174 

Research Progress on Bandgap and Non-Bandgap Reference 
Circuits 

Guanglong Zhang * 

School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, China 

* Corresponding Author Email: 22211317@bjtu.edu.cn u 

Abstract. Bandgap reference (BGR) circuits could provide stable, temperature-insensitive reference 
voltages or currents that are essential to system-level designs. These circuits are extensively 
employed in critical applications including voltage regulators, oscillators, and analog-to-digital 
converters (ADCs). Continuous advancements in bandgap reference techniques and CMOS process 
scaling have significantly enhanced the accuracy and temperature stability of modern reference 
circuits, making substantial contributions to progress in electronic engineering. This study reviews 
four classical first-order bandgap reference architectures, with particular emphasis on methods for 
second-order temperature compensation. Furthermore, it presents a detailed comparison and 
evaluation of two CMOS-only non-bandgap reference circuit topologies—namely those leveraging 
the subthreshold characteristics of MOSFETs and those employing capacitor-based biasing 
techniques. A comprehensive comparative analysis of key performance metrics between bandgap 
and non-bandgap reference circuits is provided. This research summarizes the architectural 
differences and performance trade-offs between bandgap and non-bandgap reference circuits, 
offering valuable perspectives for addressing critical challenges such as stability. 

Keywords: Bandgap Reference; Second-Order Temperature Compensation; Non-Bandgap 
Reference. 

1. Introduction 

Reference circuits serve as essential modules in mixed-signal integrated circuits which deliver a 

thermally stable reference voltage or current to the system, thereby establishing a unified reference 

value for consistent operation. These circuits are extensively utilized in a variety of critical 

applications, including voltage regulators, and oscillators. In contrast to conventional reference 

circuits, non-bandgap (CMOS-only) reference architectures offer advantages such as compact area, 

low power consumption, and structural simplicity. However, they typically suffer from limited 

accuracy and poor temperature stability. 

Recent advancements in analog and mixed-signal IC design have introduced challenges, including 

stringent power constraints, low-voltage operating ranges for data converters, and noise in reference 

voltage generation. These issues necessitate highly accurate reference voltage generators to ensure 

precise system performance. 

This paper presents a comprehensive review of reference voltage circuits. Sections 1 and 2 analyze 

classical bandgap reference architectures and topologies for second-order temperature compensation. 

Section 3 provides a comparative evaluation of non-bandgap reference circuits, highlighting their 

strengths and limitations. Finally, Sections 4 and 5 discuss recent research progress in both bandgap 

and non-bandgap reference circuits. 

2. Bandgap Reference Circuit 

2.1. Principles of Bandgap Reference Voltage Circuits 

Components in electronic circuits often exhibit temperature-dependent characteristics. For 

instance, the resistance of a resistor varies with temperature—a functional relationship known as the 

temperature coefficient. Temperature coefficients are generally categorized as positive (PTC) or 

negative (NTC), and their variation with temperature may involve higher-order functions. 



 

175 

The key to a bandgap reference voltage circuit is using two voltage components with opposing 

temperature coefficients, which cancel each other to deliver a stable reference voltage. 

2.2. Performance Metrics of Bandgap Reference Voltage Circuits 

A reference voltage (conventionally denoted as 𝑉𝑅𝐸𝐹) is delivered by complementary positive and 

negative temperature coefficient in the reference circuit leverages. Key performance metrics 

commonly evaluated for such circuits include the reference voltage value, operating voltage range, 

power consumption, temperature coefficient, and accuracy. The effective temperature coefficient (TC) 

is a widely adopted metric for characterizing reference voltage circuits. It is defined as the ratio of 

the relative change in the output quantity to the temperature change across a specified operating range. 

For reference voltage circuits, the output quantity corresponds to the reference voltage (VREF). 

The temperature coefficient is commonly normalized with respect to its value at 25°C(VREFnom). 

The expression is given as follows: 

𝑇𝐶 = 
𝑉𝑅𝐸𝐹𝑚𝑎𝑥  −  𝑉𝑅𝐸𝐹𝑚𝑖𝑛

𝑉𝑅𝐸𝐹𝑛𝑜𝑚
(1) 

2.3. Classical Bandgap Reference Voltage Circuit 

The earliest bandgap reference voltage circuit was proposed by American engineer Robert Widlar. 

A key aspect of the design involves leveraging the complementary temperature characteristics of 

bipolar devices: the negative temperature dependence inherent in the base-emitter voltage (∆𝑉𝐵𝐸), 

combined with the positive temperature dependence exhibited by the difference in base-emitter 

voltages (∆𝑉𝐵𝐸) between two transistors biased at different current densities. 

By summing these two voltage components, a "zero-temperature-coefficient" reference voltage 

was achieved. Subsequently, researchers including Kuijk, Brokaw, and Banba further optimized the 

Widlar architecture, enhancing the performance of circuit. To this day,  bandgap reference circuits 

continue to evolve based on extensions and refinements of this classical bandgap reference framework. 

2.3.1. Widlar Circuit 

The first bandgap reference voltage architecture was introduced by Widlar [1]. Its architecture is 

implemented using conventional bipolar circuitry, as illustrated in Figure 1 below. 

 

Fig 1. Circuit diagram of the Widlar Bandgap reference. 

The reference voltage 𝑉𝑅𝐸𝐹  equals the collector voltage of bipolar transistor 𝑄3. The voltage 

difference ∆𝑉𝐵𝐸 is derived from the unequal current densities in 𝑄2 and 𝑄1, which carry identical 

currents but differ in emitter junction area. 



 

176 

Transistors 𝑄1, 𝑄2, and 𝑄3 are BJTs of the same type, but Q2  has an emitter junction area 

different from those of Q1 and Q3. The emitter area of 𝑄2 is n times that of 𝑄1. Q1 and Q3 are 

of the same type with equal base-emitter voltages, as VBE1 = VBE3. The resistances of R1 and R2 

are equal. According to Ohm's law, the following relationship holds: IR1 = IR2. 

Given that Q1 and Q2 operate at same current and different area, the difference in their base-

emitter voltages(∆𝑉𝐵𝐸) can be satisfied: 

∆𝑉𝐵𝐸  =  𝑉𝐵𝐸1  −  𝑉𝐵𝐸2 = 𝑉𝑇 𝑙𝑛
𝐼𝐶1
𝐼𝑆1
 − 𝑉𝑇 𝑙𝑛

𝐼𝐶2
𝐼𝑆2
 =  𝑉𝑇 𝑙𝑛 𝑛 (2) 

Where 𝑉𝑇 denotes the thermal voltage of the BJT, which is proportional to absolute temperature. 

Assuming negligible base currents and equal resistances 𝑅2 and 𝑅3. The reference voltage 𝑉𝑅𝐸𝐹 

can be satisfied: 

𝑉𝑅𝐸𝐹 = 𝑉𝐵𝐸3 + 
𝑅2
𝑅3
𝑙𝑛 𝑛 (3) 

The following equation could ensure temperature independence of the reference voltage: 

𝜕𝑉𝑅𝐸𝐹
𝜕𝑇

= 0 (4) 

Solving equation (4) under this constraint yields a 𝑉𝑅𝐸𝐹  value of approximately 1.2 V.As 

indicated by the above equation, the temperature coefficient of 𝑉𝑅𝐸𝐹  in this circuit is mostly 

determined by resistors 𝑅2 and 𝑅3 and the emitter area ratio of the bipolar transistors. Therefore, 

satisfying Equation (4) can be achieved by adjusting the emitter area ratio and the resistance values. 

However, the Widlar bandgap reference suffers from two main limitations: first, it is challenging 

to maintain stable collector currents 𝐼𝐶1 and 𝐼𝐶2 in the two BJTs; second, the circuit lacks higher-

order temperature compensation, resulting in a relatively high temperature coefficient. 

2.3.2. Kuijk Circuit and Brokaw Circuit 

To mitigate susceptibility to power supply noise in the Widlar reference circuit, Kuijk introduced 

an operational amplifier into the original architecture [2], as shown in Fig. 2. 

 

Fig 2. Circuit diagram of the Kuijk bandgap reference. 

The added operational amplifier serves two primary purposes: it reduces power supply noise and 

delivers same voltages of nodes X and Y. By setting resistors 𝑅2  and 𝑅1  to equal values, the 

currents through them are matched, i.e., 𝐼𝑅1 = 𝐼𝑅2 . 

The emitter area of 𝑄1 is n times that of 𝑄2. The current through 𝑅3 is equal to: 



 

177 

𝐼𝑅3 = 𝑉𝑇 𝑙𝑛
𝐼𝐶1
𝐼𝑆1
 (5) 

The voltage 𝑉𝑅𝐸𝐹 must be satisfied: 

𝑉𝑅𝐸𝐹 = 𝑉𝐵𝐸1 + 
𝑅2
𝑅3
𝑙𝑛 𝑛 (6) 

Compared to the Widlar reference, the Kuijk configuration offers improved performance by 

eliminating the influence of collector current variations and reducing power supply-induced 

interference through the use of an operational amplifier. The high gain, low power consumption, and 

enhanced noise immunity of the op-amp contribute to the circuit’s broader applicability and superior 

performance. 

Based on the Kuijk circuit architecture, Brokaw proposed a novel configuration in 1974 [3]. A key 

innovation involves directly connecting the operational amplifier output to the base terminals of both 

BJTs. In contrast to the Kuijk configuration, this approach enhances feedback linearity and improves 

supply noise rejection. 

2.3.3. Banda Circuit 

In contrast to the voltage summation approach used in the three bandgap reference circuits 

mentioned above, Banba proposed a bandgap reference based on current summation in 1999 [4]. The 

circuit schematic is shown in Fig. 3. 

 

Fig 3. Circuit diagram of the Banba bandgap reference. 

This architecture offers greater design flexibility. While the previous two circuits generate a 1.25 

V reference voltage, the Banba structure can produce reference voltages below 1.25 V.  

A comparative analysis of these classical bandgap reference circuits reveals their distinct 

characteristics: The Widlar reference suffers from difficulties in stabilizing the collector currents of 

the two BJTs, high sensitivity to power supply variations, and limited accuracy. The Banba reference 

operates at low supply voltages with low power consumption and could output reference voltages 

below 1 V, but its accuracy is compromised by the op-amp’s input offset voltage The Kuijk reference 

incorporates temperature compensation, exhibits low temperature coefficient and power consumption, 

and offers high power supply rejection ratio (PSRR). Although it also introduces op-amp offset, it 

demonstrates better process compatibility and broader applicability compared to the Banba structure. 

The Brokaw reference features low output noise and low quiescent power consumption, but its line 

regulation performance is relatively poor. 



 

178 

3. Second-Order Temperature Compensation for Bandgap Reference Circuits 

In 2005, Avoinne et al. first introduced the concept of second-order compensation, which utilizes 

a quadratic curvature correction term to mitigate the high-order temperature coefficient of 

conventional first-order bandgap references, thereby achieving a significantly reduced temperature 

coefficient [5]. This compensation scheme aims to cancel the higher-order terms in the temperature 

coefficient. By introducing a compensation voltage term for higher-order nonlinearities into the 

circuit, the influence of high-order nonlinearities on the output voltage/current is effectively canceled, 

significantly enhancing the stability and accuracy of the bandgap reference . 

3.1. Fundamental Principles of Second-Order Temperature Compensation 

Taking the base-emitter voltage (𝑉𝐵𝐸) of a BJT as an example, its value can be expressed by 

Equation: 

𝑉𝐵𝐸  =  𝑉𝑇  𝑙𝑛
𝐼𝐶
𝐼𝑆

(7) 

Where 𝐼𝑆 is a higher-order function of temperature T, as given by Equation (11): 

𝐼𝑆  =  𝑏 𝑇
4−𝑚 𝑒𝑥𝑝 (

−𝐸𝑔

𝑘𝑇
) (8) 

Compared to other constant terms, it is evident that 𝑉𝐵𝐸  exhibits a higher-order dependence on 

temperature T. Classical bandgap reference circuits address the linear component of the negative 

temperature coefficient. 

Alternatively, 𝑉𝐵𝐸  can be modeled by Equation (16): 

𝑉𝐵𝐸  (𝑇) =  𝑉𝑔(𝑇) − (𝑉𝑔(𝑇𝑅) − 𝑉𝐵𝐸  (𝑇𝑅)
𝑇

𝑇𝑅
− (𝜂 − 𝛿)𝑉𝑇 𝑙𝑛

𝑇

𝑇𝑅
) (9) 

Where TR denotes the reference temperature, η represents a temperature-dependent coefficient, 

and δ is a constant associated with the temperature behavior of the current Here, η = 4−ζ, with ζ 

representing the temperature dependence of carrier properties. When δ=1, the current exhibits a 

negative temperature coefficient; when δ=0, it exhibits a positive temperature coefficient. The term 

𝑉𝑔(𝑇) denotes the bandgap voltage at temperature 𝑇, expressed in Equation (13): 

𝑉𝑔(𝑇) =  𝑉𝑔0 − 𝑏𝑇 − 𝑐 𝑇
2 (10) 

Where 𝑉𝑔0 , b, and c are characterized over a temperature interval from 150 K to 400 K [6]. 

