Time Constant

Let's Try An Analogy

Imagine a person walking up a steep hill, where the journey uphill represents a capacitor charging over time.  At the start of the climb, the person has a lot of energy and ascends the hill quickly, covering a significant portion of the distance in a short time. This initial phase is like the capacitor charging rapidly at first, with the voltage increasing quickly.  As the person gets higher up the hill, their pace slows down due to fatigue and the steepness of the hill. This is akin to the charging rate of the capacitor decreasing as it approaches its maximum charge. The person is still making progress, but each step covers less distance compared to when they started.  The time constant in this analogy is like the time it takes for the person to cover about 63.2% of the total distance to the top of the hill. Beyond this point, even though the person continues to move upwards, the rate of ascent is slower, much like how a capacitor charges more slowly as it nears its full capacity.

Time Constant in RL Circuits

• Definition: In an RL circuit, the time constant is the time required for the current to build up to 63.2% of its maximum value.

• Formula: The time constant (τ) in an RL circuit is given by:

τ = L / R

where L is the inductance in henrys (H) and R is the resistance in ohms (Ω).

Time Constant in RC Circuits

• Charging: The time constant in an RC circuit is the time required for the capacitor to be charged to 63.2% of the supply voltage.

• Discharging: Similarly, it's the time for a charged capacitor to discharge to 36.8% of its initial value.

• Formula: The time constant (τ) in an RC circuit is calculated as:

τ = R × C

where R is the resistance in ohms (Ω) and C is the capacitance in farads (F).

Dynamics Over Multiple Time Constants in RC Circuits

• After Two Time Constants:

  • The capacitor is charged to about 86.5% of the supply voltage.

  • During discharging, it drops to approximately 13.5% of its initial value.
    
    

Calculating Time Constants for Specific Circuits

1. Circuit with 100 microfarad Capacitor and 470 kilohm Resistor:

   • C = 100 × 10^-6 F

   • R = 470 × 10^3 Ω

   • τ = 470 × 10^3 × 100 × 10^-6

   • τ = 47 seconds
     
     

2. Circuit with 470 microfarad Capacitor and 470 kilohm Resistor:

   • C = 470 × 10^-6 F

   • R = 470 × 10^3 Ω

   • τ = 470 × 10^3 × 470 × 10^-6

   • τ = 221 seconds
     
     

3. Circuit with 220 microfarad Capacitor and 470 kilohm Resistor:

   • C = 220 × 10^-6 F

   • R = 470 × 10^3 Ω

   • τ = 470 × 10^3 × 220 × 10^-6

   • τ = 103 seconds
     
     

Understanding Back EMF

Back Electromotive Force (Back EMF) refers to a voltage that opposes the applied electromotive force (EMF), especially significant in inductive circuits where changing currents induce an opposing voltage.

Let's try another analogy.  Back EMF is like the action of a person walking against a strong wind. Imagine you're walking forward, and as you move, the wind starts blowing directly towards you. The harder you walk, the stronger the wind seems to blow against you, making it more difficult to move forward.  Your effort to walk forward is like the electric current provided to a motor. The wind pushing against you represents the Back EMF. Just as the wind increases in strength as you walk faster, Back EMF increases in a motor as the speed of the motor increases. It acts in opposition to the applied voltage (like the wind opposing your forward movement) and is a natural consequence of the motor's operation, much like the increased wind resistance is a natural consequence of your walking speed.

Conclusion

The time constant is an invaluable concept for understanding the dynamic behavior of RL and RC circuits. It illustrates how quickly these circuits can respond to electrical changes. The phenomenon of back EMF adds complexity, particularly in circuits with inductors, influencing the overall circuit behavior. These principles are foundational for designing and analyzing electrical and electronic systems.