Question 3.6: Design the Colpitts oscillator in Figure 3.22(a) to oscillat......
Design the Colpitts oscillator in Figure 3.22(a) to oscillate at 1 MHz.
Let C_{1}=C_{2}= = 10 nF, so that C_{1}\gg C_{b e} and C_{T}=5~\mathrm{nF}. It follows that |X_{C_{2}}|=|X_{C_{1}}|=15.9\Omega. Then, from (3.54)
\omega_{o}={\frac{1}{\sqrt{L C_{T}}}} (3.54)
L={\frac{1}{(2\pi \times 10^{6})^{2}\ S \times 10^{-9}}}=5.07\ \mu\mathrm{H}
If the coil {{Q}}_{U} is 60, then R_{s}=0.64\Omega.
The condition (3.56) requires that
\frac{g_{m}}{\omega_{o}^{2}R_{s}C_{1}C_{2}}\gt 1 (3.56)
g_{m}\gt (2\pi\times10^{6})^{2}(0.64)(10\times10^{-9})^{2}=2.52\ \mathrm{mS}
which is simple to attain.
A 2N2222 BJT can be selected and the bias point set at 8V, 10 mA, with
V_{C C} = 15V. Then,
R_{E}={\frac{7}{10 \times 10^{-3}}}=700\Omega
I_{B}={\frac{10 \times 10^{-3}}{100}}=100\ \mu\mathrm{A}
Let I_{R_{1}}\approx I_{R_{2}}=10I_{B}=1\mathrm{~mA}. Then,
R_{1}={\frac{15-7.7}{10^{-3}}}=7.3{\mathrm{~k}}\Omega
R_{2}={\frac{7 + 0.7}{10^{-3}}}=7.7\;\mathrm{kG}
The value of {g}_{m} is
g_{m}=\frac{10 \times 10^{-3}}{25 \times 10^{-3}}=400~\mathrm{mS}
which certainly satisfies the gain condition.
The coupling capacitor {{C}}_c is selected so that its reactance is negligible to that of the inductor at 1 MHz. Hence, let {{C}}_c = 0.1 μF.
The simulation of the oscillator is shown in Figure 3.23(a). The Q point values are: V_{C E}=8.13\mathrm{V} and I_{C}=9.78\ {\mathrm{mA}}. The oscillation signal is viewed across the emitter resistor. The fundamental frequency of oscillation is 1.023 MHz.
A modification to the configuration in Figure 3.23(a) is shown in Figure 3.23(c). The collector-tuned circuit at 1 MHz is used to couple the output signal and provide further filtering at the fundamental frequency. The simulation results are shown in Figure 3.23(d) where v_{B E} and i_{C} are shown. The plots of v_{B E} and i_{C} are useful with phase noise considerations.
The pulse width of the collector current affects the phase noise of the oscillator (see Chapter 2). Further considerations associated with controlling the collector pulse width and the phase noise of the oscillator are discussed in Section 3.7. The SSB phase noise of the oscillator is shown in Figure 3.23(e). The harmonic balance controller used for the calculation of the phase noise is shown. ADS recommend that for phase noise calculation in the harmonic balance controller use: Order = 7 and Oversample = 4.
