Temperature Compensated Zener Reference

#zener diodes #voltage reference #temperature compensation #breakdown voltage #avalanche effect #forward-biased diodes #resistor networks #temperature drift #circuit design #electronic components

1. Basic Operation of Zener Diodes

1.1 Basic Operation of Zener Diodes

1.2 Breakdown Mechanisms: Zener vs. Avalanche

1.3 Temperature Effects on Zener Voltage

2. Sources of Temperature Drift in Zener References

2.1 Sources of Temperature Drift in Zener References

Zener Voltage Temperature Coefficient

The temperature dependence of a Zener diode's breakdown voltage is characterized by its temperature coefficient (TC), defined as:

$$ TC = \frac{1}{V_Z} \cdot \frac{\partial V_Z}{\partial T} $$

where VZ is the nominal Zener voltage and T is temperature. The TC varies significantly with Zener voltage due to the different breakdown mechanisms:

Junction Temperature Effects

The actual junction temperature (Tj) differs from ambient due to self-heating:

$$ T_j = T_a + R_{th} \cdot P_d $$

where Ta is ambient temperature, Rth is thermal resistance, and Pd is power dissipation. For a 6.2V Zener at 5mA with Rth=200°C/W:

$$ P_d = 6.2V \times 5mA = 31mW $$ $$ \Delta T_j = 200°C/W \times 0.031W = 6.2°C $$

Current Source Variations

The bias current (IZ) affects both the operating point and temperature stability. The dynamic resistance (rz) varies with current:

$$ r_z = \frac{\Delta V_Z}{\Delta I_Z} $$

For a typical 6.2V Zener:

Package Stress and Thermal Hysteresis

Mechanical stresses from the package and bonding wires introduce additional drift mechanisms:

A typical hermetically sealed Zener shows <0.5mV hysteresis after 10 temperature cycles between -55°C and +125°C.

Long-Term Drift

Even with temperature compensation, Zeners exhibit gradual voltage shifts over time:

$$ \Delta V_Z(t) = A \cdot \log(1 + Bt) $$

where A and B are empirical constants. High-quality references specify <100ppm/1000hrs drift.

2.2 Compensation Techniques: Forward-Biased Diodes

Forward-biased diodes are widely employed to compensate for the temperature-dependent voltage drift of Zener references. The mechanism relies on the opposing thermal coefficients of the Zener breakdown voltage and the forward voltage drop of a diode. While the Zener voltage typically exhibits a positive temperature coefficient (TC), the diode forward voltage has a negative TC, enabling cancellation when properly matched.

Thermal Coefficient Matching

The temperature coefficient of a forward-biased diode is approximately −2 mV/°C, derived from the Shockley diode equation:

$$ I_D = I_S \left( e^{\frac{V_D}{nV_T}} - 1 \right) $$

where ID is the diode current, IS the saturation current, VD the forward voltage, n the ideality factor (≈1 for silicon), and VT the thermal voltage (≈26 mV at 300 K). Differentiating VD with respect to temperature yields:

$$ \frac{dV_D}{dT} = \frac{V_D - (3+n)V_T - E_g/q}{T} $$

Here, Eg is the bandgap energy (~1.12 eV for silicon) and q the electron charge. For typical silicon diodes at VD ≈ 0.65 V and T = 300 K, this reduces to the empirical −2 mV/°C.

Circuit Implementation

A practical compensation circuit stacks a forward-biased diode in series with the Zener, as shown below:

Zener Diode

The total reference voltage VREF becomes:

$$ V_{REF} = V_Z + V_D $$

For optimal compensation, the Zener’s positive TC (e.g., +3 mV/°C for a 6.2 V Zener) and the diode’s −2 mV/°C must be balanced by adjusting the diode-to-Zener current ratio or using multiple diodes in series.

Practical Considerations

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Compensation Techniques: Forward-Biased Diodes in Temperature Compensated Zener Reference
Diagram Description: The diagram would physically show the series connection between the Zener diode and forward-biased diode, illustrating their voltage contributions and current flow.

