Question 3.5: The BJT Pierce oscillator shown in Figure 3.19(a) was design......
The BJT Pierce oscillator shown in Figure 3.19(a) was designed using a 2N2907 BJT with V_{C C}=15\mathrm{V},\;R_{1}=12\mathrm{~k}\Omega,\;R_{2}=6\mathrm{~k}\Omega,\;R_{E}=500\Omega,\;C_{e}=C_{c2}=0.1~\mu\mathrm{F},\;R_{1}=C_{b}=C_{b}=0.1~\mu F,C_{1}=50~{\mathrm{nF}},\;C_{2}=500~{\mathrm{pF}},\;L=300~\mu{\mathrm{H}}, and the collector resistor R_{c} was replaced by an RFC whose inductance is 20 mH. Determine the frequency of oscillation and check if the RFC, the bypass capacitor, and the coupling capacitor were properly designed.
The 2N2907 transistor lists J_{F E} (\mathrm{min})~=~30,~I_{C}(\mathrm{max})~=~800~\mathrm{mA}, and f_{T}\,=300 MHz. An analysis of the circuit with the values given shows that the Q point is located at I_{C} = 8.16 mA and V_{{C E}} = 11V. Hence, g_{m}\, = 326 mS.
The total capacitance across L is
C_{T}=\frac{C_{1}C_{2}}{C_{1} + C_{2}}=\frac{50 \times 10^{-9}(500 \times 10^{-12})}{50 \times 10^{-9} + 500 \times 10^{-12}}=495\ \mathrm{pF}
The frequency of oscillation is
f_{o}={\frac{1}{2\pi{\sqrt{L C_{T}}}}}={\frac{1}{2\pi{\sqrt{300 \times 10^{-6}(495 \times 10^{-12})}}}}=413{\mathrm{~kHz}}
The voltage feedback is
\beta(j\omega_{o})=-\frac{C_{2}}{C_{1}}=-\frac{0.5 \times 10^{-9}}{50 \times 10^{-9}}=-0.01
Hence, the gain for oscillation is
A_{v}(j\omega_{o})\geq\frac{1}{\beta(j\omega_{o})}=-\frac{1}{0.01}=-100
Since { b}_{fe}(\mathrm{min})=30, and with R_{L}=1/b_{o e}\gt 50\ \mathrm{k\Omega}, the gain condition in (3.44) is readily satisfied.
g_{m}R_{L}\geq{\frac{C_{1}}{C_{2}}} (3.44)
The resistance R_{1}\parallel R_{2}\parallel b_{i e}\approx b_{i e} appears in parallel with the reactance of C_{1}, which is X_{C_{1}}=-\ 7.7\Omega at 413 kHz. Using a typical value of 100 for b_{f e}, the typical value of b_{f e}, is 306Ω. Hence, the loading is small.
At f_{o} the reactances of the bypass and coupling capacitors are
X_{C_{b}}=X_{C_{c}}=\frac{-1}{2\pi f_{o}C_{b}}=\frac{-1}{2\pi(413 \times 10^{3})(0.1 \times 10^{-6})}=-3.85\Omega
and the reactance of the RFC is
X_{R F C}=2\pi f_{o}L=2\pi(413\times10^{3})(20\times10^{-3})=51.9\mathrm{~k\Omega}
Also, C_{c} acts as a short circuit to the ac signal, thus properly coupling the ac signal to the resonant circuit. Since R_{E}\gg|X_{C_{b}}|, the emitter capacitor acts like a short circuit at f_{o}. A finite loss of 5 Ω was assumed for the RFC. The reactance of the coil is large and appears in parallel with r_{d}. Hence, the gain condition in (3.47) is satisfied with R_{L}\approx r_{d}.
{\frac{g_{m}}{\omega_{o}^{2}R_{s}C_{1}C_{2}}}\gt 1 (3.47)
The simulation of the oscillator is shown in Figure 3.21(b). The fundamental frequency of oscillation is calculated to be f_{o} = 424.5 kHz.
