20-Bio-B6 Analytical Biochemistry · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2013 — 04-Bio-B6 Bioinstrumentation. Three hours, open book, non-communicating calculator permitted. Six questions of equal value (25 marks each); four constitute a complete paper and only the first four appearing in the answer book are marked. All six are solved here as a complete study resource. Every question is a design/essay question (block-diagram instrumentation-system design, or descriptive explanation).
Reference texts (the books a candidate should have reviewed for this subject):
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
A piezoelectric element behaves electrically as a charge (or current) source in parallel with its own internal capacitance and a very high leakage resistance; when connected through a cable to an amplifier, the cable and amplifier-input impedances complete the model.
An applied force $F(t)$ generates a charge $Q=d\cdot F$ ($d$ = piezoelectric charge sensitivity, C/N) across the crystal, so the source is modelled as a current generator $i(t)=dQ/dt=d\cdot dF/dt$ — a piezoelectric sensor inherently responds to a changing force, not a static one. $C_a$ is the crystal's own internal (dielectric) capacitance, set by its geometry and material; together with the leakage resistance it forms the source's own internal impedance. $R_a$ is the crystal's finite internal insulation/leakage resistance (very high, $10^{10}$-$10^{14}\,\Omega$, but not infinite) — it slowly bleeds off the generated charge, which is why a piezoelectric sensor cannot measure a truly static (DC) force: the output decays with time constant $R_aC_a$. $C_c$ is the parasitic capacitance of the connecting cable between transducer and amplifier; because it appears in parallel with $C_a$ it attenuates the voltage available at the amplifier input (a longer cable directly reduces sensitivity) unless a charge amplifier is used. $R_{in}$, $C_{in}$ are the amplifier's own input resistance and capacitance, which add to $R_a$ and $C_c$ respectively in setting the overall low-frequency (high-pass) cutoff $f_c=1/(2\pi R_{total}C_{total})$ of the whole measurement chain. In practice a charge amplifier (an op-amp with capacitive feedback that holds its input at virtual ground) is preferred over a simple voltage amplifier for exactly this reason: because the input node is held near 0 V, the charge is transferred essentially entirely onto the feedback capacitor regardless of $C_c$, making the output $V_{out}\approx -Q/C_f$ largely independent of cable length — this is the standard solution to the cable-capacitance sensitivity problem inherent in the equivalent circuit above.
The reversible piezoelectric effect lets one crystal both transmit and receive ultrasound: driven by an oscillating voltage it mechanically vibrates and radiates an ultrasound pulse into the tissue (converse piezoelectric effect); after the pulse, the same crystal is switched to receive, and the mechanical vibration produced by a returning echo generates a voltage (direct piezoelectric effect). Because blood is a moving scatterer (red blood cells), the echo returning from within the vessel is Doppler-shifted by an amount proportional to blood velocity, by the Doppler equation $f_d=\dfrac{2f_0v\cos\theta}{c}$, where $f_0$ is the transmitted frequency, $v$ the blood velocity, $\theta$ the angle between the ultrasound beam and the flow direction, and $c$ the speed of sound in tissue ($\approx$1540 m/s).
Block-by-block operation. A pulser/oscillator drives the transducer, through a transmit/receive (T/R) switch, with a short burst at the ultrasound carrier frequency $f_0$ (a few MHz, chosen for adequate tissue penetration vs. resolution). The T/R switch then disconnects the pulser and connects the same crystal to the receive amplifier, so the single element alternately transmits and listens (it cannot do both simultaneously). The receiver amplifier includes a range gate: because the round-trip travel time to a reflector at depth $d$ is $t=2d/c$, the receiver is only "opened" (gated on) during the narrow time window corresponding to the depth of the artery of interest, rejecting echoes from all other depths — this is exactly what lets one transducer, unchanged, measure either a superficial or a deep artery, simply by adjusting the gate delay (a short delay selects a superficial vessel, a longer delay a deep one). The gated echo is passed to a quadrature demodulator, which extracts the Doppler frequency shift $f_d$; a converter stage applies the Doppler equation (with an operator-entered or assumed beam/flow angle $\theta$) to compute blood velocity, and the result is presented as an audible Doppler tone and/or a numeric/graphical velocity display.