2.3 Compensation Techniques: Resistor Networks

The temperature coefficient (TC) of a Zener diode is inherently nonlinear, but resistor networks can be employed to linearize its behavior over a defined operating range. The key principle relies on leveraging the opposing thermal drifts of the Zener and a series or parallel resistor network to achieve near-zero net TC.

Series-Resistor Compensation

When a Zener diode is biased through a series resistor RS, the voltage across the Zener (VZ) exhibits a TC that depends on the Zener's breakdown mechanism. For a 6.2V Zener (which has a near-zero TC at this voltage), the TC can be fine-tuned by adjusting RS to compensate for residual nonlinearities. The governing equation for the output voltage VOUT is:

$$ V_{OUT} = V_Z + I_Z \cdot R_S $$

where IZ is the Zener current. The TC of VOUT is derived by differentiating with respect to temperature T:

$$ \frac{dV_{OUT}}{dT} = \frac{dV_Z}{dT} + R_S \frac{dI_Z}{dT} + I_Z \frac{dR_S}{dT} $$

By selecting RS with a known TC (e.g., a metal-film resistor with ±50 ppm/°C), the third term can be designed to counteract the Zener's TC.

Parallel-Resistor Compensation

A shunt resistor RP placed in parallel with the Zener modifies the effective TC by altering the current distribution. The output voltage becomes:

$$ V_{OUT} = V_Z \left( \frac{R_P}{R_P + R_S} \right) $$

Differentiating with respect to temperature yields:

$$ \frac{dV_{OUT}}{dT} = \frac{R_P}{R_P + R_S} \left( \frac{dV_Z}{dT} - V_Z \cdot \frac{1}{R_P + R_S} \left( \frac{dR_P}{dT} + \frac{dR_S}{dT} \right) \right) $$

This allows for finer adjustment of the TC by balancing the thermal coefficients of RP and RS.

Practical Implementation

In precision references, a combination of series and shunt resistors is often used. For example:

A practical circuit might use a Vishay Bulk Metal® Foil resistor for RS (±2 ppm/°C) and a thin-film resistor for RP (±25 ppm/°C) to achieve a net TC below ±5 ppm/°C.

Zener RS RP
Compensation Techniques: Resistor Networks in Temperature Compensated Zener Reference
Diagram Description: The diagram would physically show the arrangement of series and parallel resistors with the Zener diode and how they connect to form the compensation network.

3. Component Selection for Optimal Stability

3.1 Component Selection for Optimal Stability

3.2 Circuit Topologies for Compensation

3.3 Practical Layout Considerations

4. Measuring Temperature Coefficient

4.1 Measuring Temperature Coefficient

The temperature coefficient (TC) of a Zener reference is a critical parameter that quantifies its voltage drift with respect to temperature variations. For precision applications, minimizing TC is essential to ensure stable reference voltage across operating conditions. The TC is typically expressed in parts per million per degree Celsius (ppm/°C).

Mathematical Definition

The temperature coefficient is derived from the fractional change in Zener voltage (VZ) per unit temperature change (ΔT):

$$ \text{TC} = \frac{1}{V_Z} \cdot \frac{\partial V_Z}{\partial T} $$

For small temperature variations, this can be approximated as:

$$ \text{TC} \approx \frac{1}{V_Z} \cdot \frac{\Delta V_Z}{\Delta T} \times 10^6 \quad \text{[ppm/°C]} $$

Measurement Methodology

To measure the TC experimentally, follow these steps:

Practical Considerations

Several factors influence TC measurement accuracy:

Compensation Techniques

Once the TC is characterized, compensation methods can be applied:

Case Study: Commercial References

Precision references like the LM399 achieve ultra-low TC (0.5 ppm/°C) by integrating a buried Zener with a temperature-stabilized substrate. The TC is minimized through careful process engineering and on-chip compensation.

$$ \text{TC}_{\text{LM399}} = 0.5 + 0.007(T - 25)^2 \quad \text{[ppm/°C]} $$

This quadratic dependence indicates that the TC is lowest near room temperature.

