Sensor Signal Calibration (Electrical & Systems Engineering) Solution

Engineers design sensor systems (like pressure transducers or temperature sensors) by chaining multiple physical operations together. For example, an industrial pressure sensor converts a physical pressure input $P$ (in psi) into a voltage signal $V$ (in volts) using a linear calibration function:

This voltage is then fed into an Analog-to-Digital Converter (ADC) that maps the voltage to a 10-bit digital value $D$ represented by the function:

1. Find the composite function $(D \circ V)(P)$ that directly maps physical pressure $P$ to the digital output $D$.
2. Solve for the exact pressure $P$ (in psi) that will cause the digital output to reach a threshold alarm value of $D = 800$.

Step-by-Step Solution:

1. Substitute $V(P)$ into the digital output function $D(V)$:
   $$(D \circ V)(P) = D(V(P)) = 204.6(0.08P + 0.5)$$
   $$(D \circ V)(P) = 16.368P + 102.3$$
2. To find the pressure corresponding to $D = 800$, set $(D \circ V)(P) = 800$ and solve for $P$:
   $$16.368P + 102.3 = 800$$
   $$16.368P = 697.7$$
   $$P = \frac{697.7}{16.368} \approx 42.63\text{ psi}$$

Meaning of the Answer & Real-Life Application:

Function composition allows systems engineers to create direct mathematical mappings from raw physical phenomena (pressure) to digital representations in firmware. Solving algebraic equations on this composition enables calibration of safety thresholds—ensuring the system triggers an emergency shutdown at exactly $42.63\text{ psi}$ when the ADC reads a digital value of $800$.