Single-Stage CMOS Operational Transconductance Amplifiers (OTAs): A Design Tutorial

Abstract

This paper presents a comprehensive design tutorial for four types of single-stage operational transconductance amplifiers (OTAs): (1) five-transistor OTAs, (2) telescopic cascode OTAs, (3) folded cascode OTAs, and (4) current mirror OTAs. These OTAs serve as fundamental building blocks in analog circuits. The operational principles of each OTA are reviewed, and the key performance metrics are derived through a hand analysis. These performance metrics encompass most crucial parameters, including small-signal parameters, frequency response, input and output swing ranges, rising and falling slew rates, nonidealities, and bias circuit simplicity. All of these metrics are verified and compared using the simulation. Furthermore, the practical applications of each OTA are summarized, and a case study demonstrates the enhancement of a neural recording amplifier’s performance through appropriate OTA selection. A thorough review of the essential building blocks will become a stepping stone to design high-performance analog amplifiers across diverse applications.

1. Introduction

Operational transconductance amplifiers (OTAs), which convert an input voltage to an output current, play crucial roles in a wide range of electronic systems. G_m-C filters, based on OTAs, have been employed in sensor interface integrated circuits (ICs) [1,2,3], biopotential recording circuits [4,5,6], and wireless communication integrated circuits (ICs) [7,8,9] to attenuate interferers with a wide tuning range of cut-off frequencies. OTA-based integrators with an inherent high gain and low-pass frequency response have been used in delta-sigma analog-to-digital converters (ADCs) [10,11,12]. Moreover, OTAs themselves have served as feedforward amplifiers of closed-loop systems such as capacitively coupled instrumentation amplifiers (CCIAs) [13,14,15] and power ICs [16,17,18,19,20] thanks to their high gain performance. The versatility and wide applicability of OTAs make them indispensable in various analog and mixed-signal circuit designs, emphasizing the paramount importance of OTA design.

Single-stage OTAs have gained significant interest due to their inherent simplicity and relatively lower power consumption compared to multi-stage OTAs, making them highly attractive for integration into ICs. Moreover, they can serve as the input stage of multi-stage operational amplifiers (OP-AMPs) and multi-stage OTAs. As a result, a comprehensive understanding of single-stage OTAs becomes crucial in the pursuit of high-performance analog circuit designs. This paper focuses on providing a comprehensive design tutorial for various types of single-stage OTAs, serving as a solid foundation for early-stage circuit designers.

To achieve this goal, this paper presents a comprehensive design tutorial covering four different types of single-stage OTAs: (1) five-transistor OTAs, (2) telescopic cascode OTAs, (3) folded cascode OTAs, and (4) current mirror OTAs. These OTAs are widely used and representative among various single-stage OTAs. Moreover, these four types are basic structures of single-stage OTAs that are included in the textbook [21,22,23]. Understanding these structures and driving performance metrics could be helpful for early-stage circuit designers to have a foundation to explore more advanced structures. For each OTA, the operation principle is reviewed, and the key performance metrics are derived through a hand analysis and then verified based on a rigorous simulation. Furthermore, the practical applications of each OTA are summarized. Finally, a case study is provided to demonstrate the tutorial’s practical implementation. A suitable OTA structure is chosen based on the analytical results, thereby improving the performance of a conventional neural recording amplifier. A thorough analysis of the basic building blocks serves as a valuable resource for designing high-performance analog amplifiers across diverse applications. It is important to note that this tutorial goes beyond conventional textbooks [21,22,23,24,25], offering a more practical and applied approach through a comparative analysis, application discussions, and a detailed case study.

The rest of this paper is organized as follows: Section 2 introduces the background of OTAs, with a primary focus on performance metrics. Section 3 elaborates on the operational principles and performance metrics derived through a hand analysis. Section 4 presents the simulation results and comparisons. In Section 5, the applications and a case study are discussed. Finally, Section 6 concludes the paper, summarizing the findings and implications of the comprehensive OTA design tutorial.

2. Background

In this section, we review the most crucial performance metrics of OTAs that will be analyzed in depth throughout the paper. In addition, we provide a brief explanation for the motivations behind the four types of single-stage OTAs: (1) five-transistor OTAs, (2) telescopic cascode OTAs, (3) folded cascode OTAs, and (4) current mirror OTAs. These performance metrics and design motivations are fundamental for understanding the characteristics and applications of each OTA type, paving the way for a comprehensive analysis and comparisons in the subsequent sections.

Figure 1 presents the electrical performance metrics of OTAs, encompassing various crucial parameters such as small-signal parameters, frequency response, input and output swing ranges, rising and falling slew rates, circuit noise, and bias circuit simplicity. The structure of the OTA plays a significant role in determining these metrics and, consequently, affects the overall performance and behavior of the amplifier. Understanding how these parameters are influenced by the OTA’s structure is vital for optimizing the design and achieving high-performance analog amplifiers across different applications.

The analysis of the small-signal parameters and frequency response is crucial in OTA design, especially considering that the input of the amplifier is typically a small AC signal. Among the small-signal parameters, two fundamental ones, effective transconductance (G_m) and output resistance (r_out), will be analyzed thoroughly. The G_m represents the ability to convert an input voltage to an output current, and the output resistance (r_out) represents the capability to deliver a constant current regardless of the load conditions. These two small-signal parameters, G_m and r_out, are dominant parameters that determine the frequency response of a single-stage OTA, which includes DC gain (A_o), 3-dB angular frequency (ω_3dB), and unity-gain angular bandwidth (ω_GB). Properly setting these parameters according to the desired signal bandwidth is essential for optimizing the OTA’s performance in specific applications.

The input and output swing ranges are also crucial performance metrics that determine the maximum and minimum allowable input and output voltage levels for the proper operation of the OTA. These swing ranges are particularly important when the OTA is placed in front of another amplifier, ensuring compatibility and preventing signal distortion. Having sufficient swing ranges allows for the OTA to handle a wide range of input signals without clipping or distortion, which is essential for maintaining the integrity of the amplified signal in practical applications.

