Question

5 (a) Briefly explain the use of quartz crystal in electrical oscillators and state three advantages of quartz crystal for such circuits. [6 marks] (b) A Colpitts crystal oscillator circuit is shown in Figure Q6. The amplifying element is an inverting gate that has been turned into a linear amplifier with voltage gain magnitude $|A| = 18$. The resultant amplifier has an output resis- tance of 1 k$\Omega$ and infinite input resistance. The intrinsic shunt capacitance of the amplifier output is $C_1 = 2$ pF. $C_3$ 10 M$\Omega$ $v_o$ Figure Q6 At the operating frequency, the quartz crystal is equivalent to an inductive reactance $jX$ in the range 10 k$\Omega$ to 35 k$\Omega$ as given below: $f$ (MHz) $X$ (k$\Omega$) 3.999900 10 3.999950 16 4.000000 22 4.000050 28 4.000100 35 (i) Draw an equivalent circuit for the oscillator suitable for small-signal ac analysis near the operating frequency. [4 marks] (ii) Show that for oscillations to occur $\omega X = \frac{C_1 + C_3}{C_1 C_3}$ and $|A| > \frac{C_3}{C_1}$. [10 marks] (iii) If $C_3 = 30$ pF, find the reactance X when $\omega \approx 8 \times 10^6 s^{-1}$. Hence, [5 marks] estimate the exact oscillation frequency.

          5 (a) Briefly explain the use of quartz crystal in electrical oscillators and state three
advantages of quartz crystal for such circuits.
[6 marks]
(b) A Colpitts crystal oscillator circuit is shown in Figure Q6. The amplifying
element is an inverting gate that has been turned into a linear amplifier with
voltage gain magnitude $|A| = 18$. The resultant amplifier has an output resis-
tance of 1 k$\Omega$ and infinite input resistance. The intrinsic shunt capacitance of
the amplifier output is $C_1 = 2$ pF.
$C_3$
10 M$\Omega$
$v_o$
Figure Q6
At the operating frequency, the quartz crystal is equivalent to an inductive
reactance $jX$ in the range 10 k$\Omega$ to 35 k$\Omega$ as given below:
$f$ (MHz)  $X$ (k$\Omega$)
3.999900     10
3.999950     16
4.000000     22
4.000050     28
4.000100     35
(i) Draw an equivalent circuit for the oscillator suitable for small-signal ac
analysis near the operating frequency.
[4 marks]
(ii) Show that for oscillations to occur
$\omega X = \frac{C_1 + C_3}{C_1 C_3}$ and $|A| > \frac{C_3}{C_1}$.
[10 marks]
(iii) If $C_3 = 30$ pF, find the reactance X when $\omega \approx 8 \times 10^6 s^{-1}$. Hence,
[5 marks]
estimate the exact oscillation frequency.

5 (a) Briefly explain the use of quartz crystal in electrical oscillators and state three
advantages of quartz crystal for such circuits.
[6 marks]
(b) A Colpitts crystal oscillator circuit is shown in Figure Q6. The amplifying
element is an inverting gate that has been turned into a linear amplifier with
voltage gain magnitude |A| = 18. The resultant amplifier has an output resis-
tance of 1 kΩ and infinite input resistance. The intrinsic shunt capacitance of
the amplifier output is C1 = 2 pF.
C3
10 MΩ
vo
Figure Q6
At the operating frequency, the quartz crystal is equivalent to an inductive
reactance jX in the range 10 kΩ to 35 kΩ as given below:
f (MHz) X (kΩ)
3.999900 10
3.999950 16
4.000000 22
4.000050 28
4.000100 35
(i) Draw an equivalent circuit for the oscillator suitable for small-signal ac
analysis near the operating frequency.
[4 marks]
(ii) Show that for oscillations to occur
ω X = (C1 + C3)/(C1 C3) and |A| > (C3)/(C1).
[10 marks]
(iii) If C3 = 30 pF, find the reactance X when ω≈ 8 × 10^6 s^-1. Hence,
[5 marks]
estimate the exact oscillation frequency.

Added by Amy Z.

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