Patent Publication Number: US-7903012-B2

Title: D/A converter and reference voltage circuit including same

Description:
This nonprovisional application claims priority under 35 U.S.C. §119(a) on Patent Application No. 2008-155987 filed in Japan on Jun. 13, 2008, the entire contents of which are hereby incorporated by reference. 
     TECHNICAL FIELD 
     The present invention relates to D/A converters and reference voltage circuits including such a D/A converter. 
     BACKGROUND ART 
     Recent reference voltage circuits for grayscale or opposite electrode application in liquid crystal displays are required to be capable of 9 or greater bit outputs and high precision monotonicity. Resistor-string D/A converters used in these reference voltage circuits are advantageously generally suitable for high speed operation and readily provides monotonicity. For 9 or greater bit outputs, however, the resistor-string D/A converter is not practical because of its increased number of wires (512 wires for 9 bit outputs), resistors, and switches. 
       FIG. 6  is a circuit diagram of a conventional resistor-string D/A converter  100 . The D/A converter  100  includes a resistor-string circuit  101  producing divided voltages from reference voltages VH and VL and a switching circuit containing switches for switching between divided voltage outputs vref&lt;0&gt; to vref&lt;(2^n)−1&gt;. 
     An analog output voltage Vout obtained by conversion of an n-bit digital signal is given by equation (1):
 
 V out= VL +( VH−VL )* k/ 2^ n   (1)
 
where k=0 to 2^n−1.
 
     The digital signal fed to the D/A converter  100  consists of n upper bits HoB&lt;n−1:0&gt; and m lower bits LoB&lt;m−1:0&gt;. 
     The D/A converter  100  in  FIG. 6  is not practical for high bit output applications due to an increased number of wires, resistors, and switches. For example, for n=9 (9-bit conversion), the D/A converter  100  needs 512 wires, 512 resistors, and 512 switches. Likewise, for n=10 (10-bit conversion), the D/A converter  100  needs 1024 wires, 1024 resistors, and 1024 switches. 
       FIG. 7  is a circuit diagram of another conventional resistor-string D/A converter  103 . The D/A converter  103  is disclosed in Japanese Patent Application Publication, Tokukaihei, No. 6-224767 (Publication Date: Aug. 12, 1994, Patent Literature 1). 
     The D/A converter  103  includes a D/A converter  104  producing a reference voltage VH_ 1  in accordance with n upper bits HoB&lt;n−1:0&gt; for output, a D/A converter  105  producing a reference voltage VL_ 1  in accordance with the n upper bits HoB&lt;n−1:0&gt; for output, a resistor-string circuit  106  producing divided voltages from reference voltages VH_ 1  and VL_ 1  in accordance with m lower bits LoB&lt;m−1:0&gt;, and a switching circuit  107  switching between divided voltage outputs vref&lt;0&gt; to vref&lt;(2^m)−1&gt;. 
     The digital signal fed to the D/A converter  103  consists of the n upper bits HoB&lt;n−1:0&gt; and the m lower bits LoB&lt;m−1:0&gt;. 
     An analog output voltage Vout obtained by conversion of an (n+m)-bit digital signal consisting of n upper bits and m lower bits is given by equation (4) (derived from equations (2) and (3)): 
                     VH_   ⁢   1     =     VL   +       (     VH   -   VL     )     *         (     1   +   1     )     /   2     ⋀   n                 (   2   )                 VL_   ⁢   1     =     VL   +       (     VH   -   VL     )     *       1   /   2     ⋀   n                 (   3   )                     Vout   =       ⁢       VL_   ⁢   1     +       (       VH_   ⁢   1     -     VL_   ⁢   1       )     *       k   /   2     ⋀   n                     =       ⁢     VL   +       (     VH   -   VL     )     *       1   /   2     ⋀   n       +                     ⁢     {         (     VH   -   VL     )     *         (     1   +   1     )     /   2     ⋀   n       -                         ⁢       (     VH   -   VL     )     *       1   /   2     ⋀   n       }     *       k   /   2     ⋀   n                 =       ⁢     VL   +       (     VH   -   VL     )     *       1   /   2     ⋀   n       +                     ⁢       (     VH   -   VL     )     *       1   /   2     ⋀   n     *       k   /   2     ⋀   m                   =       ⁢     VL   +       (     VH   -   VL     )     *         {       1   *     2   ⋀   m       +   k     }     /   2     ⋀     (     n   +   m     )                         (   4   )               
where k=0 to 2^m−1 and 1=0 to 2^n−1.
 
