Derivation
1. We’ll start by analyzing the feedback network formed by C1, C2, and L1. At resonance, the phase shift around the loop must be 0° or 360° for positive feedback to occur.
2. The impedance of L1 at resonance (XL = XC) is given by:
⇒XL = XC
⇒ωL1 = 1/ωC1 + 1/ωC2
3. Solving for ω (angular frequency) gives:
⇒ω² = 1/L1(C1 + C2)
4. The resonant frequency fres is:
⇒(fres)² = 1/L1(C1 + C2)
5. Solving for fres gives:
⇒fres = 1/√L1(C1 + C2)
This is the derived resonant frequency formula for the Colpitts oscillator. It represents the frequency at which the oscillator will naturally oscillate when powered on, provided that the circuit components are appropriately chosen and the oscillator conditions are met. Please note that this derivation assumes ideal components and neglects parasitic effects. In practice, variations in component values and non-ideal characteristics can affect the actual oscillation frequency of the Colpitts oscillator circuit.
Colpitts Oscillator Using Op-Amp
An oscillator in analog electronics is a circuit that generates a continuous and repetitive waveform output without the need for an external input signal. Oscillators are widely used in various applications, such as signal generation, frequency synthesis, clock generation, and modulation. The primary function of an oscillator is to convert direct current (DC) power from a power supply into an alternating current (AC) waveform at a specific frequency.
The generated waveform can be sinusoidal, square, triangular, or other types, depending on the oscillator’s configuration and design. Oscillators are crucial components in electronic systems, serving as timing references, frequency sources, and waveform generators. They play a significant role in communication systems, audio devices, control systems, and many other applications. Here are some key points to understand about oscillators: