Chapter 6. Capacitance and inductance

Contents 1. Capacitors 2. Inductors 3. Capacitor and inductor combinations 4. RC operational amplifier circuits 5. Application examples

Usage of inductors and capacitors Power supply board PC motherboard Cell phone inductor capacitor

1. Capacitors Capacitance is defined to be the ratio of charge to voltage difference. Used to store charges Used to store electrostatic energy If the voltage difference between the terminals of the capacitor is equal to the supply voltage, net flow of charges becomes zero.

Generation of charges : battery Electrons(-) are absorbed. (+) charges are generated Electrons(-) are generated. (+) charges are absorbed. Electrons are generated via electro-chemical reaction.

Usage of capacitors Used to store charges Used to store electrostatic energy Slow down voltage variation

Type of capacitors

Frequently used formulas on capacitors Capacitance : Current : Voltage : Power : Energy :

iυ relation of capacitors

Example 6.2 The voltage across a 5-μF capacitor has the waveform shown in Fig. 6.4a. Determine the current waveform.

Properties of capacitors Capacitor voltage cannot change instantaneously due to finite current supply. In steady state, capacitor behaves as if open circuited.

Example 6.3 Determine the energy stored in the electric field of the capacitor in Example 6.2 at t=6 ms.

Example 6.4 The current in an initially uncharged 4μF capacitor is shown in Fig. 6.5a. Let us derive the waveforms for the voltage, power, and energy and compute the energy stored in the electric field of the capacitor at t=2 ms.

2. Inductors

Two important laws on magnetic field Current B-field Current generates magnetic field (Biot-Savart Law) Current Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law) V B-field

Biot-Savart Law Faraday’s Law

Self induced voltage = The induced voltage is generated such that it opposes the applied magnetic flux. The inductor cannot distinguish where the applied magnetic flux comes from. If the magnetic flux is due to the coil itself, it is called that the induced voltage is generated by self-inductance.

Frequently used formulas on inductors Inductance : Voltage : Current : Power : Energy :

Properties of inductors Inductor current cannot change instantaneously due to finite current supply. In steady state, inductor behaves as if short circuited.

Example 6.5 Find the total energy stored in the circuit of Fig. 6.8a.

Example 6.6 The current in a 10-mH inductor has the waveform shown in Fig. 6.9a. Determine the voltage waveform.

Example 6.7 The current in a 2-mH inductor is Determine the voltage across the inductor and the energy stored in the inductor.

Example 6.8 The voltage across a 200-mH inductor is given by the expression Let us derive the waveforms for the current, energy, and power.

Capacitor and inductor specifications Standard tolerance values are ; 5%, ; 10%, and ; 20%. Tolerances are typically 5% or 10% of the specified value.

Example 6.10 The capacitor in Fig. 6.11a is a 100-nF capacitor with a tolerance of 20%. If the voltage waveform is as shown in Fig. 6.11b, let us graph the current waveform for the minimum and maximum capacitor values.

6.3 Capacitor and Inductor Combinations = =

= =

6.4 RC Operational Amplifier Circuits Op-amp differentiator

Op-amp integrator

Example 6.17 The waveform in Fig. 6.26a is applied at the input of the differentiator circuit shown in Fig. 6.25a. If R2=1 kΩ and C1=2 μF, determine the waveform at the output of the op-amp.

Example 6.18 If the integrator shown in Fig. 6.25b has the parameters R1=5 kΩ and C2=0.2μF, determine the waveform at the op-amp output if the input waveform is given as in Fig. 6.27a and the capacitor is initially discharged.