Graphical Approach to Optimization of Maximally Efficient-Gain-Boosted Feedback Amplifiers

Abstract

It is challenging to design high-gain amplifiers near the maximum oscillation frequency (f_max) of the transistors. This paper presents a comprehensive graphical approach to maximize the gain of feedback amplifiers with maximally efficient gain (G_ME) conception at near-fmax frequency. The complex gain-plane and the reflection-coefficient-plane are utilized to provide clear insights into both the gain and stability states of the two-port device while boosting G_ME. An efficient flowchart to synthesize feedback amplifiers is given, which optimizes the G_ME of a two-port device while ensuring the stability. A 210 GHz power amplifier in 40 nm CMOS was designed and optimized based on the proposed approach. The feedback circuit of the transistor pushes it to become potentially unstable and boosts GME. The measured peak small-signal gain was 10.48 dB at 195.33 GHz. The measured saturation output power and large-signal gain at 210 GHz were 3.04 dBm and 7.08 dB, respectively. The presented method could facilitate terahertz amplifier design.

1. Introduction

Terahertz (THz) frequencies of 100–300 GHz have received attention in various fields, such as high-data-rate communication, imaging, and radar [1,2,3,4]. THz amplifiers are one of the most important parts in the THz transceivers for these applications. Since THz frequencies are very close to the fmax of the active devices, the available gain from the active devices declines rapidly at these frequencies. Consequently, it is challenging to design high-gain near-f_max THz amplifiers.

It has been shown that boosting the power gain of an active two-port device (A2P) to its maximum achievable gain (G_MAX) using specific embedding is a promising solution to the power gain issue [5,6]. However, G_MAX has disadvantages. G_MAX is achieved only when the source and load impedances are conjugate-matched to the input and output terminals of the A2P simultaneously, which is possible only when the A2P is unconditionally stable. For a potentially unstable A2P, G_MAX is not defined and provides no useful design information [7]. The design of the G_MAX-based amplifier is thus limited in its unconditionally stable state.

The maximally efficient gain G_ME is another figure of merit for A2Ps [7]. It is defined as the power gain that maximizes the added power of the A2P. It is achieved by choosing a certain set of the source and load admittances. Advantages of G_ME include that it provides outstanding large-signal performances and that it is well-behaved both for stable and unstable A2Ps, which broaden the design space and bring potential for boosting the power gain beyond G_MAX. These features have attracted increasing attention from designers. The G_ME-based designs have been successfully applied in amplifiers [8,9,10] and oscillators [11,12].

Despite the excellent features of G_ME, the G_ME-based designs have not been studied comprehensively. No analytical solution has been developed yet to optimize G_ME. In amplifier design, it is crucial to ensure that both terminals of the amplifier are stable. The advantage of G_ME that it is well-behaved for the potentially unstable A2P, however, raises the risk of oscillation for a G_ME-based amplifier, making the design complex and difficult. An efficient and accurate design method for optimizing G_ME-based amplifiers is urgently needed.

In this paper, a graphical approach to optimize the G_ME of amplifiers is presented. The gain-plane and the reflection-coefficient-plane are employed to visualize both the gain and stability states of an A2P. The optimum position of G_ME on the gain-plane is determined. Embeddings are used to form a feedback circuit and create a two-dimensional movement on both planes to move the A2P to the desired position. A flowchart is developed to efficiently design the embedding that optimizes the G_ME of the A2P. The proposed process is capable of boosting the gain of a single NMOS transistor in a 40 nm CMOS process up to 7.68 dB at 200 GHz while ensuring the stability in both input and output terminals.

The remainder of this paper is organized as follows. Section 2 studies the gain-plane and reflection-coefficient-plane for a G_ME-based A2P. Section 3 derives the movement of the A2P on both planes using various embeddings. Section 4 presents the flowchart for maximizing the G_ME of the feedback amplifier. In Section 5, a two-stage embedded amplifier at 210 GHz is presented using the proposed approach, and the measurement results are presented. Finally, Section 6 concludes this paper.