Another important performance metric is the slew rate (SR). Amplifiers often handle fast input signals with frequencies higher than ω_GB, and the SR indicates how fast the OTA drives the output in this condition. As shown in Figure 1, there are two types of SRs: rising SR and falling SR. Steeper slopes of the output signal for rising and falling edges indicate an OTA that has a better performance.

The noise of the OTA is also a crucial metric when the input signal is small and/or the target system requires a high resolution. In particular, since the total noise of the system is dominant at the front end of the system, the OTA used for the front end should have a low noise level. Although it is impossible to eliminate the noise entirely, it is necessary to analyze the circuit noise to minimize it as much as possible. The same analysis can be extended to other circuit nonidealities such as offset and mismatches.

At last, considering that the advantage of single-stage OTAs lies in their simple structure, this simplicity should extend to the implementation of bias circuits. Bias circuits play critical roles in providing the necessary DC operating conditions for the OTA, ensuring its stable and optimal performance. However, if complex bias circuits are needed, this can compromise the benefits of using a single-stage OTA. Therefore, to fully leverage the advantages of single-stage OTAs, it is important to keep the bias circuits simple and straightforward. This will allow for the efficient and effective use of single-stage OTAs in analog circuit design.

A single transistor can be regarded as a voltage-controlled current source that has its own G_m and r_out. That is, for single-ended signals and a sufficiently large r_out, a single transistor can be used as a transconductance amplifier (TA) that converts an input voltage to an output current. However, for differential signaling, high load impedance, or high driving capability requirements, more advanced single-stage OTAs are required. The four types of single-stage OTAs reviewed in this paper are designed with differential signaling. The telescopic cascode OTA and folded cascode OTA achieve much higher output impedance than a single transistor. A current mirror OTA is suitable for achieving a high driving capability. In Section 3, all of these OTA types will be analyzed, and their performance metrics will be evaluated.

3. Hand Analysis

This section provides a quantitative analysis of the performance metrics of four types of single-stage OTAs, as introduced in Section 2. This analysis offers valuable insights into the capabilities and limitations of each OTA type, aiding in the selection of the most suitable one for specific applications.

3.1. Five-Transistor OTA

The five-transistor OTA shown in Figure 2 is one of the most basic and widely used OTAs. The input transistors, M₁ and M₂, form a differential transconductor. An active current mirror formed by M₃ and M₄ is loaded onto the transconductor. M₅ serves as a tail current source, providing a current to the other four transistors.

See Figure 2a to look into the design of the bias circuit and operation at equilibrium. The current source, I_b,5, is copied to M₅ via the diode-connected transistor. In general, the most clean and/or precise current, which is generated via a bandgap reference, can be copied simply by only one more diode-connected transistor. This simple bias circuit becomes one of the advantages of a five-transistor OTA compared to OTAs based on cascode structure. At the equilibrium, M₁ and M₂ generate current signals, I_D1 and I_D2, respectively. Since I_D1 is fed into M₃, the current flowing through M₄ is same as in I_D1. As shown in Figure 2a, when two input signals have the same voltage level, I_D1 and I_D2 have the same magnitude with the same direction. I_D1 and I_D2 are canceled, and no current flows to the load. In contrast, when two input signals have the same DC voltage level with a small AC differential signal, v_id, a current flows to the load due to the different directions of I_D1 and I_D2.

See Figure 2b for a small signal analysis with a differential input signal. When M₁ and M₂ are matched with the same transconductance, represented as g_m, M₁ and M₂ generate i_out = g_m(v_id/2) and −i_out. Note that the current flowing through M₄ is also i_out due to the current mirror. Two currents from M₂ and M₄ are merged together at the output node, thereby flowing 2i_out = g_mv_id to the load. In this regard, the G_m of a five-transistor OTA, G_{m,5TR_OTA}, is expressed as follows:

$$G_{m,5TR\_OTA}=g_{m1}=g_{m2}=g_m$$

(1)

where g_m1 and g_m2 are the transconductances of M₁ and M₂, respectively.

Now, let us look into the r_out of a five-transistor OTA. At the output node, r_o2 and r_o4 are connected in parallel. Therefore, when M₁ and M₂ are matched with the same output impedance, represented r_o, and when r_o2 is much higher than 1/g_m1, the r_out of a five-transistor OTA, r_{out,5TR_OTA}, is expressed as follows:

$$r_{out,5TR\_OTA}=r_{o2}||r_{o4}=\frac{r_o}{2}$$

(2)

where r_o2 and r_o4 are the output impedances of M₂ and M₄, respectively. The higher the r_o, the higher the r_{out,5TR_OTA}. A high r_{out,5TR_OTA} is required to provide a constant current regardless of the load resistance or capacitance.

Next, we analyze the frequency response of a five-transistor OTA. Since a single-stage OTA is a single-pole system, A_o, ω_3dB, and ω_GB are enough to determine the frequency response of a single-stage OTA. Moreover, ω_3dB and ω_GB as well as A_o are obtained from the G_m and r_out. A_o, ω_3dB, and ω_GB of a five-transistor OTA, represented as A_{o,5TR_OTA}, ω_{3dB,5TR_OTA}, and ω_{GB,5TR_OTA}, are expressed as follows:

$$A_{o,5TR\_OTA}=-G_{m,5TR\_OTA}·r_{out,5TR\_OTA}=-\frac{1}{2}·g_m·r_o,$$

(3)

$${\omega }_{3dB,5TR\_OTA}=\frac{1}{r_{out,5TR\_OTA}·C_{Load}}=\frac{1}{{(r}_o/2)·C_{Load}},$$

(4)

$${\omega }_{GB,5TR\_OTA}=\left|A_{o,5TR\_OTA}\right|·{\omega }_{3dB,5TR\_OTA}=\frac{g_m}{C_{Load}},$$

(5)

where C_Load is the capacitance of the load capacitor. At DC, the load capacitor is opened, generating a voltage from G_{m,5TR_OTA} and r_{out,5TR_OTA}. As the frequency increases, the load capacitor cannot be ignored, forming a pole with r_{out,5TR_OTA}.

