VCR Voltage Coefficient of Resistance – A Practical Design Guide

The voltage coefficient of resistance (VCR) is a key non‑linearity parameter of high‑value resistors that can quietly dominate error budgets in precision and high‑voltage circuits. While often ignored in low‑ohmic designs, VCR becomes critical once resistance values move into the tens of megaohms and beyond or when linearity over a wide voltage swing is required.

This article explains what VCR is, how different resistor technologies behave, how to measure and specify it correctly, and how to design circuits that remain linear despite VCR.

1. Definition and Basic Theory

Analogous to the temperature coefficient of resistance (TCR), which describes resistance change versus temperature, VCR characterizes the change of resistance versus applied voltage. For a resistor whose resistance depends on the applied voltage $U$, the VCR between two voltages $U_1$ and $U_2$ is defined as:$$VCR = \frac{R_2 – R_1}{R_1 \cdot (U_2 – U_1)}$$where $R_1$ and $R_2$ are the resistance values measured at $U_1$ and $U_2$, respectively.

The voltage coefficient of resistance (=V_cr), which is barely encountered at resistance values below 10MOhm, is a non-linear feature of resistors mainly at very high values.

Analog to the T_cr, which indicates the change of resistance with temperature, the V_cr reveals the change of resistance with applied voltage.

These resistors are non-ohmic resistors which means that the ratio between applied voltage and measured current is no longer constant for every voltage. So strictly speaking, if you use the well-known formula:

R = U / I

you have to specify the voltage at which you measure the resistance (which is not necessary for true ohmic resistors). The calculation of the V_cr is completely analogous to the calculation of the T_cr, you just have to replace the temperatures with the voltages:

Exactly speaking (and this is valid for the T_cr too) this calculus is just the linear interpolation between the two measurement points at U₁ and U₂ (green straight line in the graph below). The real behaviour is nonlinear and could look like the one which is shown with the blue curve in the graph or similar.

If you rearrange the formula from above a little bit, you see that the V_cr is nothing else but the gradient or slope of R with V normalised to R₁. It is easy to see in the graph on Figure 1. below, that the V_cr between 25V and 50V may be considerably different to the V_cr between 100V and 125V, albeit the voltage difference is in both cases ΔU = 25V.

Also, you see that when measuring the Vcr the exact value of the slope is important, and this is achieved by a big distance between the measurement voltages. Then small measurement errors of the resistance do not have such an influence.

Usually, the voltage coefficient is indicated in parts per million per volts [ppm/V] or also in percent per volt [%/V]; thick film resistors with ruthenium-oxide based conducting phase have negative V_cr’s. One example to see it more clearly:

A resistor with e.g. 1GOhm resistance value supposed to have a V_cr of lets say -1000ppm/V. Then the value of resistance will change from 1GOhm to 0,9GOhm (=-10%), if you change the voltage from 25V to 125V (voltage difference of ΔU = 100V). Analog this holds true for the T_cr.

Key points:

• VCR is usually expressed in ppm/V or %/V.
• Thick‑film resistors with ruthenium‑oxide conducting phase typically exhibit negative VCR (resistance decreases with increasing voltage).
• The calculation is a linear interpolation between two measurement points; the true $R(U)$ curve is usually non‑linear.

Example: A 1 GΩ resistor with VCR = −1000 ppm/V will change from 1 GΩ at 25 V to about 0.9 GΩ at 125 V (−10%) for a 100 V increase.

2. Standardized VCR Specification (MIL‑STD‑202 Method 309)

Different manufacturers use different test conditions, which complicates comparison between parts. MIL‑STD‑202G, Method 309, defines VCR as:

This means the voltage ratio $U_2/U_1$ is 10 and the absolute difference $U_2 – U_1$ is 0.9·$U_2$. However, this method is not widely applied in practice for two main reasons:

• Many resistors are rated up to 10 kV–100 kV or more, which demands specialized high‑voltage sources and fixtures.
• To avoid self‑heating of small SMD components, the standard calls for pulsed measurement, which increases complexity and is sensitive to RC time constants at very high resistances.

