LC Oscillators

Flashcards about LC oscillators, Colpitts oscillators, and Hartley oscillators, including key equations and concepts.

LC Oscillator

A type of oscillator using inductors (L) and capacitors (C) to produce an oscillating signal.

Pi (π) Type Feedback Network

A feedback network consisting of three impedance elements (Z1, Z2, Z3) connected in a Pi configuration.

Node equation at vi in LC oscillator analysis

vi − vo / Z3 + vi / Z1 = 0

Node equation at vo in LC oscillator analysis

vo − vi / Z3 + vo / Z2 + vo − v o / Ro = 0

Feedback factor K(jω) in LC oscillators

K (jω) = vi / v o = Z1Z2 / (Ro (Z1 + Z2 + Z3) + Z2 (Z1 + Z3))

Impedance definition (Zn)

Zn = jXn where Xn = ωL or Xn = −1/ωC

Feedback factor K(jω) with reactances

K (jω) = −X1X2 / (jRo (X1 + X2 + X3) − X2 (X1 + X3))

Phase condition for oscillation

X1 + X2 + X3 = 0

Gain condition at resonant frequency

K (ω0) = X1 / (X1 + X3) = −X1 / X2

Colpitts Oscillator

An LC oscillator configuration using two capacitors (C1, C2) and one inductor (L3).

Impedance values for Colpitts oscillator

Z1 = 1 / jωC1 , Z2 = 1 / jωC2 , Z3 = jωL3

Resonant frequency ω0 for Colpitts oscillator

ω0 = 1 / √(L3 * (C1C2 / (C1 + C2)))

Gain (A) required for Colpitts oscillator

A = C1 / C2 = R2 / R1

Hartley Oscillator

An LC oscillator configuration using two inductors (L1, L2) and one capacitor (C3).

Impedance values for Hartley oscillator

Z1 = jωL1, Z2 = jωL2, Z3 = 1 / jωC3

Resonant frequency ω0 for Hartley oscillator

ω0 = 1 / √(C3 (L1 + L2))

Gain (A) required for Hartley oscillator

A = L2 / L1 = R2 / R1