Capacitance & Inductance: Storage, Energy & Dynamic Circuits

TAKEAWAYS

• Capacitors store energy in electric fields (voltage-dependent) and block DC but allow AC; inductors store energy in magnetic fields (current-dependent) and block AC but allow DC.
• Time constants (τ) determine how fast capacitors charge/discharge or inductors reach steady state: τ = RC for capacitors, τ = L/R for inductors.
• Resonance in RLC circuits occurs when X_L = X_C, maximizing current (series) or voltage (parallel) at the resonant frequency f₀ = 1/(2π√(LC)).
• Energy equations: Capacitors store ½CV², inductors store ½LI²; both release energy when discharged.
• Applications: Capacitors filter noise (power supplies), couple signals (amplifiers); inductors block high-frequency noise (chokes), store energy (switching regulators).
• Exam focus: Sketch transient responses, derive time constants, analyze series/parallel RLC resonance, and solve energy/voltage/current relationships.

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1. Capacitors: The Electric Field Energy Banks

1.1 Definition & Basic Behavior

A capacitor is a passive two-terminal device that stores electrical energy in an electric field when charged. It consists of two conductive plates separated by a dielectric (insulating material like air, paper, or ceramic).

Key properties:

• Capacitance (C): Ability to store charge per unit voltage, measured in farads (F). $$C=\frac{Q}{V}\text{(where }Q=\text{charge in coulombs, }V=\text{voltage across plates)}$$
• Voltage-current relationship: $$i(t)=C\frac{dv(t)}{dt}\text{(current depends on rate of voltage change)}$$

Shows plate separation, dielectric, and electric field lines between plates. (Image: Geek3, CC BY-SA 4.0, via Wikimedia Commons)

1.2 Types of Capacitors & Their Symbols

1.3 Capacitors in DC Circuits: Charging & Discharging

When connected to a DC source, capacitors charge/discharge exponentially with a time constant τ = RC.

Charging Process:

1. At t = 0⁺, voltage across capacitor = 0, current = I₀ = V/R (max).
2. As capacitor charges, voltage rises, current decays.
3. At t = τ = RC, voltage reaches ~63.2% of V₀, current drops to ~36.8% of I₀.
4. At t = 5τ, capacitor is fully charged (99.3% of V₀).

Discharging Process:

1. At t = 0⁺, voltage = V₀, current = I₀ = V₀/R.
2. Voltage decays exponentially; current follows.
3. At t = τ, voltage = ~36.8% of V₀.

Worked Example: Charging a Capacitor in a Flashlight A 1000 µF capacitor is charged by a 9V battery through a 50 Ω resistor.

• Time constant (τ): $\tau =RC=1000\times 10^{-6}\times 50=0.05\text{ s}$.
• Voltage after 0.1 s: $v(t)=V_0(1-e^{-t/\tau })=9(1-e^{-0.1/0.05})\approx 8.64\text{ V}$.
• Current at t = 0.025 s: $i(t)=\frac{V_0}{R}{e}^{-t/\tau }=\frac{9}{50}{e}^{-0.025/0.05}\approx 0.1\text{ A}$.

1.4 Capacitors in AC Circuits: Reactance & Phase Shift

In AC circuits, capacitors introduce capacitive reactance (X_C), which opposes changes in voltage: $$X_C=\frac{1}{2\pi fC}\text{(in ohms, where }f=\text{frequency in Hz)}$$

• X_C decreases with frequency: At high frequencies, capacitors act like short circuits; at low frequencies, they act like open circuits.
• Phase relationship: Current leads voltage by 90° in a purely capacitive circuit.

Worked Example: Capacitor in a Smartphone Power Supply A 10 µF capacitor is used to filter noise in a 50 Hz power supply.

• X_C: $X_C=\frac{1}{2\pi \times 50\times 10\times 10^{-6}}\approx 318.3\text{ Ω}$.
• If connected to a 12V AC source, the rms current is $I=\frac{V}{X_C}=\frac{12}{318.3}\approx 0.0377\text{ A}$.

1.5 Energy Storage in Capacitors

Energy stored in a capacitor: $$W=\frac{1}{2}C{V}^2\text{(joules)}$$ Example: A 100 µF capacitor charged to 10V stores: $$W=\frac{1}{2}\times 100\times 10^{-6}\times 10^2=0.05\text{ J}$$

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2. Inductors: The Magnetic Field Energy Banks

2.1 Definition & Basic Behavior

An inductor is a passive two-terminal device that stores energy in a magnetic field when current flows through it. It consists of a coil of wire, often wound around a core (air, iron, or ferrite).