2. Gain-Plane and Reflection-Coefficient-Plane

The maximally efficient gain G_ME of an A2P is given by [7]:

$$G_{ME}=\frac{|Y_{21}{|}^2-|Y_{12}{|}^2}{4G_{11}{G}_{22}-2\mathrm{Re}(Y_{21}{Y}_{12})-2|Y_{12}{|}^2}$$

(1)

where Y_ij (i, j = 1, 2) is the Y-parameter of the A2P, and G₁₁ and G₂₂ are the real parts of Y₁₁ and Y₂₂, respectively. It is obtained when the load admittance Y_L of the A2P is set to

$$Y_L=\frac{2G_{22}{Y}_{21}}{Y_{21}+Y_{12}{}^{*}}-Y_{22}$$

(2)

The source admittance Y_S is chosen to be the complex conjugate of the input admittance:

$$Y_{in}=Y_{11}-\frac{Y_{12}(Y_{21}+Y_{12}{}^{*})}{2G_{22}}, Y_S=Y_{in}{}^{*}$$

(3)

where Y_ij are the Y parameters of the A2P. Equations (2) and (3) can also be expressed in terms of Z parameters:

$$Z_L=\frac{2R_{22}{Z}_{21}}{Z_{21}+Z_{12}{}^{*}}-Z_{22}$$

(4)

$$Z_{in}=Z_{11}-\frac{Z_{12}(Z_{21}+Z_{12}{}^{*})}{2R_{22}}, Z_S=Z_{in}{}^{*}$$

(5)

G_ME can be rewritten in terms of Rollett’s stability k and complex inverse gain λ = Y₁₂/Y₂₁:

$$G_{ME}=\frac{|1/\lambda {|}^2-1}{2(k|1/\lambda |-1)}$$

(6)

where k is expressed as a function of λ and unilateral power gain U [13]:

$$k=\frac{{\lambda }_R{}^2+{\lambda }_I{}^2+2(U-1){\lambda }_R+1}{2U\sqrt{{\lambda }_R{}^2+{\lambda }_I{}^2}}$$

(7)

where λ_R and λ_I are the real and imaginary parts of λ, respectively. Using Equation (7), G_ME can be rewritten in terms of λ and U:

$${({\lambda }_R-\frac{U-1}{2U-U/G_{ME}-1})}^2+{\lambda }_I{}^2={(\frac{U-U/G_{ME}}{2U-U/G_{ME}-1})}^2$$

(8)

Equations (7) and (8) describe the loci of constant-k contours and constant-G_ME circles on the complex λ-plane (also called the gain-plane), respectively, as shown in Figure 1a. For a given A2P, its coordinate on the gain-plane represents the gain state and is determined by its Y parameters Y₁₂/Y₂₁. The coordinate of a 40 nm NMOS transistor at 200 GHz is plotted in Figure 1a. Two observations are obtained from Figure 1a. The first is that for a given G_ME, the left intersection point of the constant-G_ME circle with the X-axis gives the largest k. Although the value of k cannot be used to evaluate the degree of stability of an A2P, it does affect the size of the stable region in the reflection-coefficient-plane. That is, the smaller the value of k, the larger the intersection area between the stability circle and the reflection coefficient plane and thus the smaller the area of the stable region, and it is more likely for the load and source reflection coefficients of a G_ME-based A2P to fall into the unstable region. Therefore, it can be concluded that the left intersection point of the constant-G_ME circle gives the best stability. The second observation is that on the X-axis, G_ME increases and k decreases along the negative direction of the axis. Consequently, the optimum position of G_ME for an amplifier is located on the X-axis. It is desired to have a A2P located on the X-axis and moving in its negative direction to find the optimized G_ME value. By applying appropriate feedback embedding to the A2P, its Y parameters can be adjusted and its coordinate can be manipulated. The manipulation method will be discussed in the next section.

