Oscillators – Page 2
LC Oscillators (Hartley & Colpitts)
LC oscillators generate sinusoidal signals using an inductor (L) and capacitor (C) resonant tank circuit.
Energy continuously transfers between:
- Magnetic field of inductor
- Electric field of capacitor
The oscillation frequency depends on the resonance of the LC circuit.
1. Resonant Frequency of LC Tank Circuit
At resonance:
XL = XC
Inductive reactance:
XL = ωL
Capacitive reactance:
XC = 1 / (ωC)
At resonance:
ωL = 1 / (ωC)
Multiply both sides:
ω² = 1 / LC
Angular frequency:
ω = 1 / √LC
Frequency:
f = 1 / (2π√LC)
2. Hartley Oscillator Derivation
Hartley oscillator uses:
- Two inductors (L1, L2)
- One capacitor (C)
Total inductance:
L = L1 + L2 + 2M
Where M is mutual inductance.
Frequency of oscillation:
f = 1 / (2π √(LC))
Substitute L:
f = 1 / (2π √((L1 + L2 + 2M) C))
Condition for oscillation:
A ≥ L2 / L1
3. Colpitts Oscillator Derivation
Colpitts oscillator uses:
- One inductor (L)
- Two capacitors (C1, C2)
Equivalent capacitance:
1/C = 1/C1 + 1/C2
Therefore
C = (C1 C2) / (C1 + C2)
Frequency of oscillation:
f = 1 / (2π √(LC))
Substitute equivalent capacitance:
f = 1 / (2π √( L (C1C2/(C1+C2)) ))
Condition for oscillation:
A ≥ C2 / C1
Quick Comparison
| Oscillator | Feedback Element | Frequency |
|---|---|---|
| RC Phase Shift | Resistor + Capacitor | 1/(2πRC√6) |
| Wien Bridge | RC Bridge | 1/(2πRC) |
| Hartley | Inductive divider | 1/(2π√LC) |
| Colpitts | Capacitive divider | 1/(2π√LC) |
.png)
No comments:
Post a Comment