1. Introduction

The goal of this project is to investigate the operating principle of a CMOS bandgap voltage reference using LTspice simulations. The project focuses on the temperature behavior of bipolar junction transistors and demonstrates how complementary temperature-dependent voltages can be combined to generate a stable reference voltage.

The simulations presented in this project include:

• CTAT (Complementary To Absolute Temperature) voltage behavior
• PTAT (Proportional To Absolute Temperature) voltage generation
• PTAT current generation
• ideal bandgap reference modeling

The final objective is to obtain a reference voltage with minimal temperature dependence over a wide temperature range.

2. Theory

2.1 CTAT and PTAT Characteristics

The operating principle of a bandgap voltage reference is based on combining two electrical quantities with opposite temperature characteristics.

The first component is the base-emitter voltage of a bipolar junction transistor:

$$V_{BE}$$

The base-emitter voltage decreases as temperature increases. This behavior is known as:

• CTAT (Complementary To Absolute Temperature)

In practical semiconductor devices, the temperature coefficient of (V_{BE}) is approximately:

$$\frac{dV_{BE}}{dT}\approx -2\text{ mV}/^\circ C$$

The second component used in bandgap references is the voltage difference between two bipolar transistors operating at different current densities:

$$\Delta V_{BE}$$

This voltage increases proportionally with temperature and is therefore referred to as:

• PTAT (Proportional To Absolute Temperature)

The PTAT voltage can be expressed as:

$$\Delta V_{BE}=V_T\ln(N)$$

where:

• V_T is the thermal voltage
• N is the emitter area ratio of the transistors

In this project, the emitter area ratio was set to:

1:8

to generate a measurable PTAT voltage component.

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2.2 Temperature Compensation

The fundamental idea behind a bandgap reference is to combine the CTAT and PTAT components in such a way that their temperature dependencies compensate each other.

The decreasing behavior of V_BE is compensated by the increasing behavior of ΔV_BE.

By properly scaling the PTAT component, it is possible to obtain a nearly constant output voltage over temperature.

This temperature compensation mechanism is the core operating principle of bandgap voltage references.

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2.3 Bandgap Equation

The output voltage of a simplified bandgap reference can be expressed as:

$$V_{REF}=V_{BE}+K\cdot\Delta V_{BE}$$

where:

• V_BE represents the CTAT component,
• ΔV_BE represents the PTAT component,
• K is a scaling coefficient.

The coefficient determines the amount of PTAT compensation added to the CTAT voltage. By selecting an appropriate value of, the temperature drift of the output voltage can be minimized.

In this project, an ideal behavioral model was used in LTspice to investigate the influence of the compensation coefficient on the temperature stability of the reference voltage.

3. Simulation Setup

3.1 LTspice Environment

All simulations presented in this project were performed using LTspice. The software was used to analyze the temperature behavior of bipolar transistors and to investigate the operating principle of a simplified CMOS bandgap reference.

The simulations focused on:

• CTAT voltage generation
• PTAT voltage generation
• PTAT current generation
• ideal bandgap voltage modeling

LTspice was selected because it provides an efficient environment for analog circuit simulation and temperature analysis.

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3.2 Temperature Sweep

A DC temperature sweep analysis was used in all simulations to evaluate circuit behavior over a wide temperature range.

The following SPICE directive was applied:

.dc temp -40 125 1

The simulation range covered temperatures from:

-40°C to 125°C

with a step size of:

1°C

This range was selected to represent typical operating conditions encountered in analog integrated circuits, automotive electronics, and industrial systems.

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3.3 Transistor Configuration

The simulations were based on the standard NPN bipolar junction transistor model:

2N3904

In all circuits, the transistors were configured as diode-connected BJTs by shorting the base and collector terminals together.

This configuration allows the direct observation of the base-emitter voltage:

$$V_{BE}$$

which is the primary CTAT component used in bandgap references.

Two transistors were used in the PTAT-related simulations in order to generate a temperature-dependent voltage difference:

$$\Delta V_{BE}$$

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3.4 Area Ratio (1:8)

To generate the PTAT voltage component, different emitter area scaling factors were applied to the transistors.

The following area ratio was used:

Q1 area = 1
Q2 area = 8

This produced an emitter area ratio equal to:

$$N=8$$

The resulting voltage difference between the two transistors can be described by:

$$\Delta V_{BE}=V_T\ln(N)$$

Using an emitter area ratio of 1:8 allowed the PTAT voltage component to become sufficiently large for clear observation during the temperature sweep simulations.

4. CTAT Simulation

4.1 Circuit Schematic

The first simulation focused on observing the temperature dependence of the base-emitter voltage of a bipolar junction transistor.

A diode-connected NPN transistor configuration was used, where the base and collector terminals were shorted together. The transistor was biased using a constant current source of:

10 µA

The following SPICE directives were used:

.dc temp -40 125 1
.op

The simulation was performed over the temperature range:

-40°C to 125°C

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4.2 VBE vs Temperature

The simulated output voltage corresponds to the transistor base-emitter voltage:

$$V_{BE}$$

The obtained results show a clear negative temperature dependence of the base-emitter voltage.