The next performance metrics, the input and output swing ranges, are illustrated in Figure 2c. Here, V_D,SAT is the minimum V_ds to maintain the saturation operation of the transistors. And V_gs is set to V_D,SAT + V_th at a minimum, where V_th is the threshold voltage of the transistor. When V_D,SAT and V_th of all transistors are same, the input voltage signal must be higher than 2V_D,SAT + V_th and lower than V_DD − V_D,SAT. In the same condition, since two NMOSs (M₂ and M₅) are operated in a saturation region, the lowest output voltage level is 2V_D,SAT. The highest output voltage level is limited to V_DD − V_D,SAT since M₄ should be operated in the saturation region. Compared to other types of single-stage OTAs, a five-transistor OTA has moderate input and output swing ranges that are neither wide nor narrow. Especially when the input DC bias voltage is relatively low, a PMOS-based five-transistor OTA is more suitable.

To look into other important performance metrics, SR_R and SR_F, unity-gain buffers are designed by five-transistor OTAs as shown in Figure 2d,e. In Figure 2d, to derive the SR_R of a five-transistor OTA, represented as SR_{R,5TR_OTA}, a step response is applied with a very short rising time. To derive the falling SR_F of a five-transistor OTA, represented as SR_{F,5TR_OTA}, a step response is applied with a very short falling time, as shown in Figure 2e. This means that the signal is much faster than the OTA bandwidth (see Figure 2d first). When a step response is applied with a rising edge, M₁ turns on and M₂ turns off. Then, all the current flowing through M₅ flows through M₁, which is copied to M₄ by the current source that is composed of M₃ and M₄. Since the current transferred to M₄ does not flow to M₂ but flows to C_Load, the output voltage rises linearly along with the step response. Conversely, when a step response is applied with a falling edge, M₁ turns off and M₂ turns on. I_D5 does not flow to M₁, M₃, and M₄, but flows to C_Load through M₂. The output voltage decreases linearly along with the step response. Note that the output voltage is a result of current accumulation on the capacitor. As a result, SR_{R,5TR_OTA} and SR_{F,5TR_OTA} are expressed as follows:

$${SR}_{R,5TR\_OTA}={SR}_{F,5TR\_OTA}=\frac{I_{D5}}{C_{Load}}.$$

(6)

The next performance metrics that we analyze are nonidealities including circuit noise, offset, and mismatches. Since the voltage-to-current conversion and amplification are mainly performed by M₁ and M₂, the nonidealities of these two input transistors become the most dominant sources that cause performance degradation. In contrast, the nonidealities of M₅ become less dominant to the overall performance since they can be canceled out at the output port with perfectly matched M₃ and M₄. As a result, the effect of the nonidealities of M₁ and M₂ on the overall performance is greater than that of M₃ and M₄, and that of M₃ and M₄ is greater than that of M₅. Therefore, in general, circuit designers reduce the performance degradation from the nonidealities by increasing the size of M₁ and M₂. Moreover, since the mismatch between M₃ and M₄ cause a systematic offset, a careful layout is needed to reduce the mismatch.

3.2. Telescopic Cascode OTA

The telescopic cascode OTA in Figure 3 is a type of single-stage OTA that achieves a higher r_out and A_o compared to those of a five-transistor OTA. In particular, the telescopic cascode OTA can obtain a high r_out and A_o without an additional stage, thereby achieving power efficiency as well.

Like in a five-transistor OTA, M₁ and M₂ generate current signals, represented as I_D1 and I_D2, respectively. And in a cascode current mirror composed of M₃, M₄, M₈, and M₉, I_D1 is copied to M₄. As shown in Figure 3a, when two input signals have the same voltage level, no current flows to the load. In contrast, when v_id is applied to the input nodes, a current flows to the load due to the different directions of I_D1 and I_D2.

See Figure 3b for a small signal analysis with a differential input signal. The principle of the voltage-to-current conversion of a telescopic cascode OTA is the same as that of a five-transistor OTA. Therefore, when M₁ and M₂ are matched with g_m, the G_m of a telescopic cascode OTA, G_{m,TC_OTA}, is expressed as follows:

$$G_{m,TC\_OTA}=g_{m1}=g_{m2}=g_m.$$

(7)

In contrast, the r_out and A_o are boosted further thanks to the cascode current mirror. At the output node, the r_out formed by two NMOSs and the r_out formed by two PMOSs are connected in parallel. Therefore, when transistors are matched with the same g_m and r_o, the r_out of a telescopic cascode OTA, r_{out,TC_OTA}, is expressed as follows:

$$r_{out,TC\_OTA}=\left(r_{o4}·g_{m9}·r_{o9}\right)||\left\{\left(\frac{1}{g_{m1}}·g_{m2}·r_{o2}\right)·g_{m7}·r_{o7}\right\}=\frac{1}{2}·g_m·{r_o}^2.$$

(8)

The r_{out,TC_OTA} is g_mr_o times higher than r_{out,5TR_OTA} in (2). The r_out is amplified by the voltage gain of a single transistor through the cascode current mirror.

Since a telescopic cascode OTA is also a type of single-pole system, A_o, ω_3dB, and ω_GB determine the frequency response. A_o, ω_3dB, and ω_GB of a telescopic cascode OTA, represented as A_{o,TC_OTA}, ω_{3dB,TC_OTA}, and ω_{GB,TC_OTA}, are expressed as follows:

$$A_{o,TC\_OTA}=-G_m·r_{out}=-\frac{1}{2}·{\left(g_m·r_o\right)}^2,$$

(9)

$${\omega }_{3dB,TC\_OTA}=\frac{1}{r_{out,TC\_OTA}·C_{Load}}=\frac{1}{{(g}_m·{r_o}^2/2)·C_{Load}},$$

(10)

$${\omega }_{GB,TC\_OTA}=\left|A_{o,TC\_OTA}\right|·{\omega }_{3dB,TC\_OTA}=\frac{g_m}{C_{Load}}.$$

(11)

A_{o,TC_OTA} in (9) is g_mr_o times higher than A_{o,5TR_OTA} in (3), while ω_{3dB,TC_OTA} in (10) is g_mr_o times lower than ω_{3dB,5TR_OTA} in (4). Note that ω_{GB,TC_OTA} is same as ω_{GB,5TR_OTA} in (5). The gain of a telescopic cascode OTA is higher than that of a five-transistor OTA for the frequency range less than ω_{3dB,5TR_OTA}/2π. In contrast, the gain of a telescopic cascode OTA is the same as that of a five-transistor OTA for the frequency range from ω_{3dB,5TR_OTA}/2π to ω_{GB,TC_OTA}.