Although the λ-plane can provide the gain state of an A2P, it gives little information about the stability of the A2P terminals. This deficiency can be filled by the reflection-coefficient-plane. For a G_ME-based A2P, the source and load admittances are completely determined by Equations (2) and (3). It is convenient to convert them to the reflection coefficient Γ and plot on the Γ-plane to observe the stability states of the A2P terminals. Figure 1b plots the coordinates of Γ_L and Γ_S for the 40 nm NMOS transistor at 200 GHz on the Γ-plane with the load and source stability circles. The load and source stability circles identify the output and input unstable regions, respectively. If Γ_L falls into the output unstable region, the real part of the input impedance of A2P become negative, and conjugated matching at the input port becomes impossible. In this case, G_ME is impossible to realize. If Γ_S falls into the input unstable region, the real part of the output impedance becomes negative, and the G_ME-based A2P becomes an oscillator.

Utilizing both the λ- and Γ-planes, the precise gain and stability states of a G_ME-based A2P can be observed simultaneously, and the G_ME-based amplifier can now be visually optimized.

3. Movement on the Planes by Feedback Circuits

3.1. Parallel Embedding

Applying feedback networks to A2P leads to the movements on both the λ- and Γ-planes. Figure 2a shows the parallel embedding (also called Y embedding). A passive component is added between the input and output terminals of the A2P to form a new embedded A2P. The Y matrix of the embedded A2P is

$$\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]=\left[\begin{array}{cc}{Y}_{11}\prime +j{B}_Y& Y_{12}\prime -j{B}_Y\\ Y_{21}\prime -j{B}_Y& Y_{22}\prime +j{B}_Y\end{array}\right]$$

(9)

where the quote denotes the Y parameters of the original A2P, and jB_Y is the admittance of the embedded component. For the λ-plane, let λ’ be the inverse gain of the original A2P; the inverse gain of the embedded A2P is written as

$$\lambda =\frac{Y_{12}}{Y_{21}}=\frac{\lambda \prime -j{B}_Y/Y_{21}\prime }{1-j{B}_Y/Y_{21}\prime }$$

(10)

When B_Y is swept, the coordinate of the embedded A2P on the λ-plane moves accordingly. It is derived by Shuhei Amakawa that the locus of the embedded A2P can be written as [6]:

$${[{\lambda }_R-\frac{r{\lambda }_R\prime -q{\lambda }_I\prime +r}{2r}]}^2+{[{\lambda }_I-\frac{q{\lambda }_R\prime +r{\lambda }_I\prime -q}{2r}]}^2={[\frac{r{\lambda }_R\prime -q{\lambda }_I\prime +r}{2r}]}^2+{[\frac{q{\lambda }_R\prime +r{\lambda }_I\prime -q}{2r}]}^2-\frac{r{\lambda }_R\prime -q{\lambda }_I\prime }{r}$$

(11)

where Y₂₁ = r + jq. Equation (11) denotes a circle on the λ-plane, as shown in Figure 2b. Note that the radius and center of the circle depend on the original A2P. This indicates that for a given A2P, the achievable position by applying parallel embedding is limited on this circle.

For the Γ-plane, the load reflection ecoefficiency Γ_L of the embedded A2P is written as

$${\Gamma }_L=\frac{Y_0-Y_L}{Y_0+Y_L}$$

(12)

where Y₀ is the port admittance. Applying Equations (2) and (9) into Equation (12), Γ_L can be rewritten as

$${\Gamma }_L=\frac{(\mathrm{a}-d{B}_Y)(e+d{B}_Y)+(b+c{B}_Y)(f-c{B}_Y)+j[(b+c{B}_Y)(e+d{B}_Y)-(\mathrm{a}-d{B}_Y)(f-c{B}_Y)]}{{(e+d{B}_Y)}^2+{(f-c{B}_Y)}^2}$$

(13a)

where

$$a+jb=Y_0(Y_{21}\prime +{Y_{12}\prime }^{*})-2G_{22}\prime Y_{21}\prime +Y_{22}\prime (Y_{21}\prime +{Y_{12}\prime }^{*})$$