As temperature increases, the value of V_BE decreases almost linearly. This behavior represents the CTAT characteristic used in bandgap reference circuits.

The simulated voltage changed approximately from:

680 mV at -40°C

to:

320 mV at 125°C

This negative temperature coefficient is one of the fundamental electrical properties of bipolar junction transistors.

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4.3 Results and Analysis

The simulation successfully demonstrated the CTAT behavior of the transistor base-emitter voltage.

The obtained characteristic confirms that:

• V_BE decreases with increasing temperature,
• the CTAT behavior is approximately linear over the analyzed temperature range,
• bipolar junction transistors can be used as temperature-dependent voltage sources in analog integrated circuits.

The CTAT characteristic generated in this simulation represents the first fundamental building block of a bandgap voltage reference.

In the following sections, this negative temperature dependence will be compensated using a PTAT component in order to obtain a temperature-stable reference voltage.

5. PTAT Simulation

5.1 Circuit Schematic

The second simulation focused on generating a PTAT (Proportional To Absolute Temperature) voltage component using two bipolar junction transistors with different emitter areas.

Two diode-connected NPN transistors were biased using identical current sources:

10 µA

The emitter area ratio was configured as:

Q1 area = 1
Q2 area = 8

This difference in emitter area produced different base-emitter voltages for the two transistors, despite both operating at the same collector current.

The following SPICE directive was used:

.dc temp -40 125 1

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5.2 ΔVBE Generation

The PTAT voltage component was generated by calculating the difference between the base-emitter voltages of the two transistors:

$$\Delta V_{BE}=V_{BE1}-V_{BE2}$$

Because the transistors have different emitter areas, they operate at different current densities. This results in different base-emitter voltages.

The PTAT voltage can be described by:

$$\Delta V_{BE}=V_T\ln(N)$$

where:

• V_T is the thermal voltage
• N is the emitter area ratio

Since the thermal voltage increases proportionally with temperature, the resulting ΔV_BE voltage also increases with temperature.

This behavior forms the PTAT component required for bandgap voltage references.

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5.3 Results and Analysis

The simulation results confirmed the expected PTAT behavior.

The lower graph shows the individual base-emitter voltages:

• V_BE1
• V_BE2

Both voltages decrease with increasing temperature, exhibiting CTAT characteristics.

However, the upper graph shows that the voltage difference:

$$V_{BE1}-V_{BE2}$$

increases approximately linearly with temperature.

The simulated PTAT voltage increased approximately from:

41 mV at -40°C

to:

72 mV at 125°C

This positive temperature coefficient confirms the successful generation of a PTAT voltage component.

The obtained PTAT characteristic represents the second fundamental building block of a bandgap voltage reference.

6. PTAT Current Generator

6.1 Circuit Schematic

The next stage of the project focused on converting the PTAT voltage component into a PTAT current.

The same transistor configuration used in the previous PTAT simulation was preserved. An additional resistor was connected between the transistor base-emitter nodes:

R1 = 1 kΩ

The resistor converted the PTAT voltage difference:

$$\Delta V_{BE}$$

into a temperature-dependent current.

Both transistors were biased using identical current sources:

10 µA

The emitter area ratio remained:

Q1 area = 1
Q2 area = 8

The following SPICE directive was used:

.dc temp -40 125 1

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6.2 IPTAT vs Temperature

The PTAT current was generated by forcing the voltage difference:

$$\Delta V_{BE}=V_{BE1}-V_{BE2}$$

across resistor.

According to Ohm’s law, the resulting current can be expressed as:

$$I_{PTAT}=\frac{\Delta V_{BE}}{R}$$

Since the PTAT voltage increases with temperature, the generated current also exhibits PTAT behavior.

The simulation monitored the current flowing through resistor:

$$I(R1)$$

The obtained current values were negative because of the current direction convention used by LTspice. However, the magnitude of the current increased with temperature, confirming the expected PTAT operation.

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6.3 Results and Analysis

The simulation results confirmed the successful generation of a PTAT current component.

The current flowing through resistor (R1) changed approximately from:

-6.99 µA at -40°C

to:

-7.32 µA at 125°C

Although the displayed current values were negative, the increasing magnitude of the current demonstrated positive temperature dependence.

This behavior confirms that the resistor successfully converted the PTAT voltage component into a PTAT current.

The generated PTAT current represents an important building block in analog integrated circuit design and is commonly used in:

• biasing circuits
• current reference generators
• bandgap references
• analog bias networks

7. Ideal Bandgap Model

7.1 Circuit Schematic

The final stage of the project focused on building an ideal behavioral model of a bandgap voltage reference.

The model combined:

• the CTAT component
• the PTAT component

in order to generate a temperature-stable reference voltage.