Bandgap references using first-order temperature compensation generally exhibit temperature 

coefficients ranging from 20 to 100 ppm/°C. By comparison, implementations incorporating second-

order compensation techniques can achieve significantly lower temperature coefficients, down to 

2.9 ppm/°C[7]. Higher-order compensation (third order and beyond) requires more complex circuitry 

and incurs higher power consumption. Considering the trade-offs between process cost and 

performance, second-order temperature compensation offers a more balanced solution. Therefore, 

this paper focuses exclusively on second-order compensation strategies, while higher-order 

approaches are not discussed. 

3.2.Introduction to Second-Order Temperature Compensation Schemes 

As indicated by Equation (13), second-order temperature compensation for 𝑉𝐵𝐸  is required. 

Advanced compensation circuits that address second and higher-order effects are referred to as 

curvature correction techniques. Current curvature correction methods primarily include: resistance 

ratio compensation, piecewise compensation, and precise compensation. 



 

179 

3.2.1. Resistance Ratio Compensation 

Resistance ratio compensation is the simplest method for higher-order temperature compensation. 

It utilizes two resistors with different temperature coefficients (TCs) to achieve the desired 

compensation [8]. A typical circuit is illustrated in Fig. 4. 

VDD

VE2

VBG

Low 
TC R3

High 
TC R4

R
1

R
5

R
6

Q
1

Q
2

R
2

 

Fig 4. Circuit diagram of the resistance compensation scheme. 

Fig. 5 illustrates a circuit incorporating two NPN transistors, denoted as 𝑄1 and 𝑄2, with an 

emitter area ratio of n:1. The circuit includes three thin-film resistors—R₁, R₂, and R₃—exhibiting 

low temperature coefficients (TC), allowing their temperature dependence to be neglected. 

Additionally, a diffusion resistor R₄ with a high temperature coefficient is employed. The values of 

R₅ and R₆ are equal. The operational amplifier forces equal potential at nodes A and B, thereby 

establishing collector currents in 𝑄1 and 𝑄2 that exhibit proportionality to absolute temperature 

(PTAT). This PTAT current is defined by the following expression: 

𝐼𝑃𝑇𝐴𝑇 = 
𝛥𝑉𝐵𝐸
𝑅1

= 𝑉𝑇  𝑙𝑛 𝑛 (11) 

Where VT denotes the thermal voltage. The resistance of R₄ equals to R40(1 + TC ∙ ∆T), where 

R40 is its nominal value at room temperature, TC denotes the temperature coefficient, and ∆T means 

the change of temperature. 

The equivalent parallel resistance of R₃ and R₄ is given by: 

𝑅3𝑙𝑙4 =  
𝑅40 𝑅3   (1 + 𝑇𝐶 · ∆𝑇)

(1 + 𝑇𝐶 · ∆𝑇)𝑅40  + 𝑅3  
 (12) 

The emitter voltage of Q₂ is equal to: 

𝑉𝐸2 = 𝐼𝑅2 (𝑅2 + 𝑅3𝑙𝑙4) (13) 

Analysis of Equation (16) indicates that by appropriately selecting the values of R₂, R₃, and R₄, the 

higher-order terms of 𝑉𝐵𝐸  can be effectively canceled, achieving low temperature drift 

characteristics. 

Compared to other techniques, this approach only introduces additional resistors to the 

conventional bandgap reference architecture, resulting in minimal increases in current consumption 

and layout area. However, it imposes stringent requirements on resistor values and their temperature 

coefficients. 



 

180 

3.2.2. Piecewise Compensation 

The piecewise compensation method involves separate high-temperature and low-temperature 

compensation circuits. The appropriate compensation circuit is selectively activated due to different 

operating temperatures. A specific circuit diagram [9] is shown in Fig. 5, which illustrates the use of 

distinct compensation circuits for high and low temperature intervals, each activated within its 

respective thermal range. 