(13b)

$$c+jd=2G_{22}\prime +Y_{21}\prime +{Y_{12}\prime }^{*}$$

(13c)

$$e+jf=Y_0(Y_{21}\prime +{Y_{12}\prime }^{*})+2G_{22}\prime Y_{21}\prime -Y_{22}\prime (Y_{21}\prime +{Y_{12}\prime }^{*})$$

(13d)

Equation (13) denotes the locus of Γ_L of the embedded A2P on the Γ_L-Plane when B_Y is swept. The locus and a group of load stability circles with the swept B_Y are plotted in Figure 2c. Similar results on Γ_S are obtained and shown in Figure 2d. It can be observed from Figure 2 that sweep of B_Y alters the distance between Γ_L and the load stability circle as well as the distance between Γ_S and the source stability circle. With certain B_Y, the input or output terminal of the embedded A2P becomes unstable. Note that when the embedded A2P is moved to the position with higher G_ME on the λ-plane, its Γ_L and Γ_S are moved closer to the load and source stability circles on the Γ-plane, respectively.

3.2. Series Embedding

Figure 3a shows the series embedding (also called Z embedding). A passive component is added in series with the common terminal of the A2P to form a new embedded A2P. The Z matrix of the embedded A2P is

$$\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]=\left[\begin{array}{cc}{Z}_{11}\prime +j{X}_Z& Z_{12}\prime +j{X}_Z\\ Z_{21}\prime +j{X}_Z& Z_{22}\prime +j{X}_Z\end{array}\right]$$

(14)

where the quote denotes the Z parameters of the original A2P, and jX_Z is the impedance of the embedded component. For the λ-plane, the locus of the embedded A2P when X_Z is swept is written as

$${[{\lambda }_R-\frac{c{\lambda }_R\prime -d{\lambda }_I\prime +c}{2c}]}^2+{[{\lambda }_I-\frac{d{\lambda }_R\prime +c{\lambda }_I\prime -d}{2c}]}^2={[\frac{c{\lambda }_R\prime -d{\lambda }_I\prime +c}{2c}]}^2+{[\frac{d{\lambda }_R\prime +c{\lambda }_I\prime -d}{2c}]}^2-\frac{c{\lambda }_R\prime -d{\lambda }_I\prime }{c}$$

(15)

where Z₂₁′= c + jd. The locus is plotted in Figure 3b.

For the Γ-plane, Γ_L of the embedded A2P is written as

$${\Gamma }_L=\frac{Z_L-Z_0}{Z_L+Z_0}$$

(16)

where Z₀ is the port impedance. Applying Equations (4) and (14) into (16), Γ_L can be rewritten as

$${\Gamma }_L=\frac{(a-d{X}_Z)(e-d{X}_Z)+(b+c{X}_Z)(f+c{X}_Z)+j[(b+c{X}_Z)(e-d{X}_Z)-(a-d{X}_Z)(f+c{X}_Z)]}{{(e-d{X}_Z)}^2+{(f+c{X}_Z)}^2}$$

(17a)

where

$$a+jb=2R_{22}\prime Z_{21}\prime -Z_{22}\prime (Z_{21}\prime +{Z_{12}\prime }^{*})-Z_0(Z_{21}\prime +{Z_{12}\prime }^{*})$$

(17b)

$$c+jd=2R_{22}\prime -Z_{21}\prime -{Z_{12}\prime }^{*}$$

(17c)

$$e+jf=2R_{22}\prime Z_{21}\prime -Z_{22}\prime (Z_{21}\prime +{Z_{12}\prime }^{*})+Z_0(Z_{21}\prime +{Z_{12}\prime }^{*})$$

(17d)

Equation (17) denotes the locus of Γ_L of the embedded A2P on the Γ_L-Plane when X_Z is swept, as shown in Figure 3c. Similar results on the Γ_S-Plane are obtained and shown in Figure 3d. It is observed that series embedding provides a phase of movements on the λ-and Γ-planes that is different from parallel embedding.

