Question 3.4: Design the BJT Pierce oscillator in Figure 3.19(a) to oscill......

A 2N2222 BJT can be used at a bias point of V_{C E} = 10V and I_{C} = 1 mA, where the minimum beta is 50 and f_{T} = 300 MHz. Letting V_{C C} = 20V and using a typical beta of 100, the resistors are designed as follows:

R_{E}={\frac{10\%V_{C C}}{I_{C}}}={\frac{0.1(20)}{10^{-3}}}=2{\mathrm{~k\Omega}}

R_{C}={\frac{V_{C C}-V_{C E}-V_{R_{E}}}{I_{C}}}={\frac{20-10-2}{10^{-3}}}=8{\mathrm{~k\Omega}}

R\,_{T{H}}=\frac{\beta R\,_E}{10}=\frac{100(2  \times  10^{3})}{10}=20\,\mathrm{k\Omega}

V_{T H}=I_{B}R_{T H}+0.7+I_{E}R_{E}=\frac{20}{100}+0.7+10^{-3}(2\times10^{3})=2.9\mathrm{V}

R_{1}=R_{T H}\frac{V_{C C}}{V_{T H}}=20\times10^{3}\frac{20}{2.9}=138\mathrm{~k\Omega}

and

R_{2}=\frac{R_{T H}}{1-\displaystyle\frac{V_{T H}}{V_{C C}}}=\frac{20  \times  10^{3}}{1-\displaystyle\frac{2.9}{20}}=23.4~\mathrm{k\Omega}

At I_{\mathbf{C}} = 1 mA: g_{m}=10^{-3}/25\times10^{-3}=40 mS and b_{i e}\gt \beta_{m i n}/g_{m}=50/0.04=1.25\,\mathrm{k\Omega}. From (3.44), with R_{L}=R_{C}, the gain condition is

g_{m}R_{L}\geq{\frac{C_{1}}{C_{2}}}                                  (3.44)

(0.04)(8\times10^{3})=320\geq{\frac{C_{1}}{C_{2}}}                          (3.48)

The input resistance is

R_{I N}=R_{1}\parallel R_{2}\parallel b_{i e}=1.18\mathrm{~k\Omega}

The reactance of {C}_{1} should be much smaller than 1.18 kΩ, say, 50 times smaller. Using a factor of 50 gives

|X_{C_{1}}|={\frac{1}{2\pi800  \times  10^{3}C_{1}}}={\frac{1.18  \times  10^{3}}{50}}

or {C}_{1} = 8.43 nF. Then, from (3.48),

C_{2}\gt {\frac{8.42  \times  10^{-9}}{320}}=26.3\mathrm{pF}

Let {C}_{2} = 50 pF.

Finally, from (3.43), with {{C}}_{T} = 49.7 pF we obtain L = 797 μH. The magnitude of the reactance of the bypass capacitor is selected to be negligible with respect to R_{E}, say C_{b}=0.1~\mu{F}. A coupling capacitor can be placed from the collector to {C}_{2}. Hence, its reactance (magnitude) must be much smaller than that of {C}_{2}, say {C}_{2} = 100 nF. It can also be connected in series with the inductor, in which case its reactance should be much smaller than that of the inductor.

\omega_{o}=\sqrt{{\frac{1}{L C_{T}}}+{\frac{1}{C_{1}\,C_{2}\,R_{L}\,b_{i e}}}}\approx\sqrt{{\frac{I}{L C_{T}}}}                                      (3.43)

The simulation and resulting waveform are shown in Figure 3.20. The fundamental frequency of oscillation is calculated to be f_{o} = 783 kHz.