Network Elements and electric circuits open the “Basics of Mechatronics” section (A.2) of the GATE Robotics and Automation (RA) 2027 syllabus, and they are a steady source of easy marks. The examined ideas are compact: Kirchhoff’s laws, network theorems (Thevenin, Norton, superposition, maximum power transfer), RC/RL transients, and AC steady-state analysis including resonance and power factor. This guide reviews each with its working formula and solves six GATE-style problems in full.
TABLE OF CONTENTS
Kirchhoff’s laws & network reduction
Kirchhoff’s current law (KCL) says the currents into a node sum to zero; Kirchhoff’s voltage law (KVL) says the voltages around a loop sum to zero. With Ohm’s law these solve any resistive network by node or mesh analysis. Series resistances add; parallel resistances combine by the reciprocal rule.
Network theorems
Thevenin’s theorem replaces any linear two-terminal network with a single voltage source Vth in series with a resistance Rth; Norton’s theorem gives the dual current-source form. Superposition handles multiple sources by summing their individual contributions. These reductions turn a messy network into a one-line calculation.
Maximum power transfer
The maximum power transfer theorem states that a source delivers maximum power to a load when the load resistance equals the source (Thevenin) resistance, RL = Rth. The power delivered at that match is Vth²/(4Rth), and the efficiency at match is exactly 50%.
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RC & RL transients
First-order circuits respond exponentially with a time constant: τ = RC for a capacitor circuit and τ = L/R for an inductor circuit. After one time constant a charging quantity reaches 63.2% of its final value; after five, it is essentially settled. This governs switching, timing and filtering.
AC steady state & resonance
In AC analysis, impedance combines resistance and reactance as Z = √(R² + X²), and the power factor is cosφ = R/Z. A series RLC circuit hits resonance when the inductive and capacitive reactances cancel, at f0 = 1/(2π√(LC)), where the impedance is purely resistive and minimum. Real (average) power is P = VI cosφ.
Worked examples (GATE-style)
Example 1 — parallel resistance
Find the equivalent resistance of a 6 Ω and a 3 Ω resistor in parallel.
Solution. R = (6×3)/(6+3) = 18/9 = 2 Ω.
Example 2 — maximum power transfer
A source has Thevenin voltage 12 V and Thevenin resistance 4 Ω. Find the load resistance for maximum power and the maximum power delivered.
Solution. Maximum power occurs at RL = Rth = 4 Ω.
Pmax = Vth²/(4Rth) = 144/16 = 9 W.
Example 3 — series-RLC resonance
A series RLC circuit has L = 10 mH and C = 1 µF. Find the resonant frequency.
Solution. f0 = 1/(2π√(LC)) = 1/(2π√(10×10−3 × 10−6)) = 1/(2π√(10−8)).
= 1/(2π × 10−4) = 1591.5 Hz.
Example 4 — impedance & power factor
An AC circuit has R = 3 Ω and inductive reactance XL = 4 Ω. Find the impedance and power factor.
Solution. Z = √(3² + 4²) = √25 = 5 Ω.
pf = R/Z = 3/5 = 0.6 lagging.
Example 5 — RL time constant
A coil of inductance 2 H is in series with a 1 kΩ resistor. Find the time constant.
Solution. τ = L/R = 2/1000 = 0.002 s = 2 ms.
Example 6 — three-phase power
A balanced three-phase load draws a line current of 10 A at 400 V (line) with a power factor of 0.8. Find the total real power.
Solution. P = √3 · VL · IL · cosφ = √3 × 400 × 10 × 0.8.
= 1.732 × 3200 = 5542.6 W ≈ 5.54 kW.
Key formulas
FORMULA SHEET
Parallel R: R1R2/(R1+R2)
Max power: RL = Rth, Pmax = Vth²/(4Rth)
Time constants: τ = RC, τ = L/R
Impedance: Z = √(R²+X²), pf = R/Z
Resonance: f0 = 1/(2π√(LC))
Three-phase: P = √3 VLILcosφ
Common mistakes to avoid
- Zeroing sources incorrectly when finding Rth — short voltage sources and open current sources.
- Forgetting the factor of 4 in Pmax = Vth²/(4Rth).
- Using τ = RC for an RL circuit — the inductor time constant is L/R.
- Adding reactance and resistance arithmetically — they combine as the hypotenuse, Z = √(R²+X²).
- Dropping the √3 in balanced three-phase power.
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Basics of Mechatronics — full solved-problem series
This guide is one part of the Section A.2 (Basics of Mechatronics) solved-problem series. Work through every sub-topic:
See also the umbrella guide, Basics of Mechatronics — Important Questions, and the complete GATE RA 2027 Syllabus.
Frequently asked questions
What is the condition for maximum power transfer?
Maximum power is transferred to the load when the load resistance equals the Thevenin resistance of the source network, RL = Rth. At this match the load receives Vth²/(4Rth) and the transfer efficiency is 50%.
What happens at resonance in a series RLC circuit?
At resonance the inductive and capacitive reactances cancel, so the impedance is purely resistive and minimum, the current is maximum, and the power factor is unity. The resonant frequency is f0 = 1/(2π√(LC)).
Why use Thevenin’s theorem?
Thevenin’s theorem collapses any complicated linear two-terminal network into a single source and series resistance, so you can analyse how different loads behave without re-solving the whole circuit each time.
This solved-problems guide is part of the complete GATE RA 2027 Syllabus overview and the Basics of Mechatronics syllabus guide.
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