Key properties:

• Inductance (L): Opposition to changes in current, measured in henries (H). $$L=\frac{N\Phi }{I}\text{(where }N=\text{number of turns, }\Phi =\text{magnetic flux)}$$
• Voltage-current relationship: $$v(t)=L\frac{di(t)}{dt}\text{(voltage depends on rate of current change)}$$

2.2 Types of Inductors & Their Symbols

2.3 Inductors in DC Circuits: Growth & Decay of Current

When connected to DC, inductors resist changes in current with a time constant τ = L/R.

Current Growth (Charging):

1. At t = 0⁺, current = 0, voltage = V₀ (max).
2. Current rises exponentially; voltage decays.
3. At t = τ = L/R, current = ~63.2% of I₀, voltage = ~36.8% of V₀.
4. At t = 5τ, current reaches ~99.3% of I₀.

Current Decay (Discharging):

1. At t = 0⁺, current = I₀, voltage = L(I₀/R).
2. Current decays exponentially; voltage follows.

Worked Example: Inductor in a Motor Starter A 100 mH inductor with 10 Ω resistance is connected to a 12V DC supply.

• Time constant (τ): $\tau =\frac{L}{R}=\frac{100\times 10^{-3}}{10}=0.01\text{ s}$.
• Current after 0.02 s: $i(t)=\frac{V_0}{R}(1-e^{-t/\tau })=\frac{12}{10}(1-e^{-0.02/0.01})\approx 1.08\text{ A}$.

2.4 Inductors in AC Circuits: Reactance & Phase Shift

In AC circuits, inductors introduce inductive reactance (X_L), which opposes changes in current: $$X_L=2\pi fL\text{(in ohms)}$$

• X_L increases with frequency: At high frequencies, inductors act like open circuits; at low frequencies, they act like short circuits.
• Phase relationship: Voltage leads current by 90° in a purely inductive circuit.

Worked Example: Inductor in a Washing Machine Motor A 50 mH inductor is used in a 60 Hz motor.

• X_L: $X_L=2\pi \times 60\times 50\times 10^{-3}\approx 18.85\text{ Ω}$.
• If connected to a 230V AC source, the rms current is $I=\frac{V}{X_L}=\frac{230}{18.85}\approx 12.2\text{ A}$.

2.5 Energy Storage in Inductors

Energy stored in an inductor: $$W=\frac{1}{2}L{I}^2\text{(joules)}$$ Example: A 200 mH inductor carrying 5A stores: $$W=\frac{1}{2}\times 200\times 10^{-3}\times 5^2=2.5\text{ J}$$

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3. RLC Circuits: The Dance of Resistance, Inductance & Capacitance

When resistors (R), inductors (L), and capacitors (C) are combined, they form RLC circuits, which exhibit resonance, transient responses, and frequency-dependent behavior.

3.1 Series RLC Circuits

In a series RLC circuit, the total impedance (Z) is: $$Z=R+j(X_L-X_C)=R+j(2\pi fL-\frac{1}{2\pi fC})$$

• Resonance occurs when X_L = X_C, i.e., $$2\pi f_{0}L=\frac{1}{2\pi f_{0}C}⟹f_0=\frac{1}{2\pi \sqrt{LC}}$$
• At resonance:
  • Impedance is minimum (Z = R).
  • Current is maximum (I = V/R).
  • Phase angle is zero (voltage and current in phase).

Worked Example: Tuning a Radio (AM Band) An AM radio tuner uses a series RLC circuit with L = 250 µH and C = variable.

• To tune to 1 MHz (f₀ = 1 × 10⁶ Hz), the required capacitance is: $$C=\frac{1}{(2\pi f_0{)}^{2}L}=\frac{1}{(2\pi \times 10^6{)}^2\times 250\times 10^{-6}}\approx 20.25\text{ pF}$$

3.2 Parallel RLC Circuits

In a parallel RLC circuit, the total admittance (Y) is: $$Y=\frac{1}{R}+j(2\pi fC-\frac{1}{2\pi fL})$$

• Resonance occurs when X_L = X_C, same as series: $$f_0=\frac{1}{2\pi \sqrt{LC}}$$
• At resonance:
  • Admittance is minimum (Y = 1/R).
  • Impedance is maximum (Z = R).
  • Voltage is maximum across the parallel branch.

