Switching sequence optimization for gradient errors compensation in the current-steering DAC design

Abstract

In this paper, an optimization method of switching sequence is proposed to compensate for the gradient errors in the current source array of the current-steering digital-to-analog converter. Combining the central symmetry method and the iterative method, the linear and the quadratic gradient errors in the current source arrays are all eliminated. Through the mathematical induction and MATLAB simulation, the proposed switching sequence shows that both the linear and quadratic gradient errors can be compensated. To verify the optimization method proposed, a 12-bit DAC was fabricated under the 55 ​nm 2.5 ​V CMOS process. The measured INL and DNL are bounded at 0.62LSB and 0.37LSB, respectively. The SFDR is more than 78 ​dB with the signal frequencies below 1 ​MHz and more than 66 ​dB in the whole Nyquist band frequency.

Introduction

In high-speed and high-resolution current-steering digital-to-analog converter (DAC) design, the segmented architectures are widely used [[1], [2], [3], [4], [5], [6]]. Generally, for a better tradeoff between the performance and the complexity of the DAC, the least significant bits (LSB’s) steer a binary weighted array, and the most significant bits (MSB’s) are the thermometer decoded and steer a unary array. The static performance is strongly dependent on the linearity of the unary array [7,8]. The linearity can be achieved by overcoming all possible random and systematic errors in segmented DAC [2,8]. The random errors are determined by the inherent matching properties of the technology used [[8], [9], [10]]. In given process technology, comparison with trimming, tuning or calibration methods, increasing the gate area of each unit element in the arrays of DAC’s is the most effective and simple method to reduce random errors. However, for each extra bit of DAC accuracy, the area of the current source array increases by a factor of four [11]. In high-resolution DAC, this method results in large dimension arrays in which gradient error has a very serious impact on linearity performance. The systematic errors, caused by the process, temperature, and electrical gradients, also occur easily in high resolution DAC in which large active area is used to reduce the random errors. Therefore, the design of DAC becomes more difficult for high resolution DAC with more than 10 bits, if there are no any trimming, tuning or calibration methods used [12].

In the gradient errors of the current steering DAC, the linear gradient errors and the quadratic gradient errors are the most serious errors affecting the DAC linearity performance [[13], [14], [15], [16]]. Many methods are proposed to reduce the effect of gradient errors [4,8]. The trimming, tuning or calibration methods are verified that these methods can effectively reduce the influence of gradient error. However, these methods often greatly increase chip area or circuit complexity. The switching sequence optimization method is also verified showing great potential to reduce the nonlinearity. More importantly, this technology is a layout optimization method that does not increase the chip area and circuit complexity. After the process of optimization, the linearity is primarily determined by the accumulated residual errors stemming from the switching sequence in the current source array. In the published literature, the centroid symmetry switching scheme is widely used to eliminate the linear gradient error [7]. Each unary current source is split into four parts with 1/4 the value in four different locations which symmetrical mirror around the X and Y axes. Through the centroid symmetry switching sequence scheme, the linear terms of the gradient errors are compensated. However, the quadratic gradient errors have no change. To reduce the quadratic errors, the quad quadrant (Q²) random walk method has been used [10,11]. Compared with the literature [7], the current sources are split into a higher number. Then a “randomize” switching sequence is used to further reduce the residual gradient errors. However, the measurement results shown that the gradient errors are only suppressed rather than removed. To further improve the linearity, this paper proposed a novel switching sequence. The linear gradient errors are still eliminated by the centroid symmetry method. The quadratic gradient errors are eliminated by converting it to gain errors instead of directly eliminating it. It is well known that gain errors do not contribute to linearity. Therefore, both the linear and quadratic gradient errors are all compensated which led to the improved linearity of the DAC.

This paper is organized as follows. In Section 2, the principle and mathematical proof of the proposed switching sequence are given. The circuit implementation of critical constituent blocks and layout strategies are described in Section 3. Measurement results are presented in Section 4. Finally, the conclusion is given in section 5.

According to Ref. [10], the gradient errors distribution across the unary matrix in the current steering DAC can be approximated by a Taylor series expansion around the center of the unary array. The gradient errors can be expressed as

$$\epsilon \left(x,y\right)=a_0+a_{11}x+a_{12}y+a_{21}{x}^2+a_{22}{y}^2+a_{23}xy+\cdots$$

where (x, y) is the coordinate of the unit in the current sources array. The linearity is mainly affected by the linear (the first order) and the quadratic (the second order) terms in Eq. (1). The effects of more than second order terms are so small that it can be usually ignored [9,10,13]. Therefore, the core goal of this paper also focuses on eliminating the effects of the linear and the quadratic terms gradient errors on the linearity.