For these reasons, many manufacturers adopt a modified approach (lower voltages, continuous measurement) while preserving a sufficient voltage span for good slope accuracy.

3. VCR by Resistor Technology and Resistance Range

Although detailed numeric ranges are often proprietary or scattered across datasheets, some qualitative trends are well established. The table below summarizes typical behaviour:

Typical VCR Behaviour by Technology (Qualitative)

Technology type	Typical resistance range where VCR matters	VCR sign & magnitude (qualitative)	Notes
Thick‑film (RuO₂) SMD	>10 MΩ, especially >100 MΩ	Negative, tens to thousands ppm/V	Strongly dependent on paste, firing profile, and geometry.​
High‑voltage thick‑film	10 MΩ–GΩ	Negative, optimized to low values	Long meander patterns and tailored pastes improve VCR.​
Thin‑film (NiCr, TaN)	Up to a few MΩ	Very low, often negligible	VCR usually not specified; linear $I$I-$V$V behaviour dominates.
Bulk metal foil	Up to a few 100 kΩ	Extremely low	Used where ultra‑linearity is required; mainly low‑ to mid‑ohmic.
Wirewound	Low to mid values	Very low, unless inductive effects	VCR rarely a concern compared with parasitic inductance.
Carbon composition	Mid to high values	Larger non‑linearities possible	Often older technology; non‑linearity can be significant.

For thick‑film high‑ohmic parts, the paste formulation, sintering temperature, and layout geometry dominate VCR behaviour.

4. Influence of Paste, Sintering, and Geometry

For thick‑film resistors, both the conductive phase and the microstructure formed during firing determine the electric field distribution in the resistor. Industrial practice shows:

• VCR strongly depends on the resistive paste material.
• Sintering temperature can shift VCR dramatically; there is often an optimum firing window that minimizes VCR without compromising stability.
• Increasing the effective length‑to‑width ratio (L/B, “number of squares”) by using meander patterns usually improves VCR.

Manufacturers internal incoming inspection as an illustrative example: several component sizes with different L/B ratios are fired at different temperatures; VCR is then measured between 10 V and 100 V to identify the optimal process window. Figure 2 shows exactly such VCR vs. sintering temperature behaviour for a single paste.

5. Measurement of Low VCR – Practical Challenges

Measuring VCR at high resistance values is considerably more demanding than measuring TCR. Several pitfalls must be addressed to obtain meaningful data.

5.1 Key Measurement Issues

1. Number of steady digits
   At very high resistances (e.g., 100 GΩ), currents are in the picoampere range, and digit hopping makes it difficult to secure enough valid digits. To resolve differences such as 10 ppm/V vs. 20 ppm/V, you may need four or five stable digits, which is challenging in industrial environments at low voltages.
2. Measurement environment
   Air humidity, airflows (HVAC, doors), and mechanical vibrations all introduce noise and drift in the setup, limiting the reliability of the last significant digit. Shielding, guarding, and environmental control are essential.
3. Measurement range changes
   When measuring between 1 V and 10 V, the current changes by nearly a factor of ten. DMMs or electrometers often change measurement range, and each range has different accuracy, linearity, and offset, which can distort the calculated VCR.
4. Self‑heating and TCR interaction
   At small packages, high voltage, and low VCR, self‑heating can cause resistance changes via TCR that are misinterpreted as VCR. External temperature changes also affect readings.
5. Instrument accuracy and limits
   Above about 1 GΩ, current measurement accuracy of electrometers at 10 pA–100 pA is commonly around 0.5–1% of reading. One percent corresponds to 10,000 ppm; even across 100 V, this leaves ~100 ppm uncertainty in each measurement point. Statements like “1 ppm/V at hundreds of megaohms” should therefore be treated critically.
6. Measurement time and RC time constant
   Large high‑value resistors have significant intrinsic capacitance $C_1$; cables and fixtures contribute external capacitance $C_2$. The time constant is $\tau = R (C_1 + C_2)$, and you should wait at least 5·$\tau$ to reach the final value, which can take many minutes.

