Lecture 18 Inductance and Capacitance ECE 205 Prof

Lecture 18 Inductance and Capacitance ECE 205 Prof. Ali Keyhani

Capacitor • A dynamic element that involves variation of an electric field produced by voltage • In its simples from it is constructed by two parallel metal plates and a dielectric material in between the plates

Capacitor • Circuit symbol: Some examples of capacitor

Capacitor Source: Wikipedia

Capacitor • When there is a voltage difference between the conductor plates, an electric filed is produced in insulator • The electric field results in charge separation with equal and opposite charges on the conducting plates • The resulting electric field:

Capacitance • Relationship between the voltage across the capacitor and the electric field: • Therefore the charge can be found : Parameter C is called Capacitance of the capacitor Unit: Farad (F)

I-V relationship Integrating the above equation yields the integral form of the i-v relationship:

Power and Energy • Capacitor power: • Stored energy: • Note: – Current in capacitor is zero unless voltage is changing – The capacitor voltage is continuous: a sudden change in capacitor voltage requires infinite current which is impossible – The capacitor absorbs power when storing energy and releases the energy when it delivers power to the circuit

Example 1 The voltage given appears across a 2μF capacitor. Find the current through the capacitor

Example 1 Solution:

Example 2 If the voltage across a 0. 5 μF capacitor is: v(t)=50[sin(5000 t)+cos(2500 t)]u(t) Find the expression for the current through the capacitor.

Example 2 Solution:

Inductor • Inductor is a dynamic element involving the time variation of a magnetic field by current • When a wire is wound into a coil the magnetic flux ɸ concentrates along the axis of the coil: • Flux linkage (webers (Wb) ):

Inductor • Using flux linkage instead of flux: • Parameter L is called inductance of the coil and its unit is henry (H) • Circuit symbol

Inductor Source: Wikipedia

i-v relationship • Faraday’s law: • This will lead us to the i-v relationship: • By integrating this equation the integral form of the characteristic is found:

Power and Energy - Power: - Energy: integrating the power equation yields energy

• Note: • The voltage across the inductor is zero unless the current is changing • Current through the inductor is continuous since a sudden change in current requires infinite voltage and power which is impossible • The inductor absorbs power and stores energy and delivers power when releases the energy to the circuit

Example 3 The voltage and current across an inductor are given. – Find the inductance. – The energy stored at time t=1 sec if the initial energy is zero.

Dynamic OP AMP Circuits • The inverting OP AMP integrator

Dynamic OP AMP Circuits • The inverting OP AMP differentiator

Example 4 The input to the circuit is vs(t)=VA cos(2000 t) and the OP AMP saturates at ± 15 V. Find: – The expression for the output – The Maximum value for VA

Equivalent Capacitance • Parallel capacitors

Equivalent Capacitance • Series capacitors

Equivalent Inductor • Parallel inductors • Series inductors