Colpitts oscillator

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Colpitts oscillator

A Colpitts oscillator, named after its inventor Edwin H. Colpitts, is one of a number of designs for electronic oscillator circuits. One of the key features of this type of oscillator is its simplicity and robustness. It is not difficult to achieve satisfactory results with little effort.

A Colpitts oscillator is the electrical dual of a Hartley oscillator. In the Colpitts circuit, two capacitors and one inductor determine the frequency of oscillation. The Hartley circuit uses two inductors (or a tapped single inductor) and one capacitor. (Note: the capacitor can be a variable device by using a varactor).

The following schematic is an example using an NPN transistor in the common base configuration. Frequency of oscillation is roughly 50 MHz:

Image:NPN_Colpitts_oscillator_collector_coil.png

The following schematic is a common collector version of the same oscillator:

Image:NPN_Colpitts_oscillator_base_coil.png

The bipolar transistor could be replaced with a JFET or other active device capable of producing gain at the oscillation frequency.

The ideal frequency of oscillation is given by this equation:

$f_0 = {1 \over 2 \pi \sqrt {L_1 \cdot \left ({ C_1 \cdot C_2 \over C_1 + C_2 }\right ) }}$

A simplified version of the formula is this:

$f_0 = {0.159 \over \sqrt {L_1 \cdot \left (C \right) }}$

$C = { C_1 \cdot C_2 \over C_1 + C_2 }$

In this simplified version, remember that L is in µH (microhenries), C is in µF (microfarads), and f is in MHz (megahertz).

Given the values in the circuit above, the equation predicts a frequency of roughly 58 MHz. The real circuit will oscillate at a slightly lower frequency due to junction capacitances of the transistor and possibly other stray capacitances.

Analysis

Colpitts oscillator model

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 impedance can be written as

$Z_{in} = \frac{v_1}{i_1}$

Where $v 1$ is the input voltage and $i 1$ is the input current. The voltage $v 2$ is given by

$v 2 = i 2 Z 2$

Where $Z 2$ is the impedance of $C 2$ . The current flowing into $C 2$ is $i 2$ , which is the sum of two currents:

$i 2 = i 1 + i s$

Where $i s$ is the current supplied by the transistor. $i s$ is a dependent current source given by

$i_s = g_m \left ( v_1 - v_2 \right )$

Where $g m$ is the transconductance of the transistor. The input current $i 1$ is given by

$i_1 = \frac{v_1 - v_2}{Z_1}$

Where $Z 1$ is the impedance of $C 1$ . Solving for $v 2$ and substituting above yields

$Z i n = Z 1 + Z 2 + g m Z 1 Z 2$

The input impedance appears as the two capacitors in series with an interesting term, $R i n$ which is proportional to the product of the two impedances:

$R i n = g m Z 1 Z 2$

If $Z 1$ and $Z 2$ are complex and of the same sign, $R i n$ will be a negative resistance. If the impedances for $Z 1$ and $Z 2$ are substituted, $R i n$ is

$R_{in} = \frac{-g_m}{\omega ^ 2 C_1 C_2}$

If an inductor is connected to the input, 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 prevous section.

For the example oscillator above, the emitter current is roughly 1 mA. The transconductance is roughly 40 mS. Given all other values, the input resistance is roughly

$R_{in} = -30 \ \Omega$

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/or smaller values of capacitance.

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:

$R i n = - g m ω 2 L 1 L 2$

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

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