Independent Sources

Independent sources are the primary energy-supplying elements in a circuit. An ideal independent voltage source and an ideal independent current source define the voltage across their terminals or the current through themselves respectively, completely independent of any other circuit variable. Understanding their ideal and practical behavior is fundamental to all network analysis.

Ideal Independent Voltage Source

An ideal independent voltage source maintains a fixed voltage Vs across its terminals regardless of the current drawn from it. This means it has zero internal resistance. In the V-I characteristic plane, it appears as a vertical straight line at V = Vs. Any amount of current can flow through it, positive or negative, and the terminal voltage remains unchanged.

The power delivered by the source is p = Vs x i(t). Since the current can be any value (determined by the external circuit), the power can range from negative to positive infinity theoretically. A positive power means the source delivers energy to the circuit. This is the defining behavior of an active element.

A practical voltage source has a small internal resistance Rs in series. The terminal voltage drops as current increases: Vterminal = Vs - (i x Rs). The ideal voltage source is the limiting case as Rs approaches zero.

Ideal Independent Current Source

An ideal independent current source forces a fixed current Is through itself regardless of the voltage that appears across its terminals. Its V-I characteristic is a horizontal line at I = Is. The terminal voltage is determined entirely by the external circuit. An ideal current source has infinite internal resistance (an open circuit in parallel with the source).

A practical current source has a finite internal resistance Rp in parallel with the ideal source. As this parallel resistance decreases, some current is shunted internally and the terminal current deviates from Is. The ideal current source is the limit as Rp approaches infinity.

Mathematical Expression

The terminal relationships are simple but important for circuit analysis. For the voltage source: v(t) = Vs for all values of i(t). For the current source: i(t) = Is for all values of v(t). When converting between practical source models, the source transformation technique applies: a voltage source Vs in series with Rs is equivalent to a current source Is = Vs/Rs in parallel with Rs, valid only for the external circuit behavior.

Given:
Practical voltage source: Vs = 12 V, internal resistance Rs = 2 ohm
Load resistance RL = 10 ohm

Why this formula applies:
Terminal voltage = source voltage minus voltage drop across internal resistance
Vterminal = Vs - i x Rs where i = Vs / (Rs + RL)

Formula:
i = Vs / (Rs + RL)
Vterminal = Vs - i x Rs

Substitution:
i = 12 / (2 + 10) = 12 / 12 = 1 A
Vterminal = 12 - (1 x 2)

Calculation:
Vterminal = 12 - 2 = 10 V
Power to load = Vterminal x i = 10 x 1 = 10 W
Power from source = Vs x i = 12 x 1 = 12 W
Power lost in Rs = i^2 x Rs = 1 x 2 = 2 W

Final Answer:
Terminal voltage = 10 V, Load power = 10 W, Internal loss = 2 W

Exam Tip: For GATE, when two ideal voltage sources of different values are connected in parallel it creates a contradiction (a circuit inconsistency). Similarly, two ideal current sources of different values cannot be connected in series. These are common trap questions in network analysis.

Mechanism: Key Behaviors

• Ideal voltage source: V = Vs for any current. Zero internal resistance. V-I graph is a vertical line.
• Ideal current source: I = Is for any voltage. Infinite internal resistance. V-I graph is a horizontal line.
• Practical voltage source: has series internal resistance Rs. Terminal voltage falls as load current increases.
• Practical current source: has parallel internal resistance Rp. Terminal current falls as load resistance increases.
• Source transformation: Vs in series with Rs converts to Is = Vs/Rs in parallel with Rs. Valid for external circuit analysis only.
• Short circuit test for voltage source: current = Vs/Rs. Open circuit test for current source: voltage = Is x Rp.

Quick Revision

• Ideal voltage source: V = Vs regardless of current. Internal resistance = 0. V-I = vertical line.
• Ideal current source: I = Is regardless of voltage. Internal resistance = infinity. V-I = horizontal line.
• Practical voltage source = ideal Vs + series Rs. Terminal voltage: Vt = Vs - i x Rs.
• Practical current source = ideal Is + parallel Rp. Terminal current: It = Is x Rp/(Rp + RL).
• Source transformation: Vs/Rs source to Is = Vs/Rs with Rp = Rs in parallel.
• Exam trap: Two unequal ideal voltage sources in parallel is an invalid circuit. Two unequal ideal current sources in series is also invalid.
• Short circuit current of a practical voltage source = Vs/Rs. This is also the equivalent current source value.