5.2 Practical Measurement Recommendations

• Use guarded, shielded fixtures and high‑insulation materials.
• Avoid auto‑range changes between $U_1$ and $U_2$; fix ranges if possible.
• Allow sufficient settling time after each voltage step (multiple time constants).
• Verify consistency by repeating measurements and, when feasible, validating with a second instrument.

6. Circuit‑Level Impact and Error Budgeting

VCR directly translates into voltage‑dependent gain or scale error in circuits where the resistor carries a significant voltage. The most obvious examples are:

• Transimpedance amplifiers (TIA) with high feedback resistors.
• High‑voltage dividers and probes.
• Electrometers, picoammeters, and leakage measurement setups.
• High‑impedance bias networks for photodiodes, PMTs, or x‑ray detectors.

6.1 Example: TIA Feedback Resistor

In a TIA, the output voltage is $V_{out} = I_{in} \cdot R_f$. If the feedback resistor $R_f$ has VCR, its effective value changes with the output voltage, and the conversion gain is no longer constant.

Consider a 1 GΩ feedback resistor with VCR = −1000 ppm/V, and a TIA operating from 1 V to 10 V output:

• ΔU = 9 V
• ΔR/R ≈ VCR · ΔU = −1000 ppm/V · 9 V = −9000 ppm ≈ −0.9%

So the effective gain changes by nearly 1% over the range, even if TCR and tolerance are ideal. For a TIA where the resistor’s maximum working voltage is 400 V (e.g., a 1206 size), measuring VCR between 40 V and 400 V per a datasheet‑like condition is not representative, because the real application only sees 1–10 V.

Design implication: for low‑voltage TIA, either:

• Measure VCR in the actual voltage range and use a correction table/curve in the system, or
• Choose a technology or value where VCR is negligible over the operating span.

6.2 Example: High‑Voltage Divider

For a divider with upper resistor $R_1$ and lower resistor $R_2$, the transfer ratio is:$$k = \frac{R_2}{R_1 + R_2}$$

If $R_1$ is large and exhibits VCR, its value will vary with the applied high voltage, causing a scale‑factor error that increases with input voltage.

A practical high‑voltage design often uses:

• Multiple series resistors to share voltage and reduce per‑resistor VCR effects.
• Carefully chosen resistor technologies (e.g., specialized HV thick‑film) with specified low VCR at rated voltage.

6.3 Combining VCR with TCR, Tolerance, and Drift

In a real error budget, you combine:

• Tolerance (e.g., ±1%)
• TCR (over temperature excursion)
• VCR (over voltage excursion)
• Long‑term drift

Conceptually, you can treat each as a percentage and combine worst‑case linearly or statistically (RSS) depending on system requirements. VCR often appears as a “hidden” term when voltage span is large; explicitly including it in spreadsheets or models prevents surprises in system‑level linearity tests.

7. Design Guidelines and Best Practices

7.1 When to Care About VCR

• Resistance values ≥10 MΩ, and especially ≥100 MΩ.
• Circuits with large voltage swings across a high‑value resistor.
• Precision measurement systems (electrometers, picoammeters, TIA front‑ends).
• High‑voltage dividers requiring linearity across their full range.

Rule‑of‑thumb: once ΔR/R from VCR over your operating voltage range approaches or exceeds your allowed gain/scale‑error budget, treat VCR as a primary design parameter.

7.2 Technology Selection Guidelines

Design scenario	Recommended technology focus	VCR considerations
Ultra‑low noise, low value (<100 kΩ)	Bulk metal foil, thin‑film	VCR typically negligible.
Precision DC gain up to a few MΩ	Thin‑film, precision metal film	VCR rarely specified, but usually small.
High‑value TIA feedback (10 MΩ–1 GΩ)	Specialized high‑ohmic thick‑film, HV chips	Look for low VCR at actual operating voltage and good TCR.
HV dividers (kV range, 10 MΩ–GΩ)	HV thick‑film chips or axial, designed for VCR	Check vendor’s HV/VCR application data and derating.
Cost‑sensitive high‑value biasing	General‑purpose thick‑film	Verify whether VCR is acceptable via measurement or prototyping.

