--- title: "Derivatives at t=0+: 1 H, 1 F, 10 V" subject: "Electrical Circuit Theory" chapter: 2 key: "semester_3/electrical_circuit_theory:initial-values-1h-1f-10v" source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/initial-values-1h-1f-10v site: BCT Engineering --- # Derivatives at t=0+: 1 H, 1 F, 10 V Electrical Circuit Theory · Chapter 2: Initial Conditions - Asked in: 2074 Chaitra, 8 marks - Video: [switch is closed at t=0. Determine 𝒊, 𝒅𝒊/𝒅𝒕, (𝒅^𝟐 𝒊)/(𝒅𝒕^𝟐 ) , (𝒅^𝟑 𝒊)/(𝒅𝒕^𝟑 ) at 𝒕=𝟎^+.](https://www.youtube.com/watch?v=RBezQaCkMyg) — Engg-Course-Made-Easy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/initial-values-1h-1f-10v - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-2 ## Question Obtain the value of i1, i2, di1/dt, di2/dt, d²i1/dt² and d²i2/dt² at t = 0⁺, if the switch is closed at t = 0 in the circuit shown in the figure below. [Figure: 10 V source; switch (closes at t = 0) and a 1 H inductor in the top wire of the left mesh (i1); a 10 Ω resistor in the middle vertical branch; a 10 Ω resistor in the top wire of the right mesh (i2) and a 1 F capacitor on the right] ## Short answer At t = 0+, i1 = i2 = 0, di1/dt = 10 A/s, di2/dt = 5 A/s, d²i1/dt² = −50 A/s², d²i2/dt² = −25.25 A/s². Inductor current and capacitor voltage start at zero; derivatives from mesh equations. ## Answer **Assumption:** the inductor and capacitor are initially relaxed. i₁ flows in the left mesh (10 V, switch, L = 1 H, common R = 10 Ω); i₂ flows in the right mesh (common 10 Ω, 10 Ω, C = 1 F). #### Mesh equations (t > 0) ```text Mesh 1: 10 = L di₁/dt + 10(i₁ − i₂) Mesh 2: 10(i₂ − i₁) + 10 i₂ + (1/C)∫i₂ dt = 0 ``` #### At t = 0⁺ - Inductor current cannot change: i₁(0⁺) = 0. - Capacitor voltage cannot change: v_C(0⁺) = 0. ```text Mesh 2: 20 i₂ − 10 i₁ + 0 = 0 → i₂(0⁺) = 0 ``` #### First derivatives ```text Mesh 1: 10 = 1·i₁′ + 10(0 − 0) → di₁/dt(0⁺) = 10 A/s Mesh 2 differentiated: 20 i₂′ − 10 i₁′ + i₂/C = 0 20 i₂′ = 10(10) − 0 → di₂/dt(0⁺) = 5 A/s ``` #### Second derivatives ```text Mesh 1 differentiated: 0 = i₁″ + 10(i₁′ − i₂′) i₁″ = −10(10 − 5) = −50 A/s² Mesh 2 differentiated twice: 20 i₂″ − 10 i₁″ + i₂′/C = 0 20 i₂″ = 10(−50) − 5 = −505 → i₂″ = −25.25 A/s² ``` | Quantity | Value at 0⁺ | |---|---| | i₁ | 0 | | i₂ | 0 | | di₁/dt | 10 A/s | | di₂/dt | 5 A/s | | d²i₁/dt² | −50 A/s² | | d²i₂/dt² | −25.25 A/s² | **Answer: as tabulated above.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is i1(0+)? A. 1 A B. 0, since inductor current cannot change instantly C. 10 A D. 0.5 A Answer: B. The inductor is initially relaxed. (Key point: Inductor current) 2. What is di1/dt at 0+? A. 0 B. 100 A/s C. 10 A/s = V/L D. 1 A/s Answer: C. With i1 = i2 = 0 mesh 1 gives 10 = L·di1/dt. (Key point: di1/dt) 3. What is di2/dt at 0+? A. 10 A/s B. 20 A/s C. 0 D. 5 A/s Answer: D. From 20 i2′ = 10 i1′ = 100 you get i2′ = 5. (Key point: di2/dt) ## Close variants set in other exams - [Derivatives at t=0+: 2 μF, 10 V, 1 H](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-values-2uf-10v-10ohm) — Electrical Circuit Theory, chapter 2, asked 2× ## Related in Electrical Circuit Theory - [Laplace: i1, i2 in unenergised two-mesh network](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-two-mesh-unenergised-100v-10ohm) — Electrical Circuit Theory, chapter 4, asked 3× - [Derivatives at t=0+: 2 H, 100 V, 1 μF](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-values-2h-100v-1uf) — Electrical Circuit Theory, chapter 2 - [Laplace: i1, i2 with 2 V charged capacitor](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-i1-i2-charged-c-2v) — Electrical Circuit Theory, chapter 4, asked 2× - [i(0+) and di/dt(0+) with sinusoidal source](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-values-ac-source-switch-k) — Electrical Circuit Theory, chapter 2 - [Classical: i1, i2 with two R-L branches](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aclassical-i1-i2-100v-10ohm-1h) — Electrical Circuit Theory, chapter 3 - [Initial values with switch open to closed (10 Ω, 15 Ω, 2 F)](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-conditions-switch-open-10-15) — Electrical Circuit Theory, chapter 2 ## Related in other subjects - [Series-parallel circuit with I1, I2, I3 phasors](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Aseries-parallel-ac-v100-z1-z2-z3) — Basic Electrical Engineering, chapter 6, asked 2× - [Mesh current method with current source](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Amesh-current-10-2a-3-50v) — Basic Electrical Engineering, chapter 2 - [Mesh analysis with 3 A source](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Amesh-analysis-3a-source) — Basic Electrical and Electronics Engineering, chapter 1