Although the cascode current mirror improves the gain and r_out further, it degrades the input and output swing ranges. Among various types of bias circuits, the bias circuits shown in Figure 3a are employed to minimize the degradation of the swing range (see M₈ and M₉ first). M₈ and M₉ are biased by a diode-connected PMOS with a current source, where the transistor size of the diode-connected PMOS is four times smaller than those of M₅ and M₉. This diode-connected PMOS generates a voltage difference of 2V_D,SAT + V_th from V_DD, which is the minimum voltage that allows for both transistors to operate in the saturation region. In this bias condition, M₈ and M₉ can be operated with minimum |V_ds| of V_D,SAT. Similarly, to allow for M₂ and M₇ to operate in the saturation region with a minimum |V_ds| of V_D,SAT, a diode-connected NMOS with a current source where the transistor size of the diode-connected NMOS is four times smaller than those of M₆ and M₇ is employed. When the V_D,SAT and V_th of all transistors are the same, the input voltage signal must be higher than 2V_D,SAT + V_th and lower than V_DD − 2V_D,SAT. In the same condition, since three NMOSs are cascoded at the output, the lowest output voltage level is 3V_D,SAT. Similarly, the highest output voltage level is limited to V_DD − 2V_D,SAT due to two cascoded PMOSs at the output node. Compared to the swing ranges of a five-transistor OTA, the telescopic cascode OTA has a more narrow swing range due to the cascade current mirror. Moreover, as shown in Figure 3a, the complexity and power consumption of bias circuits are increased. That is, the gain and r_out of a telescopic cascode OTA are improved at the sacrifice of swing ranges and the complexity of bias circuits.

The SR_R and SR_F of a telescopic cascode OTA are analyzed in the same way as a five-transistor OTA. As shown in Figure 3d,e, in the unity-gain buffer configuration, the additional cascode transistors do not affect the SR_R and SR_F. M₆, M₇, M₈, and M₉ just follow the operation of M₁, M₂, M₃, and M₄, respectively. The SR_R and SR_F of a telescopic cascode OTA, represented as SR_{R,TC_OTA} and SR_{F,TC_OTA}, are expressed as follows:

$${SR}_{R,TC\_OTA}={SR}_{F,TC\_OTA}=\frac{I_{D5}}{C_{Load}}.$$

(12)

The nonidealities of additional transistors, M₆–M₉, cause negligible performance degradation. The nonidealities of additional transistors, M₆–M₉, are transferred to the output with a low gain. Therefore, as in a five-transistor OTA, the effect of the nonidealities of M₁ and M₂ on the overall performance is greater than that of M₃ and M₄, and that of M₃ and M₄ is greater than that of M₅.

3.3. Folded Cascode OTA

The folded cascode OTA shown in Figure 4 is employed to extend the input swing ranges of the telescopic cascode OTA while maintaining a high gain and high r_out. The transformation from a telescopic cascode OTA to a folded cascode OTA involves modifying the transistor configuration and biasing arrangement. In the NMOS-input telescopic cascode OTA, the input stage consists of a differential pair, followed by NMOS cascode transistors connected to the load, as shown in Figure 3. On the other hand, the folded cascode OTA replaces the NMOS cascode transistors with four PMOS transistors configured in a folded arrangement between the input stage and the output stage. Two PMOS transistors, M₈ and M₉, serve as the cascode transistors, while the other PMOS transistors, M₃ and M₄, act as the current sources. The current sources provide bias currents to the input stage and output stage.

The modification improves the input swing range, but as shown in Figure 3a and Figure 4a, one more diode-connected transistor is required when the same wide-swing cascode current mirrors are employed. Fortunately, the bias circuit for M₈ and M₉ and the bias circuit for M₁₀ and M₁₁ can share one bias current, consuming the same power compared to the bias circuits for a telescopic cascode OTA.

See Figure 4a first to show the operation at equilibrium. The current flowing through M₉, denoted as I_D9, is the same as I_D4 − I_D2, according to Kirchhoff’s Current Law (KCL). The current flowing through M₆, denoted as I_D6, is copied to M₇, where I_D6 is I_D3 − I_D1 = I_D4 − I_D2. Therefore, like a five-transistor OTA and a telescopic cascade OTA, no current flows to the load when two input signals have the same voltage level.

The small-signal parameters are also the same as those of a telescopic cascade OTA. See Figure 4b for a small signal analysis with differential input signals. The principle of a voltage-to-current conversion of a folded cascode OTA is the same as that of a telescopic cascode OTA. Since the differential voltage inputs are converted to a current by M₁ and M₂, the G_m of a folded cascode OTA, G_{m,FC_OTA}, is expressed as follows:

$$G_{m,FC\_OTA}=g_{m1}=g_{m2}=g_m$$

(13)

where M₁ and M₂ are matched with g_m. Likewise, the r_out is boosted further thanks to the cascode current mirrors. At the output node, the r_out formed by M₇ and M₁₁ and the r_out formed by two M₂ and M₉ are connected in parallel. Since M₉ works as a common-gate amplifier from the input signal point of view like M₁₁, M₂ and M₉ also increase the r_out. Therefore, when the transistors are matched with the same g_m and r_o, the r_out of a folded cascode OTA, r_{out,TC_OTA}, is expressed as follows:

$$r_{out,FC\_OTA}=\left(r_{o4}·g_{m9}·r_{o9}\right)||\left(r_{o7}·g_{m11}·r_{o11}\right)=\frac{1}{2}·g_m·{r_o}^2.$$

(14)

Since the small-signal parameters are same as those of a telescopic cascade OTA, A_o, ω_3dB, and ω_GB of a folded cascode OTA, denoted as A_{o,TC_OTA}, ω_{3dB,TC_OTA}, and ω_{GB,TC_OTA}, are expressed as follows:

$$A_{o,FC\_OTA}=-G_m·r_{out}=-\frac{1}{2}·{\left(g_m·r_o\right)}^2,$$

(15)

$${\omega }_{3dB,FC\_OTA}=\frac{1}{r_{out,FC\_OTA}·C_{Load}}=\frac{1}{{(g}_m·{r_o}^2/2)·C_{Load}},$$

(16)

$${\omega }_{GB,FC\_OTA}=\left|A_{o,FC\_OTA}\right|·{\omega }_{3dB,FC\_OTA}=\frac{g_m}{C_{Load}}.$$

(17)

The same frequency response characteristic as that of a telescopic cascode structure is expected.