 

Fig 5. Circuit diagram of the piecewise compensation scheme. 

The compensation circuit is divided into high-temperature and low-temperature compensation 

sections. The high-temperature compensation circuit, operating in the subthreshold region, consists 

of MOSFETs M₁₈, M₁₉, M₂₀, a negative-temperature-coefficient (NTC) resistor R₆, and a bipolar 

transistor Q₃. M₁₉ mirrors the zero-temperature-coefficient (ZTC) current I₁ flowing through M₁ or 

M₂. This current passes through R₆ (NTC), generating a negative-temperature-coefficient voltage, 

which produces a high-temperature compensation current IHT that increases with temperature. The 

gate voltage VG18 of M₁₈, operating in the subthreshold region, is equal to: 

𝑉𝐺18 = 𝐼1𝑅6  +  𝑉𝐵𝐸3 (14) 

The current IHT is equal to: 

𝐼𝐻𝑇 = 𝐼0𝐾18𝑒𝑥𝑝 (
𝑉𝐺𝑆18 − 𝑉𝑇𝐻18

𝜂𝑉𝑇
) (15) 

Where K18 and VG18 are known. 

The low-temperature compensation circuit, also operating in the subthreshold region, comprises 

M₂₀, M₂₁, and a positive-temperature-coefficient (PTC) resistor R₇. M₂₁ mirrors the ZTC current I₁ 

from M₁ or M₂. This current flows through R₇ (PTC), producing a positive-temperature-coefficient 

voltage, which generates a low-temperature compensation current ILT that increases as temperature 

decreases. The gate voltage VG20 of M₂₀ is: 

𝑉𝐺20 = 𝐼1𝑅7 (16) 

The current ILT is then given by: 

𝐼𝐿𝑇 = 𝐼0𝐾18 𝑒𝑥𝑝 (
𝑉𝐺𝑆20 − 𝑉𝑇𝐻20

𝜂𝑉𝑇
) (17) 

The overall reference voltage VREF is expressed as: 



 

181 

𝑉𝑅𝐸𝐹 = (
𝑉𝐵𝐸𝑄1
𝑅1

 +  
𝑉𝐵𝐸𝑄1 −𝑉𝐵𝐸𝑄2 

𝑅2
) (𝑅3 + 𝑅4) + (𝐼𝐿𝑇  + 𝐼𝐻𝑇)𝑅4 (18) 

In this configuration, M₁₇ provides the bias branch for the first-order temperature coefficient, while 

M₁₉ mirrors the ZTC current from M₁ and M₂ to generate a negative-TC voltage across R₆ [8]. 

The uncompensated reference voltage varies from a maximum of 1.075 V to a minimum of 

1.0733 V, with a total variation of 1.7 mV and a temperature coefficient (TC) of 9.85 × 10⁻⁶/°C. After 

compensation, the reference voltage is nearly 1.075652 V, with a TC of 1.58 × 10⁻⁶/°C, indicating a 

significant improvement in temperature stability. 

3.2.3. Piecewise Compensation 

A common approach for precision compensation involves designing compensation circuits based 

on the current or voltage characteristics of bipolar transistors with different temperature behaviors, 

building upon a first-order bandgap reference architecture.  

To address this, Huang et al. proposed a method utilizing MOSFETs and BJTs to generate a 

compensating term that deals with the higher-order nonlinear component 𝑇 ln(𝑇)in the base-emitter 

voltage [10]. The circuit module for precision compensation is illustrated in Fig 6. 

VDD

R5

Vb2

m=p

m=p m=1

M1 M2

I1Vb2

Vb1

Bias 
Circuit

I2

Icomp

 

Fig 6. Circuit diagram of the precision compensation scheme. 

In Fig 6, I1 and I2 are bias currents generated by the operational amplifier and the base-emitter 

voltage difference of the transistors, respectively. Their values are given by: 

𝐼1 = 𝑃1 𝑇  
𝑘 𝑙𝑛 2

𝑞 𝑅8
(19) 

𝐼2 = 𝑃2  
𝑉𝐵𝐸2
 𝑅9

(20) 

Where 𝑃1 and 𝑃2  are mirroring ratio coefficients. The target compensation current 𝐼𝑐𝑜𝑚𝑝 is 

expressed as: 

𝐼𝑐𝑜𝑚𝑝 =  𝐼1 𝑒𝑥𝑝

(

 
 
 √

2

𝜇(𝑇)𝐶𝑜𝑥
𝑊𝑐
𝐿𝑐

 (√𝐼1 −√
1
𝑝 𝐼1  +  𝐼2)

𝑉𝑇

)

 
 
 

(21) 



 

182 

This expression indicates that 𝐼𝑐𝑜𝑚𝑝 exhibits strong temperature dependence and nonlinearity. 