3.3. Combined Embedding

Simple parallel or series embedding can only provide limited range of movement for the A2P on the λ-plane, making it difficult for an A2P to move to the optimized-G_ME position. To expand the range of movement, parallel and series embedding can be applied successively to an A2P to provide a two-step movement on the λ-plane [5,13]. However, this embedding may impose some limitations in terms of loss, DC bias, or isolation on the embedded A2P [5]. Another embedding structure that can broaden the scope of movement is cascade embedding. The idea is to cascade passive components to the A2P before parallel or series embedding is applied. The cascading components form a new cascaded A2P with the original A2P. The network parameters of the cascaded A2P are manipulated by the cascading components. It is revealed by Equations (11) and (13) that in parallel or series embedding, the locus of the embedded A2P on the λ-plane depends on the network parameters of the original A2P. Therefore, when parallel or series embedding is applied to the cascaded A2P, the locus of the final A2P can be altered by the cascading components. The range of movement on the λ-plane is thus widely expanded.

Consider a cascade–parallel embedding in Figure 4a. Two passive components are cascaded to the A2P before the parallel component is added. The Y matrix of the cascaded A2P is

$$\left[\begin{array}{cc}{Y}_{11}″& Y_{12}″\\ Y_{21}″& Y_{22}″\end{array}\right]=\left[\begin{array}{cc}\frac{j{X}_2\left|Y\right|\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }& \frac{Y_{12}\prime Y_{21}\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }\\ \frac{Y_{21}\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }& \frac{Y_{22}\prime +j{X}_2\left|Y\right|\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }\end{array}\right]$$

(18)

where jX₁ and jX₂ are the impedances of the cascaded components, and the single quote and double quotes denote the Y parameters of the original and cascaded A2P, respectively. |Y| = Y₁₂Y₂₁ − Y₁₁Y₂₂. The Y matrix of the parallel-embedded A2P is

$$\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]=\left[\begin{array}{cc}\frac{j{X}_2\left|Y\right|\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }+j{B}_Y& \frac{Y_{12}\prime Y_{21}\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }-j{B}_Y\\ \frac{Y_{21}\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }-j{B}_Y& \frac{Y_{22}\prime +j{X}_2\left|Y\right|\prime }{j{X}_2{Y}_{22}\prime -X_1{X}_2\left|Y\right|\prime +1+j{X}_1{Y}_{11}\prime }+j{B}_Y\end{array}\right]$$

(19)

For the λ-plane, replacing Y₂₁’ and λ’ in Equation (10) with Y₂₁’ and “λ” in Equation (18), the locus of the embedded A2P on the plane when B_Y is swept is written as

$$\begin{array}{cc}\hfill & {\left[{\lambda }_R-\frac{(A{r}^2-A{q}^2+2Brq){{\lambda }_R}^{\prime }-(B{q}^2-B{r}^2+2Arq){{\lambda }_I}^{\prime }+(Ar+Bq)}{2(Ar+Bq)}\right]}^2+\hfill \\ \hfill & {\left[{\lambda }_I-\frac{(B{q}^2-B{r}^2+2Arq){{\lambda }_R}^{\prime }+(A{r}^2-A{q}^2+2Brq){{\lambda }_I}^{\prime }-(Aq-Br)}{2(Ar+Bq)}\right]}^2=\hfill \\ \hfill & {\left[\frac{(A{r}^2-A{q}^2+2Brq){{\lambda }_R}^{\prime }-(B{q}^2-B{r}^2+2Arq){{\lambda }_I}^{\prime }+(Ar+Bq)}{2(Ar+Bq)}\right]}^2+\hfill \\ \hfill & {\left[\frac{(B{q}^2-B{r}^2+2Arq){{\lambda }_R}^{\prime }+(A{r}^2-A{q}^2+2Brq){{\lambda }_I}^{\prime }-(Aq-Br)}{2(Ar+Bq)}\right]}^2-\hfill \\ \hfill & \frac{(A{r}^2-A{q}^2+2Brq){{\lambda }_R}^{\prime }-(B{q}^2-B{r}^2+2Arq){{\lambda }_I}^{\prime }}{Ar+Bq}\hfill \end{array}$$