As shown in Figure 4c, the folded cascade OTA has a wider input swing range compared to the telescopic cascade OTA due to the use of a PMOS cascode transistor in the folded arrangement. In a telescopic cascode OTA, the NMOS cascode transistor, M₉, restricts the input swing range as it requires a minimum voltage headroom. However, the PMOS cascode transistor in a folded cascode OTA, M₉, allows for the input voltage to swing more positively, expanding the input range and enabling operation in a wider voltage range. In contrast, the output swing range of a folded cascode OTA is same as that of a telescopic cascode OTA since M₄, M₉, M₇, and M₁₁ require a minimum voltage headroom.

Figure 4d,e show the slewing of a folded cascode OTA for the rising and falling step responses, respectively. In general, I_D3 and I_D4 are set to be higher than I_D5 [21]. Let us first look at Figure 4d,e, where I_D3 and I_D4 are set to be same as I_D5. The rising step response turns on M₁, allowing for M₁ and M₅ to become sinking paths of I_D3. No current flows to M₈ due to I_D3 and I_D5 being equal, which turns off M₂, and the NMOS cascode current mirror is composed of M₆, M₇, M₁₀, and M₁₁. As a result, according to the KCL, the I_D4 flows to C_Load through M₉, and the rising SR of a folded cascode OTA, SR_{R,FC_OTA}, is expressed as follows:

$${SR}_{R,FC\_OTA}=\frac{I_{D4}}{C_{Load}}=\frac{I_{D5}}{C_{Load}}.$$

(18)

When the falling step response is applied, M₁ turns off and M₂ turns on, allowing for M₂ and M₅ to become sinking paths of I_D4 and turning off M₉. Therefore, the NMOS cascode current mirror composed of M₆, M₇, M₁₀, and M₁₁ sinks the current from C_Load, where the sinking current magnitude is I_D3. As a result, the falling SR of a folded cascode OTA, SR_{F,FC_OTA}, is expressed as follows:

$${SR}_{F,FC\_OTA}=\frac{I_{D3}}{C_{Load}}=\frac{I_{D5}}{C_{Load}}.$$

(19)

where I_D3 = I_D4 = I_D5, and SR_{R,FC_OTA} and SR_{F,FC_OTA} are the same as SR_{R,FC_OTA} and SR_{F,FC_OTA}, respectively.

When I_D3 = I_D4 > I_D5 and the rising step response is applied, M₂ does not turn off perfectly, flowing to I_D3 − I_D5. Therefore, SR_{R,FC_OTA} also follows (18) despite the increasing current. When I_D3 = I_D4 > I_D5 and the rising step response is applied, the sourcing current I_D4 − I_D5 flows through M₉, which has a different direction of the sinking current I_D3 flowing through M₇ and M₁₁. SR_{F,FC_OTA} also follows (19) despite the increasing current. As a result, setting I_D3 and I_D4 as larger than I_D5 to increase SR_{R,FC_OTA} and SR_{F,FC_OTA} is not effective since SR_{R,FC_OTA} and SR_{F,FC_OTA} are independent of I_D3 and I_D4.

The performance degradation due to the nonidealities of a folded cascode OTA includes the performance degradation due to telescopic nonidealities. Like the telescopic cascade amplifier, the nonidealities of cascoded transistors, M₈–M₁₁, cause negligible performance degradation due to the low gain. As in a folded cascade OTA, the effect caused by the nonidealities of M₁ and M₂ on the overall performance is greater than those caused by other transistors. However, in general, the performance degradation by two additional current sources, M₃ and M₄, are not negligible, thus reducing the nonidealities of M₃ and M₄ by increasing the gate size, and so on. This added noise is one of the disadvantages of a folded cascode OTA.

3.4. Current Mirror OTA

Telescopic and folded cascode OTAs are employed to increase the r_out, thus achieving a high A_o. However, as a trade-off, the output swing ranges of these two OTAs and one of the performance metrics related to speed, ω_GB, are limited due to the cascode output stages. Furthermore, another performance metric related to speed, SR, is also limited by the magnitude of the input bias current.

See Figure 5a to look into the design of the bias circuit and operation at equilibrium. Like a five-transistor OTA, the current source, I_b,5, is copied to M₅ via the diode-connected transistor. Since M₃ and M₄ are diode connected, the V_GS of M₃ and M₄ are determined by I₅ and the aspect ratios of M₃ and M₄. Through the current mirrors, the V_GS of other transistors are also determined without additional bias circuits. Like a five-transistor OTA, this simple bias circuit becomes one of the advantages of a five-transistor OTA compared to OTAs based on a cascode structure. At the equilibrium, M₁ and M₂ generate current signals, I_D1 and I_D2, respectively. I_D1 is fed into M₃, and I_D6 flowing through M₆ becomes K∙I_D1 via the current mirror composed of M₃ and another transistor, which is K times wider than M₃. When M₆ and M₇ have the same aspect ratios, I_D7 flowing through M₇ also becomes K∙I_D1. Note that I_D2 is equal to I_D1. Since I_D2 is amplified by the current mirror composed of M₄ and another transistor, which is K times wider than M₄, no current flows to the load. In contrast, when two input signals have the same DC voltage level with a small AC differential signal, denoted as v_id, a current flows to the load due to the different directions of I_D1 and I_D2.

