Lecture 10: Inductance & Capacitance Nilsson 6.1-6.5 ENG17 (Sec. 1): Circuits I Summer 1 2014 July 8, 2014

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Inductance in the Real World

What Does an Inductor Look Like?

What Does an Inductor Do? Inductor: electrical component that opposes any change in electrical current Associates electrical current and magnetic fields Magnetic field generated by charge in motion (i.e., current) If current changes wrt time, mag field changes wrt time Varying mag field results in voltage in linked conductor

Add’l Note Inductors can store energy through the mag fields Used in spark plugs Inductors do not generate energy, so they are still passive devices

Inductors in Circuits Symbol: L Units: [H] for henrys = [Wb/A] 𝑣=𝐿 𝑑𝑖 𝑑𝑡

Determining Voltage

Current in Inductor 𝑖 𝑡 = 1 𝐿 𝑡 0 𝑡 𝑣 𝑑𝜏+𝑖( 𝑡 0 )

Power & Energy in Inductor 𝑝=𝐿𝑖 𝑑𝑖 𝑑𝑡 𝑤= 1 2 𝐿 𝑖 2

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Capacitance in the Real World

What Does a Capacitor Look Like?

What Does a Capacitor Do? Capacitor: electrical component that consists of 2 conductors separated by dielectric “Stores” electric charge (like a battery) Associated with electric fields, due to separation of charge (voltage) If voltage varies with time, E-field varies with time Varying E-field generates displacement current in space occupied by field Capacitance relates displacement current to voltage

Capacitors in Circuits Symbol: C Units: [F] for farads = [C/V] 𝑖=𝐶 𝑑𝑣 𝑑𝑡

Capacitor Voltage, Power, & Energy 𝑣 𝑡 = 1 𝐶 𝑡 0 𝑡 𝑖 𝑑𝜏+𝑣( 𝑡 0 ) 𝑝=𝑣𝑖=𝐶𝑣 𝑑𝑣 𝑑𝑡 𝑤= 1 2 𝐶 𝑣 2

Example 𝑣 𝑡 = 0, 𝑡≤0 𝑠 4𝑡, 0𝑠 ≤𝑡 ≤1 𝑠 4 𝑒 −(𝑡−1) , 𝑡≥1 𝑠

Example Obtain the energy stored in each capacitor under DC conditions.

Example Under DC conditions, find i, vc, and iL. Also, find the energy stored in the capacitor and inductor.

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Inductors Series-Parallel

Capacitors Series-Parallel

Example

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Magnetically Coupled Coils

Self- and Mutual-Inductance

Dot Convention

Mesh Current

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Self Inductance

Mutual Inductance i.t.o. Self Inductance

Overview Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored

Energy Stored in Magnetically-Coupled Coils

Recap Inductors Capacitors Series-Parallel L/C Mutual Inductance Self Inductance Energy Stored