The linear terms of the gradient errors, ${\epsilon }^{\left(1\right)}(x,y)$ and the quadratic terms of the gradient errors, ${\epsilon }^{\left(2\right)}(x,y)$ for an element located at (x, y) can be expressed as [15].

$${\epsilon }^{\left(1\right)}(x,y)=g_l\times \cos \theta \times x+g_l\times \sin \theta \times y$$

$${\epsilon }^{\left(2\right)}(x,y)=g_q\times (b{x}^2+y^2)+a$$

where θ is the angle of the gradient and $g_l$ is the slope of the linear gradient. And $g_q$, a and b are the technology-related parameters. Ignored the effects of more than second order terms errors, the linearity is affected by the superposition of the linear terms and quadratic terms gradient errors. Therefore, the total gradient errors for an element located at (x, y) can be expressed as

$$\epsilon (x,y)={\epsilon }^{\left(1\right)}(x,y)+{\epsilon }^{\left(2\right)}(x,y)$$

According to Eq. (4), the influence of gradient errors on the linearity can be removed when the linear terms and quadratic terms gradient errors are both eliminated. In Refs. [7,9], the central symmetry method was effective in removing the linear gradient errors. Therefore, the central symmetry method is still used to eliminate the linear gradient errors in this paper. For the quadratic gradient errors, the new iterative method is proposed in this paper. All current source has the same quadratic gradient errors after the iterative optimization method. Therefore, the quadratic gradient errors of all the current sources can be treated as gain errors which have no contribution to the linearity.

For N-bit thermometer-code current sources DAC, there are 2^N current sources in the current source array of DAC. The total numbers of the current cells are 2^m*2^N when each current source has 2^m current cell, m is a natural number. As mentioned above, the centroid symmetry method and the iterative method are superimposed used to eliminate the linear and quadratic gradient errors, respectively. The principle of the centroid symmetry method is shown in Fig. 1. Each unary current source is split into four parts with 1/4 the value in four different locations which symmetrical mirror around the X and Y axes. After the optimization of the centroid symmetry method, all current sources are evenly distributed into four quadrants. Each quadrant has the same current sources. And then the iterative method is used in each quadrant to eliminate the quadratic gradient errors. Due to the current sources array in four quadrants are symmetry, there are only describe the principle of the iterative method in one quadrant. After we get the switching sequence in one quadrant, the total switching sequence for four quadrants can be also obtained by using the centroid symmetry method. The first quadrant is used in this paper to introduce the iterative method.

Assuming, the resolution of the thermometer-code current-steering DAC, N, can be expressed as

$$N=4k+j,(k=\mathrm{0,1,2},\cdots ,n;j=\mathrm{0,1,2,3})$$

when k and j both are 0, it is mean that there are no current sources. This situation does not exist for a given resolution DAC. Therefore, it will not be introduced in the following discussion. For N bit thermometer-code current-steering DAC, there are 2^4k+j current sources (2^4k+j ​= ​2^N) in which each current source has 2^m current cells. Before using the iterative method, each current source is split into four sub-current sources which distribute symmetrically in four quadrants. In the first quadrant, there are 2^4k+j sub-current sources in which each sub-current source has 2^m−2 current cells.

In the iterative technique, the sub-current source will be split again. The relation between the number of current cells, N_cells, with the resolution of DAC can be express as

$$N_{cells}=\{\begin{array}{l}{2}^j\\ 2^{2k}\\ 2^{2k+j}\end{array}\begin{array}{c};\\ ;\\ ;\end{array}\begin{array}{c}k=0,j\ne 0\\ k\ne 0,j=0\\ k\ne 0,j\ne 0\end{array}$$

All sub-current sources form a current source array which can be seen as a positive matrix, M_kj. The M_kj can be express as

$$M_{kj}=\{\begin{array}{l}{2}^j\times 2^j\\ 2^{3k}\times 2^{3k}\\ 2^{3k+j}\times 2^{3k+j}\end{array}\begin{array}{c};\\ ;\\ ;\end{array}\begin{array}{c}k=0,j\ne 0\\ k\ne 0,j=0\\ k\ne 0,j\ne 0\end{array}$$