     Equations (2) and (3) differ in “1+1” and “1”. The reason is that the D/A converter  104  receives a digital signal input containing the n upper bits HoB&lt;n−1:0&gt; and produces a reference voltage VH_ 1  that is equivalent to the digital signal containing the n upper bits HoB&lt;n−1:0&gt; plus 1 for output, whereas the D/A converter  105  receives a digital signal input containing the n upper bits HoB&lt;n−1:0&gt; and produces a reference voltage VL_ 1  that is equivalent to the digital signal containing the n upper bits HoB&lt;n−1:0&gt; for output. 
     To obtain equations (2) and (3), the D/A converter  104  and the D/A converter  105  may be the same D/A converters so that the D/A converter  104  can receive a digital signal input containing the n upper bits HoB&lt;n−1:0&gt; plus 1 and the D/A converter  105  can receive a digital signal input containing the n upper bits HoB&lt;n−1:0&gt;. 
     For example, for n=2 and m=7 (9-bit digital signal conversion), the D/A converter  103  in  FIG. 7  needs 128 wires, 128 resistors, and 128 switches. Likewise, for n=2 and m=8 (10-bit digital signal conversion), the D/A converter  103  needs 256 wires, 256 resistors, and 256 switches. The D/A converter  103  in  FIG. 7  needs fewer wires, resistors, and switches than the D/A converter  100  in  FIG. 6 . 
     The D/A converter  100  in  FIG. 6  is not practical for 9 or greater bit output applications because of its increased number of wires, resistors, and switches. 
     Although the D/A converter  103  in  FIG. 7  needs fewer wires, resistors, and switches than the D/A converter  100  in  FIG. 6 , the D/A converter  103  needs the D/A converters  104  and  105  to produce the reference voltages VH_ 1  and VL_ 1  respectively before the resistor-string circuit  106 , which adds to circuit size. Furthermore, the provision of the D/A converters  104  and  105  could lead to undesirable increases in current consumption and degradation in precision of D/A conversion. 
     SUMMARY OF INVENTION 
     The present invention has an objective of providing a small circuit size, high precision D/A converter capable of conversion of high bit digital signals. 
     A D/A converter according to the present invention is, to achieve the objective, a D/A converter for converting a digital signal consisting of n upper bits and m lower bits to an analog voltage. The converter includes: voltage dividing means containing (2´n)−1 resistors of an equal resistance connected in series, a first reference voltage and a second reference voltage being applied across both terminals of the resistors; selecting means for selecting, according to the n upper bits of the digital signal, a divided voltage produced by the voltage dividing means for output as the analog voltage; and two variable resistors connected to the terminals of the (2^n)−1 resistors connected in series, the variable resistors exhibiting resistances determined according to the m lower bits of the digital signal. 
     According to the invention, the analog voltage is produced by means of the resistances determined in the two variable resistors according to the m bits of the digital signal. The invention requires fewer wires connecting the voltage dividing means to the selecting means, fewer resistors, and fewer switches in the selecting means than the conventional D/A converter. Circuit size is thus reduced. A D/A converter is realized which is capable of converting high bit (9 to 11 bits) digital signals. 
     Unlike the conventional D/A converter, the D/A converter of the invention needs no internal D/A converters. Therefore, current consumption does not increase due to the addition of D/A converters. Precision in D/A conversion is dictated by the relative precision of the two variable resistors connected to the reference voltages. Thus, the precision in D/A conversion does not fall by as great an amount as is the case with the conventional D/A converter. A high precision D/A converter is provided. 
     Additional objectives, advantages and novel features of the invention will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       
         FIG. 1 
       
         FIG. 1  is a circuit diagram of a D/A converter in accordance with an example of the present invention. 
       