The design objective of the higher-order temperature-compensated bandgap voltage reference is to 

mitigate the 𝑇 ln(𝑇) component inherent in 𝑉𝐵𝐸  . By carefully designing the circuit, 𝐼𝑐𝑜𝑚𝑝 can be 

aligned with the 𝑇 ln(𝑇) term, thereby achieving effective compensation. Without introducing 𝐼𝑐𝑜𝑚𝑝 , 

the temperature coefficient (TC)  is 9.3 ppm/°C. After applying higher-order compensation, the TC 

is reduced to 0.611 ppm/°C, demonstrating significant improvement in nonlinear curvature correction. 

3.2.4. Other Compensation Schemes 

In 1994, Lee et al. first proposed an exponential temperature compensation scheme [11]. This 

method utilizes the temperature-dependent current gain β of bipolar transistors to generate an 

exponential compensation voltage that counteracts the higher-order curvature of 𝑉𝐵𝐸  , thereby 

enhancing its linearity. 

The current gain β exhibits an inverse exponential relationship with the emitter doping 

concentration, expressed as: 

𝛽(𝑇) =  𝛽∞  𝑒𝑥𝑝  (
−∆𝐸𝐺
𝑘𝑇

) (22) 

Where β(T) means the current gain at temperature 𝑇 , β∞ is the asymptotic current gain, ∆EG is 

the Energy bandgap reduction, and 𝑘 is the Boltzmann constant. 

This formulation indicates that as temperature TT increases, the current gain β grows exponentially, 

thereby compensating for the higher-order curvature of 𝑉𝐵𝐸  . Experimental results demonstrate that 

this approach attains a 1.45 ppm/°C temperature coefficient  without requiring additional circuitry—

only adjustments to transistor dimensions are necessary. 

4. Non-Bandgap Reference Circuits 

As the name implies, non-bandgap reference circuits do not utilize the bandgap voltage as a 

reference. Instead, as shown in Fig.7, they can be understood as CMOS-only reference circuits that 

exclude bipolar junction transistors (BJTs). Compared to bandgap reference circuits, non-bandgap 

architectures often lack effective temperature compensation mechanisms. A common implementation 

involves circuits that leverage the threshold voltage (Vth) of CMOS transistors as a reference element. 

Such implementations generally suffer from substantial temperature-induced variation owing to 

inadequate compensation mechanisms Shown below is a typical CMOS-based non-bandgap reference 

circuit [12]. 

VDD

VREF

PM1

PM2 PM5

PM3

NM1
PM4

NM2

 

Fig 7. Circuit diagram of the non-bandgap reference. 



 

183 

Circuit analysis reveals that PM₃ and NM₁ function in the subthreshold regime, whereas PM₄ 

operates in the linear region and may be represented as an equivalent resistor. The current IPTAT is 

determined as follows. Under subthreshold operation, the gate-source voltage of NM₁ complies with 

the following expression: 

 𝑉𝐺𝑆𝑁1 = 𝑉𝑇𝐻𝑁1  +  𝑚𝑉𝑇 + 𝑙𝑛 (
𝐼 𝐷𝑆𝑁1

𝑉𝑇
2𝜇𝑁1𝐶𝑜𝑥𝐾𝑁1

) (23) 

PM₄ operates in the linear region, and its equivalent resistance can be expressed: 

𝑅𝑜𝑛𝑃4 = 
1

𝜇𝑃 𝐶𝑜𝑥 𝐾𝑃4(𝑉𝐺𝑆𝑁1  − 𝑉𝑇𝐻𝑃4)
 (24) 

Let μP Cox KP4 = 𝛽𝑃4. Differentiating both sides of Equation (24) with respect to temperature 

yields:
𝜕𝑅𝑜𝑛𝑃4

𝜕𝑇
 =  

𝜕𝑉𝑇𝐻

𝜕𝑇
= 

1

𝛽𝑃4
2( 𝑉𝐺𝑆𝑃4  − 𝑉𝑇𝐻𝑃4)

