A curve plotted with time in seconds or minutes in the abscissa (x-axis) and speed in km/h as ordinate (y-axis) at different instants from start to stop of a train is known as the speed-time curve. In order words, the speed-time curve gives the relationship between the speed and time of a train during its motion. It gives complete information about the motion of a train.

The different characteristics of the train can be obtained by using a speed-time curve i.e., speed, distance, energy consumption, acceleration, etc. The motion of the train can be studied at regular intervals of time since it is not constant i.e., its motion is a function of time.

## Speed Time Curve of Main Line Service :

In mainline service, the distance between two stations or two stops is more than 10kms. This service has a long free run period. The accelerating and retarding form a small part of the total running time of the train, so their values are not so important in this service. The most requirements of mainline service are,

- High maximum speed.
- Minimum cost of the overhead structure.

The cost of mainline service can be reduced by using a single-phase overhead structure. This will reduce the number of conductors. If high voltage is used for distribution, the cost of the distribution system employing a conductor of a smaller cross-section to carry a low current is reduced. Also because of high voltage distribution, the distance between substations can be increased and the number of substations required is less.

The initial, maintenance and operating costs of ac substations are less as the substation equipment required in a single-phase ac system is less, cheap, and efficient. Single-phase ac systems and single-phase to three-phase composite systems are more suitable for this service. The below shows the speed-time curve of the mainline service.

#### The speed-time curve of mainline service consists of the following components,- Constant acceleration,
- Acceleration on speed curve,
- Free-running period,
- Coasting period,
- Braking period.

### Constant Acceleration :

Before this period commences the train is stationary or in a standstill position. The motor of the train is to be accelerated from the rest position for its motion. By varying the resistance in steps, the value of current is made constant and the voltage is increased gradually till the constant tractive effort is reached.

Thus the acceleration of the train is maintained constant during this period as indicated by the period AB in the speed-time curve. Since the train is made to move from the rest position, this period is also called as Notching period or Rheostatic acceleration.

#### The characteristics of a constant acceleration period are,- Current is constant.
- Torque is constant.
- Acceleration is constant.

### Acceleration on Speed Curve :

This period is also called speed curve running or acceleration on speed curve. During this period the full supply voltage is given. We know that speed and current are inversely proportional to each other i.e., as the speed of the train increases, the current decreases, and hence the torque also decreases.

It happens when full voltage is applied to the motor. This will cause a gradual decrease of acceleration until sufficient torque to maintain the motion is attained as shown by period BC. Although the train accelerates, its acceleration decreases with the increase in speed till the tractive effort developed by the motor becomes equal to the resistance to motion of the train.

#### The characteristics of acceleration on speed curve period are,- Voltage is maintained constant.
- Current decreases as speed increases.
- Torque decreases and it finally gets equal to the resistance of the train in motion.
- Acceleration decreases and attains zero value.

### Free Running Period :

At the end of speed-curve running or BC period, the train attains maximum speed and continues along the period CD till the coasting period. During this period the power drawn from the supply is constant.

#### The characteristics of the free-running period are,- Speed is constant.
- Power drawn is constant.
- Current is minimum.
- Torque is also minimum.

### Coasting Period :

When the train reaches or completes the end of the free-running period i.e., at point D, the coasting period starts. During this period, the power supply to the train is turned off and the motion of the train is due to its moment of inertia. During this period the speed of the train decreases because of resistance to the motion of the train as shown in period DE. The rate of decrease in speed of the train is known as retardation of coasting.

#### The characteristics of the coasting period are,- Speed of the train decreases.
- The Tractive effort of the train decreases.
- Acceleration is zero.
- Voltage and current are zero.
- The motion of the train is due to its own momentum.

### Braking Period :

After the completion of the coasting period, the train enters into the braking period. During this period the train which is in motion is brought into the standstill position by applying brakes i.e., the train comes to rest.

#### The characteristics of the braking period are,- The speed of the train decreases gradually and becomes zero.
- Acceleration is zero.
- Voltage and current are zero.
- Torque is zero.

From the above speed-time curve we can see that free-running and coasting periods are long. The maximum speed that can be attained by the train in mainline service is 160 km/hr. For this service, a single-phase system is adopted. The typical value of acceleration vary between 0.6 to 1 km/hr/sec and the typical value of retardation is 1.5 km/hr/sec.