Editing Colpitts oscillator (section)

==Theory==
[[Image:Colpitts ideal model.svg|thumb|right|250px|Figure 4: Ideal Colpitts oscillator model (common-collector configuration)]]

One method of oscillator analysis is to determine the input impedance of an input port neglecting any reactive components. If the impedance yields a [[negative resistance]] term, oscillation is possible. This method will be used here to determine conditions of oscillation and the frequency of oscillation.

An ideal model is shown to the right. This configuration models the common collector circuit in the section above. For initial analysis, parasitic elements and device non-linearities will be ignored. These terms can be included later in a more rigorous analysis. Even with these approximations, acceptable comparison with experimental results is possible.

Ignoring the inductor, the input [[Electrical impedance|impedance]] at the base can be written as

: <math>Z_\text{in} = \frac{v_1}{i_1},</math>

where <math>v_1</math> is the input voltage, and <math>i_1</math> is the input current. The voltage <math>v_2</math> is given by

:<math>v_2 = i_2 Z_2,</math>

where <math>Z_2</math> is the impedance of <math>C_2</math>. The current flowing into <math>C_2</math> is <math>i_2</math>, which is the sum of two currents:

:<math>i_2 = i_1 + i_s,</math>

where <math>i_s</math> is the current supplied by the transistor. <math>i_s</math> is a dependent current source given by

:<math>i_s = g_m (v_1 - v_2),</math>

where <math>g_m</math> is the [[transconductance]] of the transistor. The input current <math>i_1</math> is given by

:<math>i_1 = \frac{v_1 - v_2}{Z_1},</math>

where <math>Z_1</math> is the impedance of <math>C_1</math>. Solving for <math>v_2</math> and substituting above yields

:<math>Z_\text{in} = Z_1 + Z_2 + g_m Z_1 Z_2.</math>

The input impedance appears as the two capacitors in series with the term <math>R_\text{in}</math>, which is proportional to the product of the two impedances:

:<math>R_\text{in} = g_m Z_1 Z_2.</math>

If <math>Z_1</math> and <math>Z_2</math> are complex and of the same sign, then <math>R_\text{in}</math> will be a [[negative resistance]]. If the impedances for <math>Z_1</math> and <math>Z_2</math> are substituted, <math>R_\text{in}</math> is

:<math>R_\text{in} = \frac{-g_m}{\omega^2 C_1 C_2}.</math>

If an inductor is connected to the input, then the circuit will oscillate if the magnitude of the negative resistance is greater than the resistance of the inductor and any stray elements. The frequency of oscillation is as given in the previous section.

For the example oscillator above, the emitter current is roughly 1&nbsp;[[Ampere|mA]]. The transconductance is roughly 40&nbsp;[[Siemens (unit)|mS]]. Given all other values, the input resistance is roughly

:<math>R_\text{in} = -30\ \Omega.</math>

This value should be sufficient to overcome any positive resistance in the circuit. By inspection, oscillation is more likely for larger values of transconductance and smaller values of capacitance.  A more complicated analysis of the common-base oscillator reveals that a low-frequency amplifier voltage gain must be at least 4 to achieve oscillation.<ref>Razavi, B.  Design of Analog CMOS Integrated Circuits. McGraw-Hill. 2001.</ref>  The low-frequency gain is given by

:<math>A_v = g_m R_p \ge 4.</math>

[[File:Oscillator comparison.svg|thumb|Figure 5: Comparison of Hartley and Colpitts oscillators]]
If the two capacitors are replaced by inductors, and magnetic coupling is ignored, the circuit becomes a [[Hartley oscillator]]. In that case, the input impedance is the sum of the two inductors and a negative resistance given by

:<math>R_\text{in} = -g_m \omega^2 L_1 L_2.</math>

In the Hartley circuit, oscillation is more likely for larger values of transconductance and larger values of inductance.

The above analysis also describes the behavior of the [[Pierce oscillator]]. The Pierce oscillator, with two capacitors and one inductor, is equivalent to the Colpitts oscillator.<ref>Theron Jones. [http://www.maximintegrated.com/en/app-notes/index.mvp/id/5265 "Design a Crystal Oscillator to Match Your Application"] {{Webarchive|url=https://web.archive.org/web/20150122233938/http://www.maximintegrated.com/en/app-notes/index.mvp/id/5265 |date=2015-01-22 }}. Maxim tutorial 5265 Sep 18, 2012, Maxim Integrated Products, Inc.</ref> Equivalence can be shown by choosing the junction of the two capacitors as the ground point. An electrical dual of the standard Pierce oscillator using two inductors and one capacitor is equivalent to the [[Hartley oscillator]].