See Figure 4b for a small signal analysis with a differential input signal. When M₁ and M₂ are matched with the same transconductance, denoted as g_m, M₁ and M₂ generate i_out = g_m(v_id/2) and −i_out. i_out and −i_out are amplified to K∙i_out and −K∙i_out at the output node. Two amplified currents are merged together at the output node, thereby flowing 2i_out = K∙g_mv_id to the load. In this regard, the G_m of a current mirror OTA, G_{m,CM_OTA}, is expressed as follows:

$$G_{m,CM\_OTA}={K·g}_{m1}={K·g}_{m2}={K·g}_m.$$

(20)

Thanks to the current mirrors at the output stage, a higher G_m can be achieved compared to other OTAs from the same g_m1 and g_m2. However, at the expense of a higher G_m, additional current paths for amplifying the current are required, consuming higher power.

Let us look into the r_out of a current mirror OTA. At the output node, r_oK4 and r_o7 are connected in parallel, where r_oK4 is the output impedance of the transistor, which is K times wider than M₄. Therefore, two transistors at the output node have the same output impedance, denoted as r_o, and the r_out of a five-transistor OTA, denoted as r_{out,CM_OTA}, is expressed as follows:

$$r_{out,CM\_OTA}=r_{oK4}||r_{o7}=\frac{1}{2}·r_o.$$

(21)

The frequency response of a current mirror OTA can be derived from small-signal parameters. A_o, ω_3dB, and ω_GB of a current mirror OTA, denoted as A_{o,CM_OTA}, ω_{3dB,CM_OTA}, and ω_{GB,CM_OTA}, are expressed as follows:

$$A_{o,CM\_OTA}=-G_{m,CM\_OTA}·r_{out,CM\_OTA}=-\frac{K}{2}·g_m·r_o,$$

(22)

$${\omega }_{3dB,CM\_OTA}=\frac{1}{r_{out,CM\_OTA}·C_{Load}}=\frac{1}{{(r}_o/2)·C_{Load}},$$

(23)

$${\omega }_{GB,CM\_OTA}=\left|A_{o,CM\_OTA}\right|·{\omega }_{3dB,CM\_OTA}=\frac{K·g_m}{C_{Load}}.$$

(24)

Here, r_o in (21)–(23) is K times lower than r_o in (2)–(4) since r_o of a single transistor is inversely proportional to the bias current at equilibrium. In this regard, r_{out,CM_OTA} in (21) is K times lower than r_{out,5TR_OTA} in (21). As a result, theoretically, A_{o,CM_OTA} in (22) is the same as A_{o,5TR_OTA} in (3). In contrast, ω_{3dB,CM_OTA} in (23) and ω_{GB,CM_OTA} in (24) are K times lower than ω_{3dB,5TR_OTA} in (4) and ω_{GB,5TR_OTA} in (5), respectively.

A current mirror OTA has a wider output swing range compared to other types of OTAs, which becomes one of the advantages of a current mirror OTA. There are only two transistors at the output node, thus guaranteeing that those two transistors operate in the saturation region. The highest output voltage level is limited to V_DD − V_D,SAT, and the lowest output voltage level is limited to V_D,SAT. Since other single-stage OTAs have a larger number of transistors from 3 to 5, a current mirror OTA has the widest output swing ranges among single-stage OTAs. Especially when the input DC bias voltage is relatively low, a PMOS-based five-transistor OTA is more suitable. It can be said that a current mirror OTA can guarantee the rail-to-rail output swing range. The input swing range of a current mirror OTA is the same as that of a five-transistor OTA since these two OTAs have a similar structure at the input node.

The current amplification by K times also enhances the SR. As shown in Figure 5d,e, the current flowing through PMOS at the output node determines a rising SR in a current mirror OTA, denoted as SR_{R,CM_OTA}, and the current flowing through the NMOS at the output node determines a rising SR in a current mirror OTA, denoted as SR_{F,CM_OTA}. Since I_D5 is amplified by K times through current mirrors, SR_{R,CM_OTA} and SR_{F,CM_OTA} are expressed as follows:

$${SR}_{R,CM\_OTA}={SR}_{F,CM\_OTA}=K·\frac{I_{D5}}{C_{Load}}.$$

(25)

while the SRs of other OTAs are limited by the magnitude of the input bias current, the SR of a current mirror OTA is not limited by the magnitude of the input bias current. However, the higher the K, the higher the power consumption.

Despite the advantages of a current mirror OTA including a larger G_m, a wider bandwidth, a wider output swing range, and a faster SR, a current mirror OTA suffers greater performance degradation due to nonidealities compared to other structures. Nonidealities of all additional current mirrors are amplified to the output with a moderate gain. Therefore, a current mirror OTA has not been employed as a low-noise amplifier in applications that require a high resolution.

3.5. Summary

Table 1. Summary of theoretical performance metrics.

Topology	5-Transistor OTA	Telescopic Cascode OTA	Folded Cascode OTA	Current Mirror OTA
$G_m$	$g_m$	$g_m$	$g_m$	$K·g_m$
$r_{out}$	$\frac{1}{2}·r_o$	$\frac{1}{2}·g_m·{r_o}^2$	$\frac{1}{2}·g_m·{r_o}^2$	$\frac{1}{2}·r_o$
$A_o$	$-\frac{1}{2}·g_m·r_o$	$-\frac{1}{2}·{\left(g_m·r_o\right)}^2$	$-\frac{1}{2}·{\left(g_m·r_o\right)}^2$	$-\frac{K}{2}·g_m·r_o$
${\omega }_{3dB}$	$\frac{1}{{(r}_o/2)·C_{Load}}$	$\frac{1}{{(g}_m·{r_o}^2/2)·C_{Load}}$	$\frac{1}{{(g}_m·{r_o}^2/2)·C_{Load}}$	$\frac{1}{{(r}_o/2)·C_{Load}}$
${\omega }_{GB}$	$\frac{g_m}{C_{Load}}$	$\frac{g_m}{C_{Load}}$	$\frac{g_m}{C_{Load}}$	$K·\frac{g_m}{C_{Load}}$
${SR}_R$	$\frac{I_{D5}}{C_{Load}}$	$\frac{I_{D5}}{C_{Load}}$	$\frac{I_{D5}}{C_{Load}}$	$K·\frac{I_{D5}}{C_{Load}}$
${SR}_F$	$\frac{I_{D5}}{C_{Load}}$	$\frac{I_{D5}}{C_{Load}}$	$\frac{I_{D5}}{C_{Load}}$	$K·\frac{I_{D5}}{C_{Load}}$

summarizes the performance of single-stage OTAs. As mentioned earlier, the r_o in a current mirror OTA is K times lower than the r_o in other transistors since the r_o of a single transistor is inversely proportional to the bias current at equilibrium.

