- Internal Combustion Engines – Our very own engines we barely look at or into. When we picture a piston moving up and down and assume the motion is smooth and symmetric, it is actually not so. At the top and bottom of its stroke, the piston momentarily stops and reverses direction. Near the middle of the stroke, it is moving fastest. Because velocity is changing, acceleration exists at every instant. At peak acceleration, there is peak stress inside the engine. Every component is accordingly designed to survive that stress with a safety margin. Engines with higher RPM require more strength in the materials used. With the mathematical microscope of derivatives, we can model the motion of piston inside the engine and design smooth and fuel efficient engines where vibration is controlled. Modern engines are basically modelled first on computer software using the mathematics before they are actually manufactured on factory floor.
- Electric Vehicles – Very important as herein lies the present and future. In electric motors, current responds very fast, and torque must be adjusted in real time, sometimes thousands of times per second, to keep motion smooth and efficient. This is a direct application of limits and derivatives. Current changes every time we press the accelerator. If not controlled, it will result in wheel slip and jerks. Even slight increase in current results in instant acceleration and torque response. The electronic controller’s job is to find how fast is the current changing at every moment and keep stabilizing the current and torque. It detects instability before it becomes visible and ensures that torque changes gradually, not abruptly.
- Regenerative Braking in Electric Vehicles – Ever realized that applying brakes is actually adding to the driving range? Virtually all modern fully electric vehicles (EVs) use regenerative braking. It is an essential for an electric vehicle’s efficiency. It is vital and built into the driving mechanism. An electric car has a single component being the electric motor. It performs two jobs. On pressing the accelerator, electricity flows out of the battery to the motor, which spins the wheels to move the car forward. Secondly, when we lift your foot off the accelerator or press the brake pedal lightly, electronics of the car reverse the flow. The car’s momentum keeps the wheels spinning, which in turn forces the motor also to keep spinning. When the motor is forced to spin by an outside force (the wheels), it stops acting like a motor and starts acting like an electric generator. This reverses the flow of energy and the direction of torque. The electricity produced is sent back into the battery for later use. Even if braking lasts only 3–4 seconds, power is changing every millisecond so integration is essential to find out the energy recovered over the entire braking duration and what driving range was added.
- Suspension Systems – This application of calculus transforms a bumpy ride into a smooth journey. In higher-end modern cars, the suspension is electronically controlled. The suspension system is modelled using differential equations. The suspension system of a car has three parts which are the body of car resting on wheels, the spring and a damper which is basically a piston filled with oil that resists sudden motion and prevents repeated bouncing. A damper’s force opposes motion and tries to slow it down. If the spring constant is high, the spring is hard to compress which makes the car feel bumpy. If the spring constant is low, the spring is easy to compress which makes the car feel smooth but it bounces a bit more. Sports cars need more spring stiffness as they are meant for more control and not so much for comfort. On the other hand, a luxury car needs lower stiffness as it is designed for more comfort. Similarly, if a damper is tuned with low damping, car feels bouncy and with large damping, car feels stiff. So the spring and damper system together determine how much the car moves after going over a pothole and how quickly it stops bouncing. A dedicated Suspension Control Unit (SCU) is a microprocessor which reads all sensor values in real time. These values are wheel position, acceleration, speed etc. which keep changing every millisecond. This unit continuously solves the differential equations of mass–spring–damper system in real time and adjusts the damping instantly. When a car goes over a pothole or speed breaker, it goes up and down and sometimes bounces a bit. This behaviour is controlled by stiffness of spring and the damper (shock absorber). Using the derivatives of position to find velocity of the up and down movement of car, and applying the differential equations, the spring and damper are critically tuned so that the car returns to rest as quickly as possible without bouncing at all.
- In Solar and Wind Power Generation – Calculus is even used to harness free energy. Solar parks use it to maximise the electrical power produced by a solar panel at every instant. The power output is dependent on the angle between the sun’s rays and the panel surface. So how does a solar park find that perfect angle? By differentiation. The sun does not stay still. Its position in the sky changes continuously through the day as the earth rotates. This means the ideal angle is itself changing every second. For this purpose, solar parks use solar trackers which are basically motorized mounts that tilt each panel to continuously follow the sun. It is here that calculus enters to calculate how fast should the panel rotate? If it rotates too slowly, it falls behind the sun and loses efficiency. If it rotates too quickly, it wastes energy. Total energy generated over a day or year can be found out by integrating power with respect to time. This way we can estimate annual energy yield and economic returns. Similarly, Calculus is used in wind power generation to ensure extraction of maximum energy from varying wind flow. The key is safe and efficient control of turbine. Wind speed can change in seconds. So the power a turbine extracts depends on the rotor speed (how fast the blades spin) and the pitch angle (the tilt of each blade relative to the wind). A control system continuously computes the derivative of power with respect to rotor speed to find exact rotor speed at which the turbine extracts the maximum possible power. And because wind speed itself keeps changing, the optimal rotor speed is itself a moving target. Differentiation is used continuously to re-find the optimum and adjust the blade pitch angle in real time. Solar and wind power plants produce electricity that varies continuously with sunlight and wind speed, while the electric grid requires a stable frequency (50 Hz in India). In the solar inverters and wind turbine generators, calculus based control algorithms continuously compute the derivative of how fast the frequency of power generated deviates from desired frequency and accordingly makes the required corrections to maintain the frequency at 50 Hz. With varying sunlight and wind, the power generated keeps varying. Thus applying mathematics, we ensure smooth power generation that is in sync with grid frequency.
- Artificial Intelligence – Ever wondered how AI keeps improving its decisions and output? An AI hardware accelerator is a special-purpose computer chip designed to run artificial intelligence tasks much faster and more efficiently than a regular CPU. AI systems constantly keep learning and evolving to improve their decisions like recognising faces, understanding speech or recommending content. These adjustments are about how fast and in what direction things should change. At the heart of this learning lies Calculus. An AI model makes predictions, but these predictions are not perfect. The difference between the predicted result and the actual result is called the error. The goal of the system is to reduce this error step by step. This is where derivatives come into play. A derivative tells us how fast the error is changing, in which direction we should adjust the system to reduce it. Based on whether a change makes the result better or worse, the algorithm updates its internal parameters repeatedly. Predictions thus keep getting accurate with time. Special AI chips are built to do this change calculation extremely fast and with very little energy.
Leave a Reply