Worked Example: Voltage Stabilization in Power Supplies A parallel LC filter is used to stabilize voltage in a 50 Hz power supply.

• If L = 100 mH and C = 10 µF, the resonant frequency is: $$f_0=\frac{1}{2\pi \sqrt{100\times 10^{-3}\times 10\times 10^{-6}}}\approx 50.3\text{ Hz}$$
• This filters out 50 Hz noise, providing a smoother DC output.

3.3 Quality Factor (Q) & Bandwidth

The quality factor (Q) measures how selective a resonant circuit is: $$Q=\frac{f_0}{\Delta f}=\frac{X_L}{R}=\frac{X_C}{R}$$

• High Q: Narrow bandwidth, sharp resonance (used in radios).
• Low Q: Wide bandwidth, dull resonance (used in filters).

Worked Example: Q Factor of a Tuning Circuit A series RLC circuit has R = 10 Ω, L = 1 mH, C = 1 µF.

• Resonant frequency: $$f_0=\frac{1}{2\pi \sqrt{1\times 10^{-3}\times 1\times 10^{-6}}}\approx 5033\text{ Hz}$$
• Q factor: $$Q=\frac{X_L}{R}=\frac{2\pi \times 5033\times 1\times 10^{-3}}{10}\approx 31.6$$
• Bandwidth (Δf): $$\Delta f=\frac{f_0}{Q}\approx 159\text{ Hz}$$

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## In the Real World

1. Khalti & eSewa (Digital Payments)

   • Capacitors are used in power supply filtering to smooth out voltage fluctuations from the grid, ensuring stable operation of payment gateways.
   • Inductors act as chokes in switch-mode power supplies to block high-frequency noise, preventing data corruption in transactions.

2. Ncell & NTC (Telecom & Power Grid)

   • RLC filters in mobile base stations (Ncell) tune specific frequencies for signal transmission, improving call quality.
   • Power factor correction capacitors in Nepal’s grid (NTC) compensate for inductive loads (motors, transformers), reducing energy loss.

3. Pathao & Daraz (Logistics & E-Commerce)

   • Inductive sensors in automated warehouses (Daraz) detect package positions on conveyor belts for sorting.
   • Capacitive touchscreens in Pathao driver apps rely on capacitance changes to register user input.

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## Exam Tip

1. Memorize key formulas:

   • Capacitive reactance: $X_C=\frac{1}{2\pi fC}$
   • Inductive reactance: $X_L=2\pi fL$
   • Resonant frequency: $f_0=\frac{1}{2\pi \sqrt{LC}}$
   • Energy in capacitor/inductor: $W=\frac{1}{2}C{V}^2$ or $\frac{1}{2}L{I}^2$

2. Sketch transient responses:

   • Always draw exponential curves for charging/discharging capacitors/inductors, labeling τ = RC or τ = L/R.

3. Analyze resonance:

   • For series RLC, current peaks at $f_0$; for parallel RLC, voltage peaks at $f_0$.
   • Calculate Q factor and bandwidth when asked about selectivity.

4. Real-world applications:

   • Relate capacitors to filtering (e.g., power supplies, audio crossovers).
   • Relate inductors to energy storage (e.g., flyback converters in chargers) or signal tuning (e.g., radios).

5. Common pitfalls:

   • Don’t confuse X_L and X_C: $X_L$ increases with frequency; $X_C$ decreases.
   • Phase angles: Current leads voltage in capacitors, lags in inductors.
   • Units: Always check if C is in farads or µF, L in henries or mH.

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## Practice Questions (Exam-Style)

1. A 5 µF capacitor is charged to 20V and then connected across a 1 kΩ resistor. Find:

   • The time constant (τ).
   • The voltage across the capacitor after 0.01 s.
   • The energy stored initially.

2. An RLC series circuit has R = 20 Ω, L = 100 mH, C = 10 µF, and is driven by a 10V, 50 Hz source. Calculate:

   • The inductive and capacitive reactances.
   • The total impedance and phase angle.
   • The resonant frequency of the circuit.

3. Explain why capacitors are used in parallel with loads in power factor correction, while inductors are avoided.

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## Summary Table: Capacitors vs. Inductors