linear and nonlinear elements

linear and nonlinear elements

1. What Linear vs Non-linear Means (superposition test)

• Linear element: V-I relation is linear (straight line through origin) and obeys superposition + homogeneity:

i.e., if $V_1→I_1$ and $V_2→I_2$, then $c_1V_1+c_2V_2 →c_1I_1+c_2I_2$. Also $V∝I$ (Ohm's law with constant $R$).

• Examples: Ideal resistor (constant $R$), inductor (constant $L$, $V=L di/dt$ linear differential), capacitor (constant $C$). These are linear bilateral (if not saturated).
• Non-linear element: V-I is not a straight line; $V/I$ ratio changes with $V$ or $I$; does not obey superposition.
  • Examples: Diode ($I=I_0(e^{V/V_T}-1)$ exponential), transistor (exponential $I_C$ vs $V_{BE}$), iron-core inductor (saturation makes $L$ vary with $I$), filament lamp ($R$ ↑ with $T$ as $I$ heats), varistor.

2. Why Classification Matters

• Linear circuits can be solved by superposition, Thevenin/Norton, phasor analysis (all linear techniques). Non-linear cannot — need graphical, piecewise, iterative or small-signal linearization around operating point.

3. Worked Checks

(a) Resistor 10Ω: V=10V→1A, V=20V→2A (double V doubles I) ⇒ linear ✓
    Check homogeneity: if V=10V→1A, then 2·V=20V→2·I=2A ⇒ holds.

(b) Diode: V=0.5V→I≈0.1mA (hypothetical), 2·V=1.0V→I≈10mA (not 0.2mA) ⇒ non-linear ✓

(c) Iron-core coil: L at low I=10 mH, at high I core saturates L drops to 2 mH ⇒ non-linear.

(d) Superposition test: circuit with two sources V1=5V gives I1=0.5A, V2=3V gives I2=0.3A; if linear, with both V1+V2=8V, I should be 0.8A. Diode circuit fails this.

4. Exam Points

• Define linear (obeys superposition, $V∝I$ straight line) vs non-linear (curved $V-I$) — 5 marks, very frequent (often asked with example $R$ vs diode).
• Give 2 examples each (linear: $R,L,C$; non-linear: diode, transistor, iron-core) — 3-5 marks, frequent.
• Mention that $R$ can become non-linear with temperature (filament) — 3 marks.
• State that linear + bilateral ⇒ superposition applies — 3 marks.