Transient analysis RL circuit/RC circuit

 Step-by-Step Guide to Solve RL/RC Transient Circuits

When studying Network Theory or preparing for competitive exams in Electrical/Electronics Engineering, one of the most important topics is Transient Analysis of RL/RC circuits. Students often get confused about how to start and what sequence of steps to follow. This article provides a simple 4-step method with examples.

🔹 What is Transient Analysis?

When a switch is operated in a circuit containing resistors (R) and inductors (L) or capacitors (C), the voltages and currents do not change instantaneously. The study of how these quantities change with time during the transition from one steady state to another is called transient analysis.

🔹 General 4-Step Method

Step 1: Find the Initial Condition (t = 0⁻)

Before switching, assume the circuit has been in steady state for a long time.

Inductor: current cannot change suddenly → 

i (0-) = i (0+)

Capacitor: voltage cannot change suddenly → 

V(0-)= V(0+)

Step 2: Calculate the Time Constant ()

Remove the energy storage element (open the capacitor / short the inductor).

Replace independent sources: voltage sources → short, current sources → open.

Find equivalent resistance .

Time constant:

For RL circuit = L/R

For RC circuit= RC

Step 3: Find the Final Value (t → ∞)

After a long time, the circuit reaches DC steady state.

Inductor → Short Circuit

Capacitor → Open Circuit

Induction motor: torque -slip characteristics explained with numerical

Basic Terms

Slip = (Ns - Nr)/Ns

Torque (T) is the turning force produced by the motor shaft

Characteristic Curve Overview

The torque vs. slip curve has 3 regions:

1. Low Slip Region

 (Normal running condition, small slip ~0 to 0.05)

Motor runs near synchronous speed

Torque increases almost linearly with slip

Torque ≈ proportional to slip (T ∝ S)

2. Medium Slip Region

 (Moderate load increase)

Torque increases non-linearly

Approaches maximum torque (breakdown torque)

3.  High Slip Region 

(Startup or overload)

Torque starts decreasing with increase in slip

Rotor resistance dominates

Torque 

∝ 1/S (inversely proportional to slip)

Parameter	Description
Starting Torque (S = 1)	Non-zero torque, depends on rotor resistance.
Maximum Torque	Also called Breakdown Torque or Pull-out Torque.
Slip at Max Torque	S= R2/X2  independent of supply voltage.
Low Slip Region	Torque is directly proportional to slip (T ∝ S).
High Slip Region	Torque is inversely proportional to slip (T ∝ 1/S).

Numerical on induction motor 

Induction Motor Solved Numericals

 Slip and Rotor Frequency

Q: A 4-pole, 3-phase, 50 Hz induction motor runs at 1440 rpm. Find:

• Synchronous speed
• Slip
• Rotor current frequency

Solution:

Ns = (120 × f) / P = (120 × 50) / 4 = 1500 rpm

S = (Ns - Nr) / Ns = (1500 - 1440) / 1500 = 0.04 or 4%

fr = S × f = 0.04 × 50 = 2 Hz
  
 Torque Developed

Q: A 3-phase induction motor delivers 25 kW at 1470 rpm. Calculate the torque developed.

Solution:

T = (P × 60) / (2πN) = (25000 × 60) / (2π × 1470) ≈ 162.16 Nm

  

 Rotor Copper Loss

Q: A motor has rotor input of 12 kW and slip of 5%. Find rotor copper loss and mechanical power developed.

Solution:

Rotor Cu loss = S × Rotor Input = 0.05 × 12000 = 600 W

Mechanical Power = 12000 - 600 = 11400 W

  

 Rotor Input Power

Q: Motor output power = 18.5 kW, slip = 3%. Find rotor input power.

Solution:

Rotor Input = Output / (1 - S) = 18500 / 0.97 ≈ 19072.16 W

  

Starting Current vs Full Load

Q: Starting torque is 1.5 times full-load torque. Slip at full load is 4%. Find the ratio of starting current to full-load current.

Solution:

T ∝ I² × R₂ / s

1.5 = (Istart / Ifl)² × 0.04

(Istart / Ifl)² = 37.5

Istart / Ifl = √37.5 ≈ 6.12

Read also

Back emf and torque in dc motor