Measuring Temperature Coefficient in Temperature Compensated Zener Reference
Diagram Description: A diagram would visually show the experimental setup for measuring TC, including the Zener diode, temperature chamber, and measurement instruments.

4.2 Long-Term Stability and Aging Effects

The long-term stability of a temperature-compensated Zener reference is governed by material degradation mechanisms, dopant diffusion, and mechanical stress relaxation. Unlike short-term noise or thermal drift, aging effects manifest as a gradual, monotonic shift in the reference voltage over years of operation. The dominant aging mechanisms include:

Quantifying Aging Effects

The aging rate (ΔVref/Δt) follows a logarithmic time dependence, empirically modeled as:

$$ \Delta V_{ref}(t) = A \ln\left(1 + \frac{t}{\tau}\right) + B\sqrt{t} $$

where A represents the fast initial drift component (days to months), B characterizes long-term diffusion processes (years), and τ ≈ 30 days is the relaxation time constant. For precision references, manufacturers typically specify aging in ppm/√khr (parts per million per square root of thousands of hours).

Acceleration Factors

Aging rates accelerate with temperature following an Arrhenius relationship:

$$ \text{Aging Rate}(T) = R_0 \cdot e^{-\frac{E_a}{kT}} $$

where Ea ≈ 0.7-1.1 eV for silicon Zener references. Practical burn-in procedures leverage this by operating devices at 125-150°C for 168-1000 hours to stabilize the reference before calibration.

Metallurgical Considerations

The aluminum-silicon interface at bond wires develops intermetallic compounds over time, creating thermoelectric offsets. Gold-doped references exhibit superior stability due to:

Package Stress Analysis

Hermetic ceramic packages with matched CTE (Coefficient of Thermal Expansion) substrates maintain <0.5 ppm/year drift, while plastic encapsulated devices may exhibit 2-5 ppm/year shifts due to:

$$ \sigma(t) = \sigma_0 e^{-t/\tau_\sigma} + \sigma_\infty $$

where τσ ranges from 500-2000 hours for epoxy relaxation. Kovar leads introduce additional stress from differential thermal contraction during temperature cycling.

Historical Case Study: LM399 Evolution

The National Semiconductor LM399 (1978) demonstrated how substrate doping profile optimization reduced aging from 20 ppm/year to 4 ppm/year. Key improvements included:

4.3 Calibration Techniques for Precision Applications

Calibrating a temperature-compensated Zener reference requires careful consideration of both static and dynamic error sources. The primary objective is to minimize drift due to thermal hysteresis, aging, and load variations while maintaining long-term stability.

Nulling the Temperature Coefficient

The temperature coefficient (TC) of a Zener reference is minimized by adjusting a trimming resistor in series with the compensation network. For a first-order approximation, the TC can be expressed as:

$$ \text{TC} = \alpha + \beta \cdot \Delta T $$

where α is the linear coefficient and β is the quadratic term. A practical implementation involves:

Dynamic Load Regulation Calibration

Load regulation errors arise from finite output impedance and thermal gradients. A precision current sink can be used to characterize the reference under varying loads (e.g., 1mA to 10mA). The output impedance Zout is given by:

$$ Z_{\text{out}} = \frac{\Delta V_{\text{out}}}{\Delta I_{\text{load}}} $$

Compensation is achieved by adding a low-TC series resistor or active buffering.

Long-Term Stability and Aging Compensation

Zener references exhibit logarithmic aging, approximated by:

$$ \Delta V(t) = k \cdot \log(1 + t/t_0) $$

where k is the aging coefficient and t0 is a time constant. Accelerated aging at elevated temperatures (e.g., 125°C for 168 hours) followed by recalibration reduces long-term drift.

Noise Reduction Techniques

Low-frequency noise (1/f noise) can be mitigated through:

For high-precision applications, a lock-in amplifier can be used to characterize noise spectra and optimize filtering parameters.

5. Key Research Papers on Zener References

5.1 Key Research Papers on Zener References

5.2 Recommended Books on Voltage References

5.3 Online Resources and Datasheets