         FIG. 2 
       
         FIG. 2  is a circuit diagram of a variable resistor for m=3 (m is the number of lower bits contained in a digital signal) in a D/A converter in accordance with an example of the present invention. 
       
         FIG. 3 
       
         FIG. 3  is a circuit diagram of a D/A converter in accordance with another example of the present invention. 
       
         FIG. 4 
       
         FIG. 4  is a circuit diagram of a variable resistor for m=3 in a D/A converter in accordance with another example of the present invention. 
       
         FIG. 5 
       
         FIG. 5  is a circuit diagram of a D/A converter in accordance with a further example of the present invention. 
       
         FIG. 6 
       
         FIG. 6  is a circuit diagram of a conventional resistor-string D/A converter. 
       
         FIG. 7 
       
         FIG. 7  is a circuit diagram of another conventional resistor-string D/A converter. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     The following will describe an embodiment of the present invention by means of examples 1 to 3 and in reference to  FIGS. 1 to 5 . 
     Example 1 
       FIG. 1  is a circuit diagram of a D/A converter  1  in accordance with example 1. The D/A converter  1  includes a resistor-string circuit  2  producing divided voltages from reference voltages VH and VL and a switching circuit  3  containing switches for switching between divided voltage outputs vref&lt;0&gt; to vref&lt;(2^n)−1&gt;. 
     The resistor-string circuit  2  includes (2^n)−1 resistors R, a variable resistor  4 , and a variable resistor  5 . The reference voltage VH is applied to a terminal of the variable resistor  4 , and the reference voltage VL is applied to a terminal of the variable resistor  5 . The (2^n)−1 resistors R are connected in series between the other terminal of the variable resistor  4  and the other terminal of the variable resistor  5 . 
     A digital signal input to the D/A converter  1  consists of n upper bits HoB&lt;n−1:0&gt; and m lower bits LoB&lt;m−1:0&gt;. The n upper bits HoB&lt;n−1:0&gt; consists of bits H n−0  to H 0 . The m lower bits LoB&lt;m−1:0&gt; consists of bits L m−1  to L 0 . In the notation H n−1  to H 0  and L m−1  to L 0 , greater subscripted numerals indicate higher bits. The bit H n−1  is the most significant bit, and the bit L 0  is the least significant bit. 
     The resistance RH of the variable resistor  4  and the resistance RL of the variable resistor  5  are determined according to the m lower bits LoB&lt;m−1:0&gt; of the digital signal and given respectively by equations (5) and (6).
 
 RH =(2^ m−k )/(2^ m )* R   (5)
 
 RL=k /(2^ m )* R   (6)
 
where k=0 to 2^m−1. From equations (5) and (6), it is appreciated that RH+RL=R.
 
     The divided voltage output vref&lt;0&gt; is the output at the contact of the variable resistor  5  and a resistor R. The divided voltage output vref&lt;2^n−1&gt; is the output at the contact of the variable resistor  4  and a resistor R. For x=1 to 2^n−2, vref&lt;x&gt; is the output at the contact of the x-th and (x+1)-th resistors R as counted from the side of the variable resistor  5 . For example, vref&lt;1&gt; is the output at the contact of the first and second resistors R as counted from the side of the variable resistor  5 . 
     The switching circuit  3  selects one of the divided voltage outputs vref&lt;0&gt; to vref&lt;2^n−1&gt; according to the n upper bits HoB&lt;n−1:0&gt; of the digital signal. As an example, if n=2 and the upper two bits HoB&lt;1:0&gt; are representing a binary “10”, the switching circuit  3  selects vref&lt;2&gt;, hence Vout=vref&lt;2&gt; because “10” in binary is “2” in decimal notation. 
     The analog output voltage Vout, obtained by conversion of an (n+m)-bit digital signal consisting of n upper bits and m lower bits, is given by equation (7):
 
 V out= VL +( VH−VL )*( RL+R* 1)/{ RH+RL+R *(2^ n− 1)}  (7)
 
Equation (8) is derived by substituting RH and RL of equations (5) and (6) to equation (7).
 
