Initial Conditions

In transient analysis, the behavior of a circuit immediately after switching depends entirely on the initial conditions of its energy-storing elements. Understanding what an inductor and capacitor do at the instant of switching, denoted as t=0+, is the first and most critical step in solving any transient circuit problem in network analysis.

Core Concept Explanation

The concept of initial conditions arises from a fundamental physical law: energy cannot change instantaneously. A capacitor stores energy in its electric field as (1/2)CV², and an inductor stores energy in its magnetic field as (1/2)LI². Since energy cannot jump from one value to another in zero time, the voltage across a capacitor and the current through an inductor must remain continuous across the switching instant.

The notation t=0- refers to the instant just before the switch operates, and t=0+ refers to the instant just after. The key rules that govern circuit behavior at t=0+ are straightforward: the voltage across a capacitor at t=0+ equals its voltage at t=0-, and the current through an inductor at t=0+ equals its current at t=0-. These are the initial conditions of the circuit.

To find the circuit state at t=0-, you analyze the circuit under DC steady-state conditions before switching. In DC steady state, a capacitor acts as an open circuit (no current flows through it) and an inductor acts as a short circuit (no voltage drop across it). This lets you compute Vc(0-) and IL(0-) directly.

Once the switch operates at t=0, you replace each capacitor with a voltage source of value Vc(0+) and each inductor with a current source of value IL(0+) to draw the equivalent circuit at t=0+. From this equivalent circuit, you can find all other voltages and currents in the network at that instant using standard circuit analysis methods such as KVL, KCL, or nodal analysis.

Mathematical Expression

The governing equations for capacitor voltage and inductor current are derived from their terminal characteristics. For a capacitor, the current-voltage relationship is given by ic = C × (dVc/dt). Since current can be finite, voltage must be continuous, giving the condition Vc(0+) = Vc(0-).

For an inductor, the voltage-current relationship is VL = L × (diL/dt). Since voltage can be finite, current must be continuous, giving iL(0+) = iL(0-). A discontinuity in inductor current would require an infinite voltage impulse, which is physically unrealizable in a circuit with finite sources.

At t=0+, any element that is not a capacitor or inductor (such as resistors and independent sources) can change instantaneously. So the currents through capacitors and voltages across inductors can change abruptly at t=0+, even though Vc and iL themselves cannot. This distinction is frequently tested in GATE.

Practical Understanding

Consider a simple series RC circuit where a charged capacitor discharges through a resistor after a switch is closed. Before switching, the capacitor holds voltage V0 and no current flows. The moment the switch closes, the capacitor maintains its voltage V0 at t=0+ and the circuit sees a current of V0/R at that instant. This initial current then decays exponentially with time constant tau = RC.

For an RL circuit, if the inductor carries current I0 before switching, it continues to force that same current I0 through whatever path is available at t=0+. If the switch opens suddenly and there is no freewheeling path, the inductor generates a very large voltage spike to maintain its current, which is the basis for flyback phenomena seen in relay coils and power electronics circuits.

Solved Numerical Example

A series circuit consists of a 10V DC source, a 4 ohm resistor, a 2H inductor, and a 3 ohm resistor, all connected in a loop with a switch. The switch has been closed for a long time and is opened at t=0. Find the current through the inductor and the voltage across the 3 ohm resistor at t=0+. Before switching, the circuit is in DC steady state, so the inductor is a short circuit. The two resistors are in series, carrying a current of 10/(4+3) = 10/7 A. At t=0+, the inductor maintains this current.

Given:
DC Source = 10V, R1 = 4 ohm, L = 2H, R2 = 3 ohm
Switch opened at t = 0

Why this formula applies:
At DC steady state (t=0-), inductor = short circuit. iL(0-) = V/(R1+R2).
At t=0+, iL(0+) = iL(0-) by continuity of inductor current.

Formula:
iL(0-) = Vs / (R1 + R2)

Substitution:
iL(0-) = 10 / (4 + 3) = 10/7 A

Calculation:
iL(0+) = iL(0-) = 10/7 ≈ 1.43 A
Voltage across R2 at t=0+ = iL(0+) × R2 = (10/7) × 3 = 30/7 ≈ 4.29 V

Final Answer with units:
iL(0+) = 1.43 A, V_R2(0+) = 4.29 V

Exam Tip: At t=0-, inductor = short circuit and capacitor = open circuit for DC steady state. At t=0+, replace capacitor with a voltage source equal to Vc(0-) and inductor with a current source equal to iL(0-). Forgetting this replacement is the most common GATE mistake in transient problems.

• At t=0-, DC steady state means capacitor is open circuit and inductor is short circuit. Use this to find Vc(0-) and iL(0-).
• At t=0+, replace the capacitor with a voltage source of magnitude Vc(0-) and inductor with a current source of magnitude iL(0-).
• Analyze the resulting resistive circuit at t=0+ using KVL, KCL, or node voltage method to find all unknown quantities.
• Currents through capacitors and voltages across inductors can jump at t=0+, but Vc and iL themselves cannot.
• If a capacitor is uncharged (Vc=0) it behaves as a short circuit at t=0+. If an inductor carries no initial current it behaves as an open circuit at t=0+.

Quick Revision

• Vc(0+) = Vc(0-): Capacitor voltage cannot change instantaneously due to energy continuity.
• iL(0+) = iL(0-): Inductor current cannot change instantaneously due to energy continuity.
• At t=0-, L = short circuit, C = open circuit for DC steady state analysis.
• At t=0+, L is replaced by current source iL(0-), C is replaced by voltage source Vc(0-).
• Uncharged capacitor at t=0+ = short circuit; inductor with zero initial current at t=0+ = open circuit.
• GATE trap: Confusing which element acts as open/short at DC steady state vs at t=0+ is a frequent error.
• Key formula: iC = C(dVc/dt) and VL = L(diL/dt) are the basis for continuity conditions.