(20a)

where

$$A=(1-X_1{B}_{11}\prime -X_2{B}_{22}\prime -X_1{X}_2{G}_{\Delta }\prime )$$

(20b)

$$B=(X_1{G}_{11}\prime +X_2{G}_{22}\prime -X_1{X}_2{B}_{\Delta }\prime )$$

(20c)

$$\left|Y\right|\prime =Y_{11}\prime \prime Y_{22}\prime \prime -Y_{12}\prime \prime Y_{21}\prime \prime =G_{\Delta }\prime +j{B}_{\Delta }\prime$$

(20d)

Y_ij’ = G_ij’ + jB_ij’ is the Y parameter of the original A2P. Note that now the center and radius of the circle are a function of X₁ and X₂. Therefore, by manipulating the value of X₁ and X₂, the achievable coordinates for the embedded A2P are now widely expanded, as shown in Figure 4b.

For the Γ-plane, replacing Y_ij’ in Equation (13) with Y_ij’ in Equation (18), the loci of Γ_L of the embedded A2P when X₁, X₂, and B_Y are swept can be derived. Again, similar results can be obtained for the loci of Γ_S. Figure 4c and 4d plot the loci on the Γ_L- and Γ_S-planes with different X₁ and X₂, respectively.

4. Flowchart for G_ME Optimization

With the knowledge of visually analyzing the gain and stability states of a G_ME-based A2P and the methods for manipulating the movements of the A2P on the λ-plane and Γ-plane, a design process for optimizing the G_ME of an amplifier is developed, as shown in Figure 5.

According to above analysis, the optimal position of G_ME of an amplifier, at which the value of G_ME reaches its maximum while both the input and output ports of the amplifier remain stable, is located on the X-axis of the λ-plane. Consequently, the main idea of optimization is to move A2P onto the X-axis and then gradually move it towards the negative direction of the X-axis until one of Γ_L or Γ_S reaches the stability circle. In this state, any slight movement of A2P towards the negative direction of the X-axis on the λ-plane will push Γ_L or Γ_S into the unstable region on the Γ-plane. This means that G_ME has reached its maximum value for an amplifier and cannot be improved any further; i.e., the G_ME is optimized. Note that one decision point is added between step 1 and step 2 in the flowchart to prevent the case that the A2P is already unstable after being moved onto the X-axis on the λ-plane.

To exhibit the details of the proposed process of optimizing G_ME, an example of a 40 nm CMOS G_ME-based amplifier is designed and optimized at 200 GHz. A common-source NMOS transistor with a gate width of 32 μm is used as the original A2P. The G_max and k of the transistor are 3.58 dB and 1.19, respectively.

In step 1, an inductor and a capacitor are parallelly embedded to the transistor to move the embedded transistor onto the X-axis on the λ-plane. Γ_L and Γ_S of the embedded transistor are calculated using Equations (2) and (3) and confirmed to be within the stable region. In step 2, two cascading inductors are added to the transistor and form a cascade–parallel embedding to broaden the scope of movement of the transistor on the λ-plane. The values of the embedding components are adjusted to move the transistor towards the negative direction of the X-axis on the λ-plane. The movements of Γ_L and Γ_S on the Γ-plane are observed simultaneously. When Step 2 is repeated for the third time, Γ_S reaches the source stability circle on the Γ-plane, and thus the transistor reaches the optimum position for G_ME on the λ-plane. The movement of the transistor on the λ-plane and the movements of Γ_L and Γ_S of the transistor on the Γ-plane are shown in Figure 6b–6d, respectively. Note that it is Γ_S that reaches the stability circle in this example. The values of the embedding components and G_ME of the transistor following each step are shown in

Table 1. The values of the embedding components and G_ME of the embedded transistor following each step in the process.

Steps	C1 (fF)	L1 (pH)	L2 (pH)	L3 (pH)	GME (dB)
1	200	76	\	\	4.48
2	200	49.5	5	5	5.42
2	200	40	7	7	6.09
2	200	29	9.5	9.5	7.68

. The G_ME of the embedded transistor is boosted to 7.68 dB. The k of the embedded transistor is now 0.601, and G_max is no longer available as the transistor is in the potential unstable state. The stronger robustness of G_ME compared to G_max is thus reflected. The circuit schematic of the optimum embedded transistor is shown in Figure 6a. The Y_L and Y_S of the embedded transistor are calculated using Equations (2) and (3), and the output and input matching networks are designed accordingly.