Two diode-connected bipolar junction transistors were used with the following emitter area ratio:

Q1 area = 1
Q2 area = 8

Both transistors were biased using equal current sources:

5 mA

A behavioral voltage source was used to generate the reference voltage according to the following equation:

$$V_{REF}=V_{BE2}+10\cdot(V_{BE1}-V_{BE2})$$

The following SPICE directive was used:

.dc temp -40 125 1

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7.2 Behavioral Model

The behavioral voltage source was implemented in LTspice using a mathematical expression that combines the CTAT and PTAT voltage components.

The generated reference voltage can be expressed as:

$$V_{REF}=V_{BE}+K\cdot\Delta V_{BE}$$

where:

• V_BE represents the CTAT component
• ΔV_BE represents the PTAT component
• K is the PTAT scaling coefficient

In this project, the scaling coefficient was set to:

K = 10

The behavioral model allowed direct investigation of the temperature compensation principle without implementing a full transistor-level bandgap circuit.

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7.3 VREF Optimization

Several scaling coefficient values were investigated during the simulation process in order to minimize the temperature dependence of the output voltage.

The selected coefficient:

K = 10

provided a nearly flat output voltage characteristic over the analyzed temperature range.

The final simulated reference voltage was approximately:

1.19 V

over the entire temperature sweep from:

-40°C to 125°C

This result demonstrates successful compensation between the CTAT and PTAT components.

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7.4 Results and Analysis

The simulation results confirmed the operating principle of a bandgap voltage reference.

The green and blue traces represent:

• V_BE1
• V_BE2

which both decrease with temperature due to their CTAT behavior.

The red trace represents the generated reference voltage:

$$V_{REF}$$

The obtained output voltage remained nearly constant across the entire temperature range, demonstrating successful temperature compensation.

The simulation confirmed that:

• the PTAT component compensated the CTAT behavior
• a nearly temperature-independent reference voltage was obtained
• the simplified behavioral model successfully reproduced the operating principle of a bandgap reference

The generated output voltage is close to the typical bandgap reference voltage used in analog integrated circuits.

8. Results Summary

8.1 CTAT Summary

The first simulation demonstrated the CTAT behavior of the bipolar junction transistor base-emitter voltage.

The obtained results confirmed that:

$$V_{BE}$$

decreases approximately linearly as temperature increases.

The simulated base-emitter voltage changed from approximately:

680 mV at -40°C

to:

320 mV at 125°C

This negative temperature coefficient represents the fundamental CTAT component used in bandgap reference circuits.

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8.2 PTAT Summary

The second simulation demonstrated the generation of a PTAT voltage component using two bipolar transistors with different emitter areas.

The voltage difference:

$$\Delta V_{BE}=V_{BE1}-V_{BE2}$$

increased approximately linearly with temperature.

The obtained PTAT voltage increased from approximately:

41 mV at -40°C

to:

72 mV at 125°C

The PTAT behavior was achieved using an emitter area ratio equal to:

1:8

The following simulation stage successfully converted the PTAT voltage into a PTAT current using a resistor-based current generation method.

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8.3 VREF Stability

The final behavioral model successfully combined the CTAT and PTAT components to generate a nearly temperature-independent reference voltage.

The implemented reference equation was:

$$V_{REF}=V_{BE2}+10\cdot(V_{BE1}-V_{BE2})$$

The final simulated output voltage remained approximately constant at:

1.19 V

over the entire temperature range from:

-40°C to 125°C

The obtained results confirmed successful temperature compensation between the CTAT and PTAT components.

The simulations demonstrated the fundamental operating principle of a bandgap voltage reference and showed how complementary temperature characteristics can be combined to achieve stable voltage generation in analog integrated circuits.

9. Conclusion

This project investigated the operating principle of a simplified bandgap voltage reference using LTspice simulations.

The performed simulations demonstrated:

• the CTAT behavior of the bipolar transistor base-emitter voltage
• the generation of a PTAT voltage component using different transistor emitter areas
• the generation of a PTAT current using a resistor-based approach
• the temperature compensation mechanism used in bandgap references

The final behavioral model successfully combined the CTAT and PTAT components to generate a nearly temperature-independent reference voltage.

The obtained reference voltage remained close to:

1.19 V

over the temperature range:

-40°C to 125°C

The simulations confirmed that proper scaling of the PTAT component can effectively compensate for the negative temperature coefficient of the transistor base-emitter voltage.

This project also demonstrated several fundamental concepts used in analog integrated circuit design, including:

• bipolar transistor temperature behavior
• PTAT and CTAT signal generation
• temperature compensation techniques
• analog bias generation principles

Although the implemented model was simplified and based on ideal behavioral elements, it successfully reproduced the core operating principle of practical bandgap reference circuits used in modern analog and mixed-signal integrated circuits.

The project provides a strong foundation for further work involving:

• transistor-level bandgap implementations
• current mirrors
• startup circuits
• analog bias networks
• precision analog integrated circuit design