200 + V₂T / 7200 – V₁t₃ / 7200 – V₂t₁ / 7200

Now t = V₁/ α and t₃ = V₂ / β

S = { T ( V₁ + V₂ ) / 7200 } – ( V₁ × V₂ ) / 7200 β – ( V₁ × V₂ ) / 7200 α

S = { T ( V₁ + V₂ ) / 7200 } – V₁V₂ / 7200 β – V₁V₂ / 7200 α

7200S = T ( V₁ + V₂ ) – V₁V₂ ( 1 / α + 1 / β )

From equation ( 2 )

t₂β_C= V₁ – V₂

V₂ = V₁ – t₂ β_C

V₂ = V₁ – β_C ( T – t₁ – t₃)

V₂ = V₁ – β_C ( T – V₁ / α – V₂ / β )

V₂ = V₁ – β_C T + β_C V₁ / α + β_C V₂ / β

V₂ – β_C V₂ / β = V₁– β_C T + β_C V₁ / α

V₂ ( 1 – β_C / β ) = V₁ – β_C ( T + V₁ / α )

V₂ = V₁ – β_C ( T + V₁ / α ) / ( 1 – β_C / β )

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