 V out= VL +( VH−VL )*( k/ 2^ m*R+R* 1)/{(2^ m−k )/(2^ m )* R+k /(2^ m )* R+R *(2^ n− 1)}  (8)
 
By dividing both the denominator and the numerator in the second term on the right side of equation (8) by R, equation (9) is obtained:
 
 V out= VL +( VH−VL )*{ k /(2^ m )+1}/{(2^ m−k )/(2^ m )+ k /(2^ m )+(2^ n− 1)}  (9)
 
By substituting equation (10) into the right side of equation (8), equation (11) is obtained.
 
(2^ m−k )/(2^ m )+ k /(2^ m )=1  (10)
 
 V out= VL +( VH−VL )*{ k /(2^ m )+1}/{1+(2^ n− 1)}  (11)
 
Arranging equation (11) yields equation (12).
 
 V out= VL +( VH−VL )*{ k /(2^ m )+1}/2^ n   (12)
 
The output voltage Vout is given by equation (13) which is derived by multiplying the denominator and the numerator in the second term on the right side of equation (12) by 2^m.
 
 V out= VL +( VH−VL )*( k+ 1*2^ m )/2^( m+n )  (13)
 
     A D/A converter capable of 9 or greater bit digital signal conversion is to be built around the D/A converter  1 . Assume in the D/A converter  1 , for example, that n=2 and m=7 for a 9-bit digital signal conversion, in which case there are 2^7=128 wires connecting the resistor-string circuit  2  to the switching circuit  3 , 2^7−1=127 resistors R, and 2^7=128 switches in the switching circuit  3 . 
     Likewise, for n=2 and m=8 for a 10-bit digital signal conversion in the D/A converter  1 , there are 2^8=256 wires connecting the resistor-string circuit  2  to the switching circuit  3 , 2^8−1=255 resistors R, and 2^8=256 switches in the switching circuit  3 . 
     Likewise, for n=2 and m=9 for a 11-bit digital signal conversion in the D/A converter  1 , there are 2^9=512 wires connecting the resistor-string circuit  2  to the switching circuit  3 , 2^9−1=511 resistors R, and 2^9=512 switches in the switching circuit  3 . 
     As described in the foregoing, the output voltage Vout is produced by means of the resistances RH and RL of the variable resistors  4  and  5  determined according to the m-bit digital signal. The D/A converter  1  therefore requires fewer wires connecting the resistor-string circuit  2  to the switching circuit  3 , fewer resistors R, and fewer switches in the switching circuit  3  than the conventional D/A converter  100  shown in  FIG. 6 . Circuit size is thus reduced. A D/A converter is realized which is capable of converting high bit (9 to 11 bits) digital signals. 
     Unlike the conventional D/A converter  103  shown in  FIG. 7 , the D/A converter  1  needs no internal D/A converters. Therefore, current consumption does not increase due to the addition of D/A converters. Precision in D/A conversion is dictated by the relative precision of the variable resistors  4  and  5  connected to the reference voltages VH and VL. Thus, the precision in D/A conversion does not fall by as great an amount as is the case with the D/A converter  103 . A high precision D/A converter is provided. 
       FIG. 2  is a circuit diagram of the variable resistors  4  and  5  when m=3 in the D/A converter  1 . In  FIG. 2 , the variable resistor  4  includes parallel resistors  6  to  9  and p-channel MOSFETs (metal-oxide-semiconductor field-effect transistor)  16  to  18 . The variable resistor  5  includes parallel resistors  10  to  12  and n-channel MOSFETs  19  to  21 . Similarly to  FIG. 1 , (2^n)−1 resistors R are connected in series between the variable resistors  4  and  5 . 
     The parallel resistors  6 ,  7 , and  10  each contain eight resistors R connected in parallel to exhibit a resistance of R/8. The parallel resistors  8  and  11  each contain four resistors R connected in parallel to exhibit a resistance of R/4. The parallel resistors  9  and  12  each contain two resistors R connected in parallel to exhibit a resistance of R/2. 
