Chapter 6: Capacitors and Inductors

Inductor characteristics

1. An inductor acts like a short circuit to dc (the voltage across an inductor is 0 when the current is constant) 2. The current through an inductor cannot change instantaneously since an important property of the inductor is its opposition to the change in current flowing through it. 3. Like the ideal capacitor, the ideal inductor does not dissipate energy, and the energy stored in it can be retrieved at a later time. The inductor takes power from the circuit when storing energy and delivers power to the circuit when returning previously stored energy. 4. A practical, nonideal inductor has a significant resistive component, since the inductor is made of a conducting material with some resistance. This resistance is winding resistance, and appears in series with the inductance of the inductor, and Rw makes it both an energy storage and dissipation device. The nonideal inductor also has a winding capacitance Cw due to the capacitive coupling between the conducting coils. Cw is very small and can be ignored in most cases, except at high frequencies.

Capacitor and inductor shared special properties

1. The capacity to store energy makes them useful as temporary voltage or current sources, can generate a large amount of current or voltage for a short period of time. 2. Capacitors oppose any abrupt change in voltage, while inductors oppose any abrupt change in current. Inductors are useful for spark/arc suppression, and for converting pulsating DC voltage into relatively smooth DC voltage. 3. Capacitors and inductors are frequency sensitive, this property makes them useful for frequency discrimination.

What affects capacitance?

1. The surface area of the plates 2. The spacing between the plates 3. The permittivity of the material

Capacitor characteristics

1. When the voltage across a capacitor is not changing with time (i.e. dc voltage), the current through the capacitor is 0, thus a capacitor is an open circuit to dc 2. The voltage on the capacitor must be continuous, so the voltage on it cannot change abruptly 3. The ideal capacitor does not dissipate energy, it takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit. 4. A real, nonideal capacitor has a parallel-model leakage resistance (resistor parallel to capacitor) which may be as high as 100M-ohms and can be neglected for most practical applications

Series equivalent capacitance

1/Ceq=1/C₁+1/C₂+1/C₃... Ceq=C₁C₂C₃/(C₁C₂+C₂C₃+C₁C₃)

Parallel equivalent inductance

1/Leq=1/L₁+1/L₂+1/L₃... Leq=L₁L₂L₃/(L₁L₂+L₁L₃+L₂L₃)

Integrator

An op amp circuit whose output is proportional to the integral of the input signal

Differentiator

An op amp circuit whose output is proportional to the rate of change of the input signal

Capacitance equation

C=Q/V=εA/d

Parallel equivalent capacitance

Ceq=C₁+C₂+C₃...

Capacitor

Consists of two conducting plates separated by an insulator (or dielectric)

Series equivalent inductance

Leq=L₁+L₂+L₃...

Instantaneous power delivered to the capacitor

P=Vi=CV*dV/dt

Inductor

Passive element designed to store energy in its magnetic field. It consists of a coil of conducting wire

Inductance

The property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H)

Capictance

The ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F)

Voltage-current relation of the capacitor

V=1/C*[t₀ to t]∫idt+v(t₀)

Voltage across inductor

V=L*di/dt

Output voltage of integrator

V₀=-1/RC*[0 to t]∫Vin(t)dt

Output voltage of differentiator

V₀=-RC*dvin/dt

Current-voltage relationship of inductor

i=1/L*[t₀ to t]∫v(t)dt+i(t₀)

Current-voltage relationship of the capacitor

i=C*dV/dt

Energy stored in capacitor

w=½CV²

Energy stored in inductor

w=½Li²