Introduction
Analog circuits do not operate in ideal, temperature-invariant environments. Transistor junction voltages, resistor values, and carrier mobility all vary with temperature, resulting in drift of voltage references, bias currents, offsets, and gains. In many applications—such as data converters, sensors, and power-management systems—temperature drift directly limits achievable accuracy.
Although modern simulators provide temperature sweeps, designers still benefit from hand analysis that answers two fundamental questions:
- Which parameters or devices dominate temperature drift?
- How does circuit topology amplify or suppress these effects?
This article introduces a rule-based temperature coefficient analysis method that requires only basic calculation and intuitive circuit inspection.
Absolute Temperature Coefficient (ATC) vs. Relative Temperature Coefficient (RTC)
Before analyzing circuits, it is important to distinguish between two commonly used forms of temperature coefficient.
Absolute Temperature Coefficient
The ATC measures how an output quantity changes directly with temperature:
Typical units include mV/°C, µA/°C, or Ω/°C. ATC is most useful when absolute error matters, such as voltage-reference drift.
Relative Temperature Coefficient
The RTC normalizes the drift to the nominal value:
It is usually expressed in ppm/°C or %/°C and is particularly suitable for circuits in which the absolute output value is adjustable, such as adjustable-output (VOUT) regulator circuits.
Categorizing Circuit Temperature Behavior
Most analog circuits can be reduced to one of two functional categories for temperature analysis.
Category 1: Additive and Subtractive Functions
Definition
An additive or subtractive function has the form:
Example
A band-gap voltage reference is a classic example:
where:
- VBE is complementary-to-absolute temperature (CTAT)
- ΔVBE is proportional-to-absolute temperature (PTAT)
Temperature-Coefficient Rule
For additive functions:
Key Insight:
ATC can be added algebraically. If one term has a significantly larger temperature coefficient than the others, it dominates the overall drift.
Practical Design Implication
Temperature independence is achieved by canceling positive ATC terms with negative ATC terms. This principle is fundamental to band-gap references and many compensated bias generators.
Category 2: Multiplicative and Ratio Functions
Definition
A multiplicative or ratio function takes the form:
Example
A current reference based on a voltage-to-resistor ratio:
Temperature-Coefficient Rule
Applying first-order differentiation:
where the sign depends on multiplication or division.
Key Insight:
RTC terms can be added algebraically. For multiplicative circuits, temperature drift is minimized by canceling positive RTC terms with negative RTC terms.
For ratio-based circuits, temperature drift is minimized by matching the RTCs of the numerator and denominator in both magnitude and polarity.
Applying the Method: A Reference Current Example
Consider a simple reference current generator circuit as seen in Figure 1:
Figure 1. Simple reference current generator using bipolar transistor’s base-emitter voltage (VBE) and resistor to generate reference current.
with:
- VBE ≈ −2mV/°C (CTAT)
- RTCR2 = +100ppm/°C
The RTC of the current reference becomes:
For example, with VBE ≈ 0.7V, the RTC due to VBE can be approximated as follows:
By applying Equation 11 and RTCR2 to Equation 10:
Based on Equation 12, IREF exhibits a strong negative temperature dependence
Selecting a resistor with an RTC comparable in magnitude and polarity to the numerator’s RTC can significantly reduce overall temperature drift.
Why This Method Works
This approach is effective because:
- First-order temperature effects dominate over most operating ranges
- Higher-order nonlinearities are typically small compared to first-order effects and are often negligible in applications that do not require very high precision
- The analysis directly reflects the combined behavior of circuit elements
Most importantly, it allows designers to predict temperature performance before simulation or silicon validation.
Design Guidelines Summary
- Use ATC when analyzing circuits with additive or subtractive behavior
- Use RTC when analyzing multiplicative or ratio-based circuits
Conclusion
Temperature-coefficient analysis does not need to be complex to be effective. By classifying circuits into additive and multiplicative categories and applying simple mathematical calculations, engineers can gain rapid insight into temperature behavior. This method bridges the gap between device characteristics and practical circuit design, enabling faster convergence to stable, temperature-robust solutions.
About the Author
Sherwin Clemente is an analog IC design engineer in the Industrial, Power, and Precision Group, handling safe power protection products at Analog Devices Philippines. He joined ADI in September 2019. Prior to joining ADI, he worked for four years as a design and development engineer at a semiconductor company, focusing on high-voltage switching regulators. He holds a bachelor’s degree in electronics engineering from the University of La Salette, Inc.