2  (25) 

From Equation (25), it is evident that since 
∂VTH

∂T
 is negative, the equivalent resistance has a 

negative temperature coefficient, meaning its value decreases as temperature increases. Given that 

the currents in the two branches of the bias circuit are equal, i.e.,  IDSN1  = IDSP4  =  IPTAT , the 

following holds: 

 𝑉𝐺𝑆𝑁1 = 𝐼 𝐷𝑆𝑃4 ∙ 𝑅𝑜𝑛𝑃4  (26) 

Substituting Equation (24) and IDSN1  = IDSP4 =  IPTAT into Equation (26): 

𝐼𝑃𝑇𝐴𝑇 = 

 𝑉𝑇𝐻𝑁1 +𝑚𝑉𝑇 𝑙𝑛 (
 𝐼𝑃𝑇𝐴𝑇

𝑉𝑇
2𝜇𝑁1𝐶𝑜𝑥𝐾𝑁1

)

𝑅𝑜𝑛𝑃4
 (27)

 

To produce the reference voltage, the current is replicated into the NM₂ branch via current 

mirroring. Assuming the aspect ratios of PM₁, PM₂, and PM₅ are 1 : 1 : a, the current mirror gain is 

1 : a, yielding the current through NM₂ as: 

𝐼𝐷𝑆𝑁2 =  𝑎𝐼𝑃𝑇𝐴𝑇 (28) 

Based on the subthreshold I-V characteristics, the gate-source voltage of NM₂, i.e., VREF is equal 

to: 

𝑉𝑅𝐸𝐹 =  𝑉𝐺𝑆𝑁2 = 𝑉𝑇𝐻𝑁2 +𝑚𝑉𝑇 𝑙𝑛 (
𝑎 𝐼𝑃𝑇𝐴𝑇

𝑉𝑇
2𝜇𝑁2𝐶𝑜𝑥𝐾𝑁2

) (29) 

Let μN2CoxKN2 = 𝛽𝑁2. Differentiating both sides of Equation (28) with respect to temperature T 

gives: 

𝜕𝑉𝑅𝐸𝐹
𝜕𝑇

=  
𝜕𝑉𝑇𝐻𝑁2
𝜕𝑇

+𝑚
𝑘

𝑞
𝑙𝑛 (

𝑎𝐼𝑃𝑇𝐴𝑇

𝑉𝑇
2𝛽𝑁2

) +𝑚𝑉𝑇 ∙
𝑉𝑇
2𝛽𝑁2

𝑎𝐼𝑃𝑇𝐴𝑇
∙
𝑎𝑉𝑇

2𝛽𝑁2
𝜕𝐼𝑃𝑇𝐴𝑇
𝜕𝑇

− 2
𝑘
𝑞 𝑉𝑇𝛽𝑁2𝐼𝑃𝑇𝐴𝑇

(𝑉𝑇
2𝛽𝑁2)2

(30) 

In the above equations, VTH denotes the threshold voltage; 𝑚 represents the subthreshold slope 

factor; k stands for the Boltzmann constant; Cox indicates the gate oxide capacitance; 𝑞 means the 

electron charge; VT = 
𝑘 𝑇

𝑞
 defines the thermal voltage; μ corresponds to the carrier mobility; and Kn 

= (𝑊/𝐿)𝑛 gives the MOS aspect ratio. 

Satisfying the condition 
𝜕VREF

𝜕𝑇
= 0 in Equation (37) is challenging, indicating that non-bandgap 

circuits exhibit limited temperature stability. However, their simple structure and fully CMOS-



 

184 

compatible implementation result in low power consumption, making them suitable for basic 

reference voltage applications in everyday scenarios. 

Additionally, some studies have proposed alternative approaches, such as using capacitive biasing 

with active body diodes to generate PTAT and CTAT voltages, replacing traditional BJT-based 

methods to design BJT-free sub-bandgap reference circuits [13]. A specific circuit diagram is shown 

in Fig. 8. 

VDD

C3

C1

C2

M5

M6

M3 M4

VSS

M1 M2

D1

Tc

Ts2

Ts1 Tc

B

VREFTa

Cbg

 

Fig 8. Circuit diagram of the BJT-free sub-bandgap reference. 