A five-transistor OTA has the simplest architecture among the four types of single-stage OTAs. Compared to a five-transistor OTA, telescopic cascode and folded cascode OTAs achieve a larger r_out, which results in a higher A_o. A current mirror OTA achieves higher ω_3dB, ω_GB, SR_R, and SR_F compared to other types of OTAs. That is, most of the performance metrics related to the speed of OTAs are improved thanks to the current amplification via current mirrors. Moreover, a higher G_m is also achieved. However, these improvements come at the expense of power consumption. Another advantage of a current mirror OTA is the wide output swing range. Particularly, regardless of the power consumption, it can obtain the widest output swing range thanks to a small number of transistors.

4. Results

Four types of single-stage OTAs, which were analyzed in Section 3, were designed and simulated in a 180 nm complementary metal–oxide–semiconductor (CMOS) process with a V_DD of 1.8 V to verify the theoretical analysis. For a fair performance comparison, the input transistors of OTAs, M₁ and M₂, have the same g_m with the same bias current of 1 µA.

Table 2. Transistor sizes of designed single-stage OTAs.

	5-Transistor OTA	Telescopic Cascode OTA	Folded Cascode OTA	Current Mirror OTA
M1(W/L)	1 µ/1 µ	1 µ/1 µ	1 µ/1 µ	1 µ/1 µ
M2(W/L)	1 µ/1 µ	1 µ/1 µ	1 µ/1 µ	1 µ/1 µ
M3(W/L)	4 µ/1 µ	4 µ/1 µ	8 µ/1 µ	4 µ/1 µ
M4(W/L)	4 µ/1 µ	4 µ/1 µ	8 µ/1 µ	4 µ/1 µ
M5(W/L)	2 µ/1 µ	3 µ/1 µ	2 µ/1 µ	2 µ/1 µ
M6(W/L)	N/A	1 µ/1 µ	1 µ/1 µ	3 µ/1 µ
M7(W/L)	N/A	1 µ/1 µ	1 µ/1 µ	3 µ/1 µ
M8(W/L)	N/A	4 µ/1 µ	4 µ/1 µ	N/A
M9(W/L)	N/A	4 µ/1 µ	4 µ/1 µ	N/A
M10(W/L)	N/A	N/A	1 µ/1 µ	N/A
M11(W/L)	N/A	N/A	1 µ/1 µ	N/A

shows the transistor sizes of the designed single-stage OTAs. V_D,SAT is set to about 100 mV so that all transistors operate in the strong inversion region.

Figure 6 shows the simulation results of a designed five-transistor OTA. Figure 6a shows the frequency response, and Figure 6b shows the rising slewing of an OTA with SR_R when C_Load is set to 1 pF. Figure 6c shows an input-referred noise, which is one of the representative nonidealities of an OTA. Figure 7, Figure 8, and Figure 9 show the same performance metrics of a telescopic cascode OTA, a folded cascode OTA, and a current mirror OTA, respectively.

Table 3. Summary of simulated performance metrics.

	5-Transistor OTA	Telescopic Cascode OTA	Folded Cascode OTA	Current Mirror OTA
Current bias	ID5 = 2 µA	ID5 = 2 µA	ID3 = ID4 = 2 µA ID5 = 2 µA	ID5 = 2 µA (K = 3)
Power dissipation	3.6 µW	3.6 µW	7.2 µW	14.4 µW
$A_o$	42.2951 dB	73.4576 dB	75.9819 dB	42.9838 dB
${\omega }_{3dB}$	2π·20 kHz	2π·511.79 Hz	2π·400 Hz	2π·54.8545 kHz
${\omega }_{GB}$	2π·2.642 MHz	2π·2.472 MHz	2π·2.567 MHz	2π·7.407 MHz
${SR}_R$	2.028 MV/s	2.156 MV/s	1.868 MV/s	5.335 MV/s
${SR}_F$	1.944 MV/s	1.8469 MV/s	1.879 MV/s	5.499 MV/s
Integrated input-referred noise	0.110 mVrms	0.125 mVrms	0.181 mVrms	0.200 mVrms

summarizes the simulated performance metrics of four types of single-stage OTAs. The folded cascode OTA consumes two times more power than five-transistor and telescopic cascode OTAs due to additional current sources when I_D3 = I_D4 = I_D5. The K of a current mirror OTA is set to 3, which results in a power consumption that is four times higher.

Compared to a five-transistor OTA and current mirror OTA, two cascode OTAs achieve much a larger A_o thanks to a higher r_out. Here, a five-transistor OTA and current mirror OTA obtain similar A_o values. This is because G_{m,CM_OTA} is K times higher than G_{m,5TR_OTA}, but r_{out,CM_OTA} is K times lower than r_{out,5TR_OTA}. According to the simulation related to the DC operating condition, r_{out,CM_OTA} is 7.8 Mohm and r_{out,5TR_OTA} is 2.8 Mohm as expected, which results in similar A_o values.

Compared to a five-transistor OTA, two types of cascode OTAs have a smaller ω_3dB due to a higher r_out while maintaining a similar ω_GB. On the contrary, compared to a five-transistor OTA, a current mirror OTA with K = 3 improves the ω_3dB and ω_GB by about three times thanks to the current amplification of additional current mirrors. For the same reason, a current mirror OTA with K = 3 improves the SR_R and SR_F by about three times, which are verified by the simulation results.

