EEE1012 Introduction to Electrical & Electronics Engineering Chapter 3: Capacitors and Inductors by Muhazam Mustapha, August 2010

Learning Outcome By the end of this chapter students are expected to: Understand the formula involving capacitors and inductors and their duality Be able to conceptually draw the I-V characteristics for capacitors and inductors

Chapter Content Units and Measures Combination Formula I-V Characteristics

Units and Measures

Capacitors Capacitors are electric devices that store static electric charge on two conducting plates when voltage is applied between them. Energy is stored as static electric field between the plates. +++++++++++++++++++++++++++ Electrostatic Field −−−−−−−−−−−−−−−−−−−−−−−−−−−

Capacitance, Charge & Voltage Capacitance: The value of a capacitor that maintains 1 Coulomb charge when applied a potential difference of 1 Volt across its terminals. Q = CV Q = charge, C = capacitance (farad), V = voltage

Inductors Inductors are electric devices that hold magnetic field within their coils when current is flowing through them. Energy is stored as the magnetic flux around the coils. Magnetic Field

Inductance, Magnetic Flux & Current Inductance: The value of an inductor that maintains 1 Weber of magnetic flux when applied a current of 1 Ampere through its terminals. Φ = LI Φ = magnetic flux, L = inductance (henry), I = current

Combination Formula – Duality Approach

Inductors Combination Inductors behave (or look) more like resistors. Hence, circuit combination involving inductors follow those of resistors. Series combination: LT = L1 + L2 + L3 L1 L2 L3 Parallel combination: L1 L2 L3

Inductors Combination Series: Current is the same for all inductors Total flux is simple summation I1 I2 I3 Φ1 Φ2 Φ3 IT = I1 = I2 = I3 ΦT = Φ1 + Φ2 + Φ3 Φ1 I1 Parallel: Total current is simple summation flux is the same for all inductors Φ2 I2 Φ3 I3 IT = I1 + I2 + I3 ΦT = Φ1 = Φ2 = Φ3

Capacitors Combination The inverse of resistors are conductors; and the dual of inductors are capacitors. If inductors behave like resistors, then capacitors might behave like conductors – in fact they are. Series combination: Parallel combination: C1 C2 C3 C1 C2 CT = C1 + C2 + C3 C3

Capacitors Combination Series: Charge is the same for all capacitors Total voltage is simple summation Q1 Q2 Q3 V1 V2 V3 QT = Q1 = Q2 = Q3 VT = V1 + V2 + V3 Parallel: Total charge is simple summation Voltage is the same for all capacitors Q1 V1 Q2 V2 Q3 V3 QT = Q1 + Q2 + Q3 VT = V1 = V2 = V3

I-V Characteristics

Capacitors At the instant of switching on, capacitors behave like a short circuit. Then charging (or discharging) process starts and stops after the maximum charging (discharging) is achieved. When maximum charging (or discharging) is achieved, i.e. steady state, capacitors behave like an open circuit. Voltage CANNOT change instantaneously, but current CAN.

Capacitors I-V relationship and power formula of a capacitor

Capacitors Charging Current: time constant, τ = RC R i i C V t τ 2τ 3τ 4τ 5τ Charging period finishes after 5τ

Capacitors Charging Voltage: time constant, τ = RC R v C v V V t τ 2τ 3τ 4τ 5τ Charging period finishes after 5τ

Capacitors Discharging Current: time constant, τ = RC R i C V i Discharging period finishes after 5τ t τ 2τ 3τ 4τ 5τ time constant, τ = RC

Capacitors Discharging Voltage: time constant, τ = RC R v C v V V t τ 2τ 3τ 4τ 5τ Discharging period finishes after 5τ

Inductors At the instant of switching on, inductors behave like an open circuit. Then storage (or decaying) process starts and stops after the maximum (minimum) flux is achieved. When maximum (or minimum) flux is achieved, inductors behave like a short circuit. Current CANNOT change instantaneously, but voltage CAN.

Inductors I-V relationship and power formula of a inductor

Inductors Storing Current: time constant, τ = L/R R i i L V t τ 2τ 3τ 4τ 5τ Storing period finishes after 5τ

Inductors Storing Voltage: time constant, τ = L/R R v L v V V t τ 2τ 3τ 4τ 5τ Storing period finishes after 5τ

Inductors Decaying Current: time constant, τ = L/R R i i L V t τ 2τ 3τ 4τ 5τ Decaying period finishes after 5τ

Inductors Decaying Voltage: time constant, τ = L/R R L v V i Discharging period finishes after 5τ t τ 2τ 3τ 4τ 5τ time constant, τ = L/R −V