8. Layout, PCB, and System‑Level Effects

At very high resistances, parasitics and leakage can dominate over true resistor behaviour.

• Leakage paths: Contamination, flux residues, and moisture on the PCB surface can add parallel leakage, effectively reducing apparent resistance and masking or distorting VCR behaviour.
• Creepage and clearance: High voltage plus contamination can cause corona or partial discharge, adding non‑linear leakage paths.
• Guarding: Guard rings driven at similar potential as high‑impedance nodes help suppress leakage currents into sensitive measurement points.

For meaningful VCR characterization in‑circuit, ensure that PCB layout, materials, and cleaning procedures are designed to support gigaohm‑level impedances.

9. Reliability, Aging, and Environmental Influences

VCR can also evolve over life. Qualitative considerations:

• Humidity: Increased humidity can change surface conduction and in some cases alter the effective field distribution within the resistive film.
• Mechanical and thermal stress: Microcracks or damaged terminations can locally concentrate electric fields and increase effective VCR.
• Long‑term load and surges: High‑voltage stress, especially near maximum rating, may cause gradual changes in microstructure that shift VCR over time.

Because the same microstructural features that affect VCR also influence stability, regular requalification (e.g., during supplier changes or process changes) is advisable for tight‑linearity applications.

10. Practical VCR Measurement Setup (Conceptual)

A typical DC measurement arrangement for VCR on a high‑value resistor includes:

• Stable DC voltage source capable of generating both $U_1$ and $U_2$.
• Electrometer or picoammeter to measure current, from which resistance is derived.
• Guarded test fixture reducing leakage and capacitance.
• Shielded cables and an environmentally controlled bench.

Measurement sequence example:

1. Connect resistor in guarded fixture; short input to discharge any residual charge.
2. Apply $U_1$, wait ≥5·$\tau$, measure current $I_1$ and compute $R_1 = U_1 / I_1$.
3. Apply $U_2$, again wait ≥5·$\tau$, measure $I_2$ and compute $R_2$.
4. Compute VCR from the formula above.
5. Repeat to check repeatability; if necessary, reverse voltage polarity to evaluate symmetry.

11. Application‑Specific Recommendations

11.1 TIA Design

• Use resistor values only as high as necessary to achieve required gain; avoid unnecessary gigaohm values.
• Prefer resistor series with explicit VCR characterization or from vendors with strong HV/thick‑film expertise.
• If error budgets are tight, measure VCR at the real TIA output voltage range and store correction factors in firmware.

11.2 High‑Voltage Dividers

• Distribute voltage across multiple series resistors to limit per‑resistor electric field and VCR contribution.
• Validate linearity across the full voltage scale using calibrated equipment, not just at a single point.
• Consider temperature‑compensated design if TCR and VCR interact significantly.

12. Conclusion

VCR is a non‑linear, voltage‑dependent resistance change that becomes essential once resistors reach tens of megaohms or see significant voltage stress. For thick‑film high‑ohmic resistors, VCR is strongly influenced by paste formulation, sintering conditions, and layout geometry, and can reach values that clearly affect gain, scale factor, and linearity in precision and high‑voltage circuits.

Accurate VCR measurement is challenging due to picoampere‑level currents, instrument limits, RC time constants, and environmental sensitivity. Designers should therefore treat extremely low VCR claims at very high resistance levels with healthy scepticism and, when in doubt, perform their own verification.

In practical design, VCR should be considered alongside TCR, tolerance, and long‑term drift in the error budget, especially for TIA feedback resistors and high‑voltage dividers. By choosing suitable resistor technologies, understanding datasheet conditions, and designing proper measurement and PCB environments, engineers can keep VCR under control and achieve predictable, linear system behaviour.

FAQs: Voltage Coefficient of Resistance (VCR)