Leveraging the superior electrical characteristics and strong process compatibility of body diodes, 

Eberlein et al. proposed a resistor-free sub-bandgap reference circuit in 2020, which replaces 

traditional BJTs with body diodes and incorporates capacitive biasing. The capacitor is initially 

charged to a voltage exceeding the diode’s built-in potential (>0.8 V) and subsequently discharged 

through the diode. After a brief transient period, the diode voltage exhibits a logarithmic dependence: 

𝑉𝐷(𝑡)  ≅  𝜂 ∙ 𝑉𝑇 ∙ 𝑙𝑛 (
𝐶 ∙ 𝜂 ∙ 𝑉𝑇
𝐼𝑆 ∙ 𝑡

) (31) 

where η is the diode ideality factor, 𝑉𝑇 is the thermal voltage (proportional to temperature), 𝐼𝑆 is 

the reverse saturation current, 𝐶  is the capacitance, and 𝑡 is the discharge time. Over multiple 

decades of discharge time, 𝑉𝐷(𝑡) strictly adheres to this logarithmic behavior and is nearly 

independent of the initial bias condition. 

The circuit operates by charging and discharging capacitors 𝐶1 and 𝐶2 to generate a diode voltage 

𝑉𝐷with CTAT (Complementary to Absolute Temperature) characteristics. Through reverse charge 

sharing between capacitors, the bandgap reference voltage VREF is derived as: 

𝑉𝑅𝐸𝐹 = 𝑉𝐷(𝑡𝑠2) −
𝐶1

𝐶1 + 𝐶3
𝑉𝐷(𝑡𝑠1) (32) 

This scheme achieves an output voltage 𝑉𝑅𝐸𝐹 ≈ 370 𝑚𝑉 , operates within a supply 

voltage 𝑉𝐷𝐷 range of 0.85 – 1.05 𝑉, consumes only 18  𝑛𝑊 (including switch driving circuitry), and 

occupies a chip area of 1680 𝜇𝑚2, demonstrating significant advantages over comparable solutions. 

Four historically significant circuits—the Widlar reference, the Banba reference, the series 

capacitive-biased reference, and the piecewise-compensated reference—are analyzed in this paper. 

The key performance metrics are summarized in Table 1. 

The results indicate that non-bandgap reference circuits, represented by the series capacitive-

biased architecture, achieve smaller area and lower power consumption. In contrast, bandgap-based 

references maintain superior performance in temperature coefficient (TC) tuning and operating 

temperature range. 



 

185 

Furthermore, with ongoing technological advancements, modern reference circuits continue to 

evolve toward higher process compatibility and improved accuracy, enabling robust integration into 

advanced analog and mixed-signal systems. 

Table 1. Four circuits compared. 

 Widlar[1] Banba[4] piecewise-compensated[9] series capacitive-biased[13] 

Power ~1.5A 2.2μA 20.44μA 18nW 

∆𝑉𝑅𝐸𝐹(mV) 20 3 0.28 2.7 

TC(ppm/°C) ~200 N/A 1.58 ~44 

Area(𝑚𝑚2) N/A 0.1 N/A 0.0016 

5. Development Status 

With technological advancements, bandgap circuits have witnessed numerous innovations. These 

include self-calibration techniques for fine-tuning the reference voltage to achieve high-precision 

output [14], the integration of super source followers to enhance voltage stability, and the adoption 

of novel processes and materials to improve overall performance. Meanwhile, modern applications 

impose increasingly stringent requirements on power consumption and stability of bandgap 

references [15]. Challenges such as inherent random offset in operational amplifiers and current 

mirror mismatches in CMOS-based bandgap circuits remain to be addressed. In the future, with 

continued progress in CMOS, BiCMOS, and quantum technologies, the temperature stability and 

process precision of bandgap reference circuits are expected to advance significantly. 

6. Conclusion 

Voltage references based on bandgap and non-bandgap principles are critical components in 

analog circuit design. To facilitate a comprehensive understanding of reference circuits, this paper 

chronologically reviews the evolution of voltage reference architectures, provides an in-depth 

analysis of various typical second-order temperature compensation schemes, and highlights both 

breakthroughs and ongoing challenges in bandgap circuit development. Furthermore, a comparative 

study of non-bandgap reference circuits (CMOS-only) is presented, evaluating the advantages and 

disadvantages of BJT-based and BJT-free approaches. It is hoped that this work offers valuable 

insights into future research in Bandgap reference circuit design. 

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