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ENGR5001 / Week 3
ENGR201 2018 B3Past paperOld spec ENGR2012:125 marks30 min

ENGR201 Engineering Analysis 2018 B3[VALID]

The quartz crystal is a small electronic component used in timing applications and sensing. It comprises a thin piece of quartz, sandwiched between two vapour-deposited electrodes, which oscillates in response to an r.f. excitation. The electrical characteristics of the crystal can be represented by the equivalent circuit in Figure B3: a resistor RR, inductor LL and capacitor CC in series, driven by a voltage v(t)v(t).

Figure B3: series R, L, C equivalent circuit of the quartz crystal, driven by v(t).
Figure B3: series R, L, C equivalent circuit of the quartz crystal, driven by v(t).
Formulas you may need
  • Component laws vR=Riv_R = Ri, vL=Ldidtv_L = L\dfrac{di}{dt}, vC=1C∫i dtv_C = \dfrac{1}{C}\displaystyle\int i\,dt; Kirchhoff's voltage law (learn this)
  • Laplace with zero initial conditions: didt→sI\dfrac{di}{dt} \to sI, ∫0ti dt→Is\displaystyle\int_0^t i\,dt \to \dfrac{I}{s}; unit step →1s\to \dfrac{1}{s} (learn this)
  • e−αtsin⁡ωt↔ω(s+α)2+ω2e^{-\alpha t}\sin\omega t \leftrightarrow \dfrac{\omega}{(s + \alpha)^2 + \omega^2} (learn this; printed in the ENGR201 Laplace table)
  • Second order form s2+2ζωns+ωn2s^2 + 2\zeta\omega_n s + \omega_n^2, damped frequency ωd=ωn1−ζ2\omega_d = \omega_n\sqrt{1 - \zeta^2} (learn this)
  1. (a)
    Show that the differential equation describing the rate of voltage change with time in terms of current for the quartz crystal is dvdt=Ld2idt2+Rdidt+iC\frac{dv}{dt} = L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{i}{C} explaining the origin of each term in your answer.
    [4]Third
  2. (b)
    Using Laplace transformation, derive an expression describing the current response I(s)I(s) in terms of the voltage stimulus V(s)V(s), given that the crystal is initially quiescent.
    [8]2:2
  3. (c)
    If the quartz crystal is subject to an impulse voltage stimulus in the form of a unit step function at t=0t = 0, determine an expression for the current response i(t)i(t).
    [8]
  4. (d)
    Given that R=10 ΩR = 10\ \Omega, L=1L = 1 mH and C=4C = 4 pF, determine the time at which the current reaches 1/e1/e of its initial value following the impulse. Approximately how many cycles will the quartz oscillator have undergone at this time?
    [5]2:2