Linearization, Nulling, and Digital Adjustment

Linearization Techniques for Non-Linear Transducers

Many physical sensors exhibit non-linear physical responses that cannot be corrected by simple two-point zero and span adjustments.

Common Non-Linear Characteristics in Metrology

1. Thermocouples (Seebeck Effect): The thermoelectric voltage $V$ generated across dissimilar metal junctions is non-linear across temperature $T$, modeled by high-order polynomials:

$$V = \sum_{i=0}^n c_i T^i = c_0 + c_1 T + c_2 T^2 + c_3 T^3 + \dots + c_n T^n$$

1. Platinum Resistance Thermometers (RTDs): Governed by the Callendar-Van Dusen equation for temperatures above $0^\circ\text{C}$:

$$R_T = R_0 \left( 1 + A \cdot T + B \cdot T^2 \right)$$

Where $A \approx 3.9083 \times 10^{-3\ \circ}\text{C}^{-1}$ and $B \approx -5.775 \times 10^{-7\ \circ}\text{C}^{-2}$. 3. Piezoresistive Pressure Transducers: Exhibit parabolic non-linearity across pressure $P$:

$$P = a_0 + a_1 V + a_2 V^2$$

Multi-Point Linearization Methods

To adjust and standardize non-linear instruments, technicians perform multi-point calibrations across the operating range:

• Segmented Linear Interpolation: The calibration span is subdivided into discrete segments (e.g., 5 or 10 calibration points: $0\%, 20\%, 40\%, 60\%, 80\%, 100\%$). The instrument firmware calculates local slope and offset values for each segment.
• Polynomial Curve Fitting: Calibration data points are processed via least-squares regression to generate high-order correction coefficients stored directly in instrument memory.

---

Digital and Firmware Adjustments: Closed-Case Calibration

Modern digital test equipment (such as 6.5- to 8.5-digit multimeters, multifunction calibrators, and frequency counters) has replaced mechanical trimpots with closed-case digital calibration.

Closed-Case Calibration Process Architecture

1. The instrument is allowed to achieve full thermal equilibrium (typically 1 to 2 hours powered on with covers sealed).
2. Traceable standard values (e.g., $+10.000000\text{ V}$, $0.000000\text{ V}$, $-10.000000\text{ V}$) are applied to the input terminals.
3. The technician or automated software sends a secure digital calibration command across the communications bus.
4. The instrument's internal microprocessor reads its analog-to-digital converter (ADC), computes the exact numerical offset ($b$) and gain error ($m$), and writes new calibration coefficients into write-protected EEPROM.
5. During routine measurement, the firmware automatically applies these constants in real-time:

$$x_{\text{corrected}} = \frac{x_{\text{raw}} - b_{\text{EEPROM}}}{m_{\text{EEPROM}}}$$

---

Realistic Calibration Scenario: Industrial Pressure Transmitter Alignment

Scenario: A technician is calibrating a 4–20 mA gauge pressure transmitter rated for $0\text{ to }100.00\text{ psi}$.

• At $0.00\text{ psi}$, nominal current output is $4.000\text{ mA}$.
• At $100.00\text{ psi}$, nominal current output is $20.000\text{ mA}$.
• In this hypothetical model, $k=0.15$ means a 1.0 mA observed change at the full-scale endpoint during a SPAN adjustment changes the zero endpoint by 0.15 mA. A ZERO change translates both endpoints equally. This definition describes the feedback example, not the dimensioned gain coefficient of the transfer function. Assume endpoint tolerance ±0.020 mA and verify all required intermediate points after adjustment.

Step 1: As-Found Verification

• Apply $0.00\text{ psi}$: Transmitter indicates $4.200\text{ mA}$ (Error: $+0.200\text{ mA}$; OOT).
• Apply $100.00\text{ psi}$: Transmitter indicates $19.600\text{ mA}$ (Error: $-0.400\text{ mA}$; OOT).
• The technician documents these complete As-Found readings on the data sheet.

Step 2: First Alignment Pass

• Apply $0.00\text{ psi}$: Adjust ZERO trimpot until output is exactly $4.000\text{ mA}$ (Shift: $\Delta b = -0.200\text{ mA}$).
• Apply $100.00\text{ psi}$: Output now reads $19.400\text{ mA}$. The technician adjusts the SPAN trimpot to bring output to $20.000\text{ mA}$ (Shift: $\Delta m = +0.600\text{ mA}$).

Step 3: Verification of Interactive Cross-Talk

• Re-apply $0.00\text{ psi}$: Due to cross-coupling, the output has shifted:

$$\text{Output} = 4.000\text{ mA} + (k \cdot \Delta m) = 4.000\text{ mA} + (0.15 \times 0.600\text{ mA}) = 4.090\text{ mA}$$

• The zero has shifted out of specification by $+0.090\text{ mA}$ due to span adjustment!

Step 4: Second Alignment Pass (Convergence)

• Adjust ZERO trimpot from $4.090\text{ mA}$ to $4.000\text{ mA}$ (Shift: $-0.090\text{ mA}$).
• Re-apply $100.00\text{ psi}$: Output reads $19.910\text{ mA}$. Adjust SPAN trimpot to $20.000\text{ mA}$ (Shift: $+0.090\text{ mA}$).
• Re-apply $0.00\text{ psi}$: Output reads $4.0135\text{ mA}$. Minor touch-up brings zero to $4.000\text{ mA}$.
• Full upscale and downscale verification confirms all points ($0\%, 25\%, 50\%, 75\%, 100\%$) are well within the $\pm 0.020\text{ mA}$ tolerance limit. Record final As-Left data.

---

Common Calibration Traps & CCT Exam Pitfalls