The switching sequence in M_kj is working like this: when k ​= ​0 and j≠0, M_kj is formed by four M_k(j-1), each sub-current source in M_kj is split into 2^j cells; when k≠0 and j ​= ​0, M_kj is a 2^3k rows and 2^3k columns positive matrix which stems from M_(k-1)3 split, each sub-current source in M_kj is split into 2^2k cells; when k≠0 and j≠0, M_kj is formed by a 2^3k rows and 2^3k columns positive matrix in which each region is equal to the matrix M_0j, each sub-current source in M_kj is split into 2^2k+j cells. For example, when N is 1, it can get that k is 0 and j is 1. The current sources array in the first quadrant is a positive matrix, M₀₁, which has 2 (2¹) rows and 2 (2¹) columns. Each sub-current source is split into 2 ​cells, as shown in Fig. 2(a). When N is 2, it is clear that k is 0 and j is 2. The current sources array is formed by 4 (2²) rows and 4 (2²) columns matrix (M₀₂), as shown in Fig. 2(b). The matrix M₀₂ is formed with four M₀₁ regions (referred to matrix A-B). Each sub-current source is split into four cells. When N is 3, we can get that k is 0 and j is 3. The current sources are divided into 8 (2³) rows and 8 (2³) columns matrixes (M₀₃), as shown in Fig. 2(c). The matrix M₀₃ is formed with four M₀₂ regions (referred to matrix i-ii). Each sub-current source is split into eight cells. When N is 4, k is 1 and j is 0. The current sources are divided into 16 (2⁴) rows and 16 (2⁴) columns matrixes (M₁₀), the switching sequence M₁₀ comes from M₀₃, as shown in Fig. 2(d). Each sub-current source is split into four cells. The arrangement of M₁₀ is

$$\begin{array}{l}{0}_{11}in{M}_{10}is{0}_{11}in{M}_{03},\\ 0_{44}in{M}_{10}is{0}_{44}in{M}_{03},\\ 0_{66}in{M}_{10}is{0}_{66}in{M}_{03},\\ 0_{77}in{M}_{10}is{0}_{77}in{M}_{03},\\ 1_{22}in{M}_{10}is{0}_{22}in{M}_{03},\\ 1_{33}in{M}_{10}is{0}_{33}in{M}_{03},\\ 1_{55}in{M}_{10}is{0}_{55}in{M}_{03},\\ 1_{88}in{M}_{10}is{0}_{88}in{M}_{03},\\ 2_{21}in{M}_{10}is{1}_{21}in{M}_{03},\\ \cdots \\ {15}_{18}in{M}_{10}is{7}_{18}in{M}_{03}.\end{array}$$

where N_xy is the current cell located at x row and y column in the matrix. When N is 5, it can get that k is 1 and j is 1. The current sources are divided into 8 (2³) rows and 8 (2³) matrix (M₁₁), as shown in Fig. 2(e). The matrix M₁₁ is formed with sixty-fourth M₀₁ regions (referred to matrix A-P). Each region in the matrix M₁₁ is equal to the matrix M₀₁. Each sub-current source is split into 8 ​cells. Similarly, other current sources array can be got in the same method as described above.

After achieving the switching sequence in the first quadrant through the iterative method as described above, the quadratic gradient errors are eliminated, the conclusions will be shown later. For the total switching sequence of the current sources array, we only turn the sub-current sources in the first quadrant around the X and Y axes. Finally, the linear and quadratic quadrant errors can be eliminated in the whole current sources array.

As analyzed above, the linear gradient errors in the current sources array can be eliminated by the centroid symmetry method. Therefore, we only prove that quadratic gradient errors can be eliminated through the iterative method.

According to Eq. (3), in the first quadrant, the quadratic gradient errors of the current cell “0” in M₀₁ in Fig. 2(a) are

$${\epsilon }_{M_{01}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{01}}^{\left(2\right)}\left(\mathrm{2,1}\right)+{\epsilon }_{M_{01}}^{\left(2\right)}\left(\mathrm{1,2}\right)=g_q\times \left(5b+5\right)+2a$$

where (x, y) is the coordinate of the unit in the matrix. Similarly, the quadratic gradient errors of the current cell “1” in M₀₁ in Fig. 2(a) are

$${\epsilon }_{M_{01}}^{\left(2\right)}\left(1\right)={\epsilon }_{M_{01}}^{\left(2\right)}\left(\mathrm{1,1}\right)+{\epsilon }_{M_{01}}^{\left(2\right)}\left(\mathrm{2,2}\right)=g_q\times \left(5b+5\right)+2a$$

From Eqs. (8), (9), it can be got that

$${\epsilon }_{M_{01}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{01}}^{\left(2\right)}\left(1\right)$$