     The reference voltage VH is applied to a terminal of the parallel resistor  6 . The other terminal of the parallel resistor  6  is connected to a terminal of the parallel resistor  7  and the source of the p-channel MOSFET  16 . The other terminal of the parallel resistor  7  is connected to a terminal of the parallel resistor  8 , the drain of the p-channel MOSFET  16 , and the source of the p-channel MOSFET  17 . The other terminal of the parallel resistor  8  is connected to a terminal of the parallel resistor  9  and the source of the p-channel MOSFET  18 . The other terminal of the parallel resistor  9  is connected to a terminal of a resistor R and the source of the p-channel MOSFET  18 . 
     A terminal of the parallel resistor  10  and the drain of the n-channel MOSFET  19  are connected to a terminal of a resistor R other than the resistor R to which the parallel resistor  9  is connected. The other terminal of the parallel resistor  10  is connected to a terminal of the parallel resistor  11 , the source of the n-channel MOSFET  19 , and the drain of the n-channel MOSFET  20 . The other terminal of the parallel resistor  11  is connected to a terminal of the parallel resistor  12 , the source of the n-channel MOSFET  20 , and the drain of the n-channel MOSFET  21 . The reference voltage VL is applied to the other terminal of the parallel resistor  12  and the source of the n-channel MOSFET  21 . 
     The output of the inverter  13  is connected to the gate of the p-channel MOSFET  16  and the gate of the n-channel MOSFET  19 . The output of the inverter  14  is connected to the gate of the p-channel MOSFET  17  and the gate of the n-channel MOSFET  20 . The output of the inverter  15  is connected to the gate of the p-channel MOSFET  18  and the gate of the n-channel MOSFET  21 . 
     The least significant bit L 0  is supplied to the input of the inverter  13 . A lower bit L 1  is supplied to the input of the inverter  14 . A lower bit L 2  is supplied to the input of the inverter  15 . 
     Suppose, as an example, that k=6, that is, (L 2 , L 1 , L 0 )=(1, 1, 0), in which case the output of the inverter  13  is High, and the output of the inverter  14  and the output of the inverter  15  are Low. That turns off the p-channel MOSFET  16  and the n-channel MOSFETs  20  and  21  and turns on the p-channel MOSFETs  17  and  18  and the n-channel MOSFET  19 . 
     The resistance RH of the variable resistor  4  is therefore equal to 2R/8 as shown in equation (14) below:
 
 RH=R/ 8+ R/ 8=2 R/ 8  (14)
 
     The resistance RL of the variable resistor  5  is equal to 6R/8 as shown in equation (15) below:
 
 RL=R/ 4+ R/ 2=6 R/ 8  (15)
 
     The RH and RL shown in equations (5) and (6) respectively are readily obtained by giving weights of 2^m to the variable resistors  4  and  5  as above. 
     Table 1 shows relationship between (L 2 , L 1 , L 0 ), the resistance RH of the variable resistor  4 , the resistance RL of the variable resistor  5 , and RH+RL. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 (L2, L1, L0) 
                 RH 
                 RL 
                 RH + RL 
               
               
                   
                   
               
             
            
               
                   
                 (0, 0, 0) 
                 8R/8 
                 0 
                 R 
               
               
                   
                 (0, 0, 1) 
                 7R/8 
                 R/8 
                 R 
               
               
                   
                 (0, 1, 0) 
                 6R/8 
                 2R/8 
                 R 
               
               
                   
