Circuit Elements Review

Circuit elements are the fundamental building blocks of every electrical network. A clear understanding of the classification of circuit elements into active vs passive and linear vs non-linear is essential not only for GATE but also for applying network theorems correctly, as most theorems have specific applicability conditions tied to these properties.

Active vs Passive Elements

A passive element is one that cannot deliver more energy to the circuit than it has absorbed from it over all time. Resistors, inductors, and capacitors are passive. A resistor always dissipates energy. An inductor and capacitor store energy and can return it, but they cannot generate new energy.

An active element is capable of delivering more energy to the circuit than it receives at its terminals, because it has an internal energy source. Voltage sources, current sources, transistors, and operational amplifiers are active elements. They supply power to the circuit from an external energy source such as a battery or DC supply.

A practical test: if the average power delivered by an element over all time is negative (it delivers more power than it absorbs), it is active. If it is always non-negative (it absorbs or at most returns stored energy), it is passive.

Linear vs Non-Linear Elements

A linear element satisfies both the homogeneity property (scaling the input scales the output by the same factor) and the additivity property (response to a sum of inputs equals the sum of individual responses). Together these form the superposition principle. Resistors (V = IR), ideal inductors (V = L dI/dt), and ideal capacitors (I = C dV/dt) are linear.

A non-linear element does not satisfy superposition. Its V-I relationship is not a straight line through the origin. Diodes, transistors, and varistors are non-linear. The diode follows the Shockley equation: I = Is (e^(V/VT) - 1), which is clearly exponential and non-linear.

Most network analysis theorems (Superposition, Thevenin, Norton) apply strictly only to linear networks. When a circuit contains non-linear elements, these theorems cannot be directly applied without linearization techniques like small-signal models.

Mathematical Expression

The power absorbed by a two-terminal element is p(t) = v(t) x i(t), using the passive sign convention where current enters the positive terminal. If this product is positive, the element absorbs power. If negative, the element delivers power to the rest of the circuit. The total energy absorbed over all time is the integral of p(t) from minus infinity to plus infinity.

Given:
A resistor R = 10 ohm, current i(t) = 2 sin(100t) A
Verify linearity and calculate power

Why this formula applies:
For a linear resistor, V = IR satisfies V(ki) = kV(i) for any constant k
Power is p(t) = v(t) x i(t) = i^2 x R

Formula:
p(t) = i^2(t) x R

Substitution:
p(t) = [2 sin(100t)]^2 x 10

Calculation:
p(t) = 4 sin^2(100t) x 10
p(t) = 40 sin^2(100t) W
Average power = 40/2 = 20 W (since avg of sin^2 = 0.5)

Final Answer:
Average power absorbed by resistor = 20 W (always positive, confirms passive element)

Exam Tip: GATE frequently asks whether a specific element is active or passive using power calculation. An element is active if it can deliver net energy. Superposition applies only to linear circuits. A transformer is passive but it is also linear and bilateral.

Mechanism: Key Properties Summary

• Passive elements (R, L, C): Cannot supply more energy than they receive. Net power absorption is always non-negative for resistors.
• Active elements (voltage source, current source, transistor): Have an internal energy source. They supply energy to the circuit.
• Linear elements: Satisfy superposition. V-I relationship is a proportional equation (for resistor) or a linear differential equation (for L and C).
• Non-linear elements: Diode, BJT, MOSFET. Require graphical or iterative methods for DC analysis. Superposition cannot be applied directly.
• Bilateral elements: Those whose V-I relation is the same regardless of current direction. Resistors are bilateral. Diodes are unilateral.
• Lumped elements: Entire electrical effect is concentrated at one point. Circuit analysis assumes lumped elements unless the physical size is comparable to wavelength.

Quick Revision

• Passive: cannot deliver more energy than absorbed. Active: has internal energy source, can deliver power.
• Linear: obeys superposition (homogeneity + additivity). Non-linear: does not.
• Resistor, inductor, capacitor are passive and linear (ideal case).
• Diode and transistor are active (transistor) or passive but non-linear (diode).
• Superposition theorem applies only to linear circuits.
• Exam trap: An inductor or capacitor is passive even though it stores energy. Passive means it cannot supply more energy than it absorbed.
• Bilateral vs unilateral: resistor is bilateral, diode is unilateral (current flows mainly in one direction).