In this optimization method, the amplifier is pushed to the edge between stability and instability in order to obtain the maximum G_ME at the design frequency. This may lead to a small bandwidth of the amplifier. In practical design, compromising between high gain and large bandwidth is necessary. The final G_ME-based amplifier and the simulated S parameters are shown in Figure 7. After the compromise for bandwidth, a peak small signal gain of 6.7 dB is obtained. Both S₁₁ and S₂₂ are less than 0 dB at a large frequency range from 150 GHz to 250 GHz, indicating that both the input and output terminals of the amplifier stay stable.

5. Implemented Amplifier and Measurement Results

Using the proposed G_ME optimization process, a two-stage G_ME-based differential embedded amplifier is designed and fabricated in a 40 nm CMOS technology at 210 GHz.

The G_ME core of the amplifier is shown in Figure 8a. Two identical nmos transistors are used to form a differential transistor pair. The transistor has a channel length of 40 nm and a total width of 32 μm with 32 fingers. The bias condition is V_GS = 600 mV; V_DS = 900 mV. The fmax of the transistor is 284 GHz. The G_max and k of the transistor are 3.58 dB and 1.19, respectively. A pair of cross-connected capacitors and four transmission lines with characteristic impedance of 50 ohm are applied to the transistor pair to form the embedding and boost G_ME. The neutralizing capacitor C_n of 19.5 fF equivalently introduces a negative admittance of −0.027 j ohm across gate and drain terminals of the transistor [13], as shown in Figure 8b. C_n, TL₁, and TL₂ together form a cascade–parallel embedding and boost the G_ME of the transistor to a desired value of 6.53 dB, as shown in Figure 8c. The movements of Γ_L and Γ_S of the core by the embedding are shown in Figure 8d.

The circuit schematic of the amplifier is shown in Figure 9a. The input/output matching and interstage matching are implemented using transmission lines with characteristic impedance of 50 ohm and metal-oxide-metal capacitors. Figure 9b shows the micrograph of the amplifier. A pair of baluns are introduced to the input and output terminals for single-ended-differential conversion. The chip size is 0.73 × 0.79 mm². As shown in Figure 9c, the amplifier has a maximum measured small-signal gain of 10.48 dB at 195.33 GHz; the small-signal gain at 210 GHz is 9.53 dB. The large-signal measurement is shown in Figure 9d; the measured maximum output power is 3.04 dBm, and the large-signal gain is 7.08 dB at 210 GHz. The power consumption is 41.4 mW with a 0.9 V supply voltage.

There are two main reasons that lead to the differences between the simulation and experiment results. The first reason is that there are differences between the simulated circuit characteristics and the actual circuit characteristics, including differences in transistors, transmission lines, MOM capacitors, and so on. These differences lead to changes in the matching state of the amplifier, resulting in differences between the simulated and measured S parameter results in Figure 9c. The second reason is that the baluns for single-ended-differential conversion in both input and output terminals of the amplifier introduce additional insertion loss, resulting in performance degradation in the measured results.

6. Conclusions

This paper presents a graphical process for optimizing the maximally efficient gain G_ME of a near-f_max A2P utilizing both the gain-plane and the reflection-coefficient-plane. The optimum G_ME is obtained by moving the A2P to a specific position on the X-axis of the gain-plane while maintaining Γ_L and Γ_S of the A2P within the stable regions of the reflection-coefficient-plane. The movements on both planes are achieved by parallel, series, or cascaded embedding components. These embeddings are calculated and visually analyzed. Details of the optimizing process are exhibited. To show the feasibility of the process, a two-stage embedded amplifier is implemented in a 40 nm CMOS technology with a measured small-signal gain of 10.48 dB at 195.33 GHz.