=== Working Principle ===
A Colpitts oscillator is an electronic circuit that generates a sinusoidal waveform, typically in the radio frequency range. It uses an inductor and two capacitors in parallel to form a resonant tank circuit, which determines the oscillation frequency. The output signal from the tank circuit is fed back into the input of an amplifier, where it is amplified and fed back into the tank circuit. The feedback signal provides the necessary phase shift for sustained oscillation.<ref>{{Cite web |last=Ayushi |date=2023-10-04 |title=Colpitts Oscillator - Principle, Working, Circuit Diagram |url=https://www.electricalvolt.com/colpitts-oscillator/ |access-date=2023-12-27 |website=Electrical Volt |language=en-us}}</ref>

The working principle of a Colpitts oscillator can be explained as follows:

* When the power supply is switched on, the capacitors <math>C_1</math> and <math>C_2</math> start charging through the resistor <math>R_1</math> and <math>R_2</math>. The voltage across <math>C_2</math> is coupled to the base of the transistor through the capacitor <math>C_\text{in}</math>.
* The transistor amplifies the input signal and produces an inverted output signal at the collector. The output signal is coupled to the tank circuit through the capacitor <math>C_\text{out}</math>.
* The tank circuit resonates at its natural frequency, which is given by:

:<math>f = \frac{1}{2 \pi \sqrt{LC_t}}</math>

Where:

* f = frequency of oscillation
* L = inductance of the inductor
* <math>C_t</math> = total capacitance of the series combination of <math>C_1</math> and <math>C_2</math>, given by:

:<math>C_t = \frac{C_1 C_2}{C_1 + C_2}</math>

* The resonant frequency is independent of the values of <math>C_1</math> and <math>C_2</math>, but depends on their ratio. The ratio of <math>C_1</math> and <math>C_2</math> also affects the feedback gain and the stability of the oscillator.
* The voltage across the inductor L is in phase with the voltage across <math>C_2</math>, and 180 degrees out of phase with the voltage across <math>C_1</math>. Therefore, the voltage at the junction of <math>C_1</math> and <math>C_2</math> is 180 degrees out of phase with the voltage at the collector of the transistor. This voltage is fed back to the base of the transistor through <math>C_\text{in}</math>, providing another 180 degrees phase shift. Thus, the total phase shift around the loop is 360 degrees, which is equivalent to zero degrees. This satisfies the Barkhausen criterion for oscillation.
* The amplitude of the oscillation depends on the feedback gain and the losses in the tank circuit. The feedback gain should be equal to or slightly greater than the losses for sustained oscillation. The feedback gain can be adjusted by varying the values of <math>R_1</math> and <math>R_2</math>, or by using a variable capacitor in place of <math>C_1</math> or <math>C_2</math>.<ref>{{Cite web |last=Ayushi |date=2023-10-04 |title=Colpitts Oscillator - Principle, Working, Circuit Diagram |url=https://www.electricalvolt.com/colpitts-oscillator/ |access-date=2023-12-27 |website=Electrical Volt |language=en-us}}</ref>

The Colpitts oscillator is widely used in various applications, such as RF communication systems, signal generators, and electronic testing equipment. It has better frequency stability than the Hartley oscillator, which uses a tapped inductor instead of a tapped capacitor in the tank circuit.<ref>{{Cite web |date=2009-10-12 |title=Colpitts Oscillator Circuit diagram & working. Frequency equation. Colpitts oscillator using opamp |url=https://www.circuitstoday.com/colpitts-oscillator |access-date=2023-12-27 |website=Electronic Circuits and Diagrams-Electronic Projects and Design |language=en-US}}</ref> However, the Colpitts oscillator may require a higher supply voltage and a larger coupling capacitor than the Hartley oscillator.<ref>{{Cite web |date=2023-10-26 |title=Colpitts Oscillators {{!}} How it works, Application & Advantages |url=https://www.electricity-magnetism.org/colpitts-oscillators/ |access-date=2023-12-27 |website=Electricity - Magnetism |language=en-us}}</ref>

===Oscillation amplitude===
The amplitude of oscillation is generally difficult to predict, but it can often be accurately estimated using the [[describing function]] method.

For the common-base oscillator in Figure&nbsp;1, this approach applied to a simplified model predicts an output (collector) voltage amplitude given by<ref>Chris Toumazou, George S. Moschytz, Barrie Gilbert. [https://books.google.com/books?id=VoBIOvirkiMC&dq=the+tank+voltage+amplitude+is+calculated+to+be&pg=PA568 Trade-Offs in Analog Circuit Design: The Designer's Companion, Part 1].</ref>

:<math>
V_C = 2 I_C R_L \frac{C_2}{C_1 + C_2},
</math>

where <math>I_C</math> is the bias current, and <math>R_L</math> is the load resistance at the collector.

This assumes that the transistor does not saturate, the collector current flows in narrow pulses, and that the output voltage is sinusoidal (low distortion).

This approximate result also applies to oscillators employing different active device, such as [[MOSFET]]s and [[vacuum tubes]].