From Eq. (10), it is can be got that current cell “0” and “1” have the same quadratic gradient errors in Fig. 2(a) after switching sequence optimization. In the same way, the quadratic gradient errors of current cells in M₀₂ in Fig. 2(b)are

$${\epsilon }_{M_{02}}^{\left(2\right)}\left(0\right)+{\epsilon }_{M_{02}}^{\left(2\right)}\left(1\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(2\right)+{\epsilon }_{M_{02}}^{\left(2\right)}\left(3\right)$$

And from Eq. (10), it can be obtained that

$${\epsilon }_{M_{02}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(1\right)$$

$${\epsilon }_{M_{02}}^{\left(2\right)}\left(2\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(3\right)$$

Finally, from Eq. (10), (11), (12), (13)), the quadratic gradient errors of four cells in Fig. 2(b) can be obtained that

$${\epsilon }_{M_{02}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(1\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(2\right)={\epsilon }_{M_{02}}^{\left(2\right)}\left(3\right)$$

Through a similar calculation, for matrix M₀₃ in Fig. 1(c), the quadratic gradient errors can be got

$${\epsilon }_{M_{03}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{03}}^{\left(2\right)}\left(1\right)=\cdots ={\epsilon }_{M_{03}}^{\left(2\right)}\left(7\right)$$

For matrix M₁₀ in Fig. 1(d), the quadratic gradient errors are

$$\begin{array}{l}{\epsilon }_{M_{10}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{1,8}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{4,5}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{6,3}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{7,2}\right)\\ ={\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{2,7}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{3,6}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{5,4}\right)+{\epsilon }_{M_{10}}^{\left(2\right)}\left(\mathrm{8,1}\right)\\ ={\epsilon }_{M_{10}}^{\left(2\right)}\left(1\right)\end{array}$$

Therefore, from Eq. (11), (12), (13), (14), (15), (16)), it is easily obtained that

$${\epsilon }_{M_{10}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{10}}^{\left(2\right)}\left(1\right)=\cdots ={\epsilon }_{M_{10}}^{\left(2\right)}\left(15\right)$$

Through similar calculation, for matrix M_kj, it is easily obtained that the quadratic gradient errors can be got as

$${\epsilon }_{M_{kj}}^{\left(2\right)}\left(0\right)={\epsilon }_{M_{kj}}^{\left(2\right)}\left(1\right)=\cdots ={\epsilon }_{M_{kj}}^{\left(2\right)}\left(n\right)$$

From Eqs. (10), (14), (15), (17), (18), in the first quadrant, the current cell in the sub-current sources array has the same quadratic gradient errors after the iterative switching sequence optimization. Due to the symmetry method in the four quadrants, the sub-current sources in other three quadrants also have the same quadratic gradient errors with the first quadrant. Therefore, after the optimization process by the iterative method, all current sources have the same quadratic gradient errors in the whole current source array. The same quadratic gradient errors can be seen as the gain errors which do not affect the linearity in DAC. Finally, both the linear and quadratic gradient errors are compensated after switch sequence optimization proposed in this paper.

To verify the above conclusions, MATLAB simulation was implemented. As shown in Fig. 3, three methods (The centroid switching scheme, the Q² random walk switching scheme and this paper proposed) are compared. The three methods have the same initial error distribution. ${\epsilon }_{sp}^{\left(1\right)}$ and ${\epsilon }_{sp}^{\left(2\right)}$ represent the linear and quadratic gradient errors, respectively. The ${\epsilon }_{res}^{\left(1\right)}$ and ${\epsilon }_{res}^{\left(2\right)}$ represent the linear and quadratic residual errors after the switching sequence process, respectively. The centroid switching scheme as shown in Fig. 3(a), each thermometer-code current sources are split into four part (shown in red) which are symmetrical distribution around the center of the geometry. From the residual error distribution in the right part of Fig. 3(a), it is observed shown that the linear gradient errors are eliminated, but the quadratic gradient errors have no change. As shown in Fig. 3(b), the Q² random walk switching scheme is used. From the simulation results, we can get that the linear terms are also eliminated where the quadratic gradient errors are suppressed. The simulation results in Fig. 3(a) and (b) are consistent with the previous literature results [9]. In Fig. 3(c), through using the switching scheme proposed in this paper, the simulation results show that the linear terms are eliminated, and all the current sources have the same residual errors (the quadratic gradient errors). According to the previous analysis, these same residual errors can be treated as the gain errors which have no contribute with linearity in DAC. Therefore, the MATLAB simulation results also show that the switching sequence scheme proposed in this paper can eliminate the gradient errors of the current steering DAC.