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).
Figures 1-3 show simple examples of bipolar junction transistor based Colpitts oscillators. The BJTs can be replaced with any active device, such as a JFET or a MOSFET, capable of producing gain at the oscillation frequency.
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Theory
Oscillation frequency
The ideal frequency of oscillation for the circuits in Figures 1 and 2 are given by the equation:
- <math>
f_0 = {1 \over 2 \pi \sqrt {L \cdot \left ({ C_1 \cdot C_2 \over C_1 + C_2 }\right ) }} </math> where the series combination of C1 and C2 creates the effective capacitance of the LC tank. Real circuits will oscillate at a slightly lower frequency due to junction capacitances of the transistor and possibly other stray capacitances.
Instability criteria
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 <math>Z_{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 \left ( v_1 - v_2 \right )</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_{in} = Z_1 + Z_2 + g_m Z_1 Z_2</math> The input impedance appears as the two capacitors in series with an interesting term, <math>R_{in}</math> which is proportional to the product of the two impedances: <math>R_{in} = g_m \cdot Z_1 \cdot Z_2</math> If <math>Z_1</math> and <math>Z_2</math> are complex and of the same sign, <math>R_{in}</math> will be a negative resistance. If the impedances for <math>Z_1</math> and <math>Z_2</math> are substituted, <math>R_{in}</math> is <math>R_{in} = \frac{-g_m}{\omega ^ 2 C_1 C_2}</math> 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 previous 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 <math>R_{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/or 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 four to achieve oscillation.[1] The low frequency gain is given by: <math>A_v = g_m \cdot R_p \ge 4</math> 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_{in} = -g_m \omega ^ 2 L_1 L_2</math> In the Hartley circuit, oscillation is more likely for larger values of transconductance and/or larger values of inductance.
Oscillation amplitude
The amplitude of oscillation is generally difficult to find, but it can often be accurately estimated using the describing function method.
References
- ^ Razavi, B. Design of Analog CMOS Integrated Circuits. McGraw-Hill. 2001.
- Lee, T. The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge University Press. 2004.