The integrated input-referred noises of OTAs are compared to effect caused by nonidealities of transistors. The input-referred noises in Figure 6c, Figure 7c, Figure 8c, and Figure 9c are integrated from 1 Hz to ω_GB/2π. The integrated noise of a telescopic cascode OTA is slightly higher than that of a five-transistor OTA due to cascaded transistors. However, the increase is relatively small because the effect caused by cascoded transistors on the output is negligible. The integrated noise of a folded cascode OTA is higher than that of a telescopic cascode OTA because the effect caused by additional current sources on the output cannot be negligible. A current mirror OTA obtains the largest integrated noise, even with a higher power consumption, due to the current amplification.

5. Discussions and Case Study

Figure 10 presents a comparative analysis of the four types of single-stage OTAs using a spider map, considering both the theoretical and simulated performance metrics summarized in Table 1 and Table 3. The five-transistor and telescopic cascode OTAs stand out for their lower power consumption compared to folded cascode and current mirror OTAs with a K of 3. The five-transistor OTA exhibits the simplest structure based on fewer transistors, minimizing the negative effects caused by the nonidealities. The telescopic cascode and folded cascode OTAs achieve a higher r_out, resulting in a higher A_o at the sacrifice of swing ranges. While the folded cascode structure improves the input swing range, its output swing range remains limited like that of the telescopic cascode OTA. On the other hand, the current mirror OTA excels in terms of ω_GB, SR_R, and SR_F, but consumes more power. However, it offers the advantage of a wide output swing range even with a relatively small number of transistors at the output node.

The analysis results provide valuable guidance in choosing a suitable OTA structure for specific applications. The five-transistor OTA has the simplest structure, which makes the design easy. It can be employed as a feedforward amplifier in a closed-loop system, but it has been employed as the first stage of multi-stage OP-AMPs [26,27,28,29]. Despite its simplicity and moderate performance, since it is difficult for the five-transistor OTA to achieve a high A_o, additional stages are adopted to increase the A_o. The telescopic cascode OTA is a good choice when additional DC current paths are challenging to implement due to a limited power budget, but a high A_o is required. Similar structures based on cascode output stages without additional current sources have been effectively used as feedforward amplifiers in low-power instrumentation amplifiers (IAs) [30] and unity-gain buffers in biomedical applications [31]. The folded cascode OTA has been adopted as the feedforward amplifier of IAs in biopotential recording ICs and sensor interface ICs [32,33,34,35] and as an error amplifier of low-dropout regulators (LDOs) [16,17] to utilize a high A_o and a wide input swing range. Compared to telescopic cascode OTAs, using folded cascode OTAs may require a more generous power budget for the system. Moreover, in order to further increase the A_o, the folded cascode structure has been adopted as the first stage of a multi-stage OP-AMP [36]. Lastly, the current mirror OTA has been used as the error amplifier of DC-DC converters and LDOs [18,19,20] or as the feedforward amplifier of neural recording amplifiers [13,14]. In power converters, the error amplifier often needs to be fast since the output voltage or the load condition can be changed rapidly. In this case, the current mirror OTA could be more suitable than other structures.

At last, a case study is provided to demonstrate the practical application of the tutorial’s findings. An appropriate OTA structure is selected based on the analysis results, thereby improving the performance of the conventional neural recording amplifier. In [13], a neural recording amplifier was designed based on a CCIA with the target gain of 40 dB. Here, the current mirror OTA is used as the feedforward amplifier of a CCIA. For the case study, we designed a current mirror OTA based on the transistor sizes in Table 2, and then designed a CCIA based on the current mirror OTA.

Figure 11a shows the closed-loop and open-loop gains of a CCIA based on a current mirror OTA. The closed-loop gain does not reach 40 dB in the mid-band range due to the low A_o. To address this, we applied the insights from our previous analysis and utilized a telescopic cascode OTA, which offers a significantly higher A_o with a much lower power consumption. Figure 11b shows the closed-loop and open-loop gains of a CCIA based on a telescopic cascode OTA, achieving nearly a 40 dB closed-loop gain in the mid-band range thanks to the higher A_o. Additionally, the current consumption of the telescopic cascode OTA is only one-fourth of that of the current mirror OTA with a K of 3. Furthermore, the integrated input-referred noise of a CCIA based on a current mirror OTA is 78 μV_rms, while a CCIA based on a telescopic cascode OTA achieves 66 μV_rms of the integrated input-referred noise. The noise is integrated from 1 Hz to 10 kHz. This indicates that the noise, which is one of important parameters of neural recording amplifiers, can also be improved by adopting a telescopic cascode OTA. In [13], in order to increase the A_o, a cascode output stage was employed in a current mirror OTA. In this structure, the A_o can be increased to as much as the A_o of a telescopic cascode OTA, but this structure would not lead to an improved power consumption and noise compared to a telescopic cascode OTA.

The operation principle, key performance metrics, and practical applications of each OTA have been explored with a case study. The performance comparison based on the measurement results can become a promising future work. Moreover, exploring other advanced structures such as current-reusing OTAs [37] and recycling folded cascode OTAs [33] based on this design tutorial can also become a promising and valuable future work.

6. Conclusions

This paper offers a comprehensive design tutorial for four different types of single-stage OTAs: five-transistor OTAs, telescopic cascode OTAs, folded cascode OTAs, and current mirror OTAs. The performance metrics of each OTA are derived through a hand analysis and verified through a simulation. Among the four types of single-stage OTAs, the five-transistor OTA stands out for its simple structure, minimizing the negative effects of the nonidealities by reducing the number of transistors. On the other hand, telescopic cascode and folded cascode OTAs achieve higher r_out and A_o at the expense of swing ranges. The input swing range can be improved by the folded cascode structure at the expense of power consumption, but the output swing range of a folded cascode OTA is still limited like that of a telescopic cascode OTA. A current mirror OTA excels in terms of bandwidth and SRs but consumes more power. Additionally, it offers a wide output swing range thanks to the small number of transistors at the output node. The insights gained from this tutorial will serve as a valuable resource for designing high-performance analog amplifiers in diverse applications.