                 (0, 1, 1) 
                 5R/8 
                 3R/8 
                 R 
               
               
                   
                 (1, 0, 0) 
                 4R/8 
                 4R/8 
                 R 
               
               
                   
                 (1, 0, 1) 
                 3R/8 
                 5R/8 
                 R 
               
               
                   
                 (1, 1, 0) 
                 2R/8 
                 6R/8 
                 R 
               
               
                   
                 (1, 1, 1) 
                 R/8 
                 7R/8 
                 R 
               
               
                   
                   
               
            
           
         
       
     
     Example 2 
       FIG. 3  is a circuit diagram of a D/A converter  22  in accordance with example 2. The D/A converter  22  has the same configuration as the D/A converter  1  of example 1 and additionally includes a switching control circuit  23 . The D/A converter  22  is characterized by its ability to control switching between the resistance RH of the variable resistor  4  and the resistance RL of the variable resistor  5  according to the least significant bit LoB&lt;0&gt; (least significant bit L 0 ) and thereby produce output voltages with greater precision. 
     For example, if no switching control is carried out when the least significant bit LoB&lt;0&gt;=0 and a (n+m)-bit digital signal consisting of n upper bits plus m lower bits is to be converted to analog voltage, the output voltage Vout produced is given by equation (13) similarly to example 1.
 
 V out= VL +( VH−VL )*( k+ 1*2^ m )/2^( m+n )  (13)
 
where k=0 to 2^m−1 and 1=0 to 2^n−1.
 
     Switching control is carried out by repeatedly toggling the least significant bit LoB&lt;0&gt; between 1 and 0 as an example, in which case the output voltage Vout toggles between an output voltage Vout_H for the least significant bit LoB&lt;0&gt;=1 and an output voltage Vout_L for the least significant bit LoB&lt;0&gt;=0. 
     The output voltage Vout_L is equal to the Vout in equation (13) and calculated from equation (16). As the least significant bit LoB&lt;0&gt; increases from 0 to 1, “k” in RH given by equation (5) and RL given by equation (6), both discussed in example 1, is replaced by “k+1.” As “k” is replaced by “k+1” in equation (16), the output voltage Vout_H is calculated from equation (17).
 
 V out —   L=VL+{VH−VL }*( k+ 1*2^ m )/2^( n+m )  (16)
 
 V out —   H=VL +( VH−VL )*{( k+ 1)+1*2^ m}/ 2^( n+m )  (17)
 
     Letting “a” represent the period in which LSB&lt;0&gt;=1 and 1−a represent the period in which LSB&lt;0&gt;=0 where 0≦a≦1, the output voltage Vout_SW produced as a result of switching control is averaged out and given by equation (18).
 
 V out —   sw=VL+{a*V out —   H +(1− a )* V out —   L}   (18)
 
Expanding the third term on the right side of equation (18) and collecting terms containing “a” yields equation (19).
 
 V out —   sw=VL+V out —   L+a *( V out —   H−V out —   L )  (19)
 
Substituting equations (16) and (17) into equation (19) and collecting terms containing (VH−VL) yields equation (20).
 
 V out —   sw= 2 VL +( VH−VL )*{( k+ 1*2^ m )/2^( n+m )+ a/ 2^( n+m )}  (20)
 
Factoring out 2^(n+m) in the second term on the right side of equation (20) and sorting out the numerator yields equation (21).
 
 V out —   sw= 2 VL +( VH−VL )*{1*2^ m +( k+a )}/2^( n+m )  (21)
 
Since “a” is added to the term of the variable “k” which is related to the m lower bits, n+m bit or greater resolution is achieved depending on the value “a” (0≦a≦1).
 
     For example, if a=0.75, that is, if the period in which LSB&lt;0&gt;=1 is 75%, the output voltage Vout_SW produced as a result of switching control is given by equation (22) which is equivalent to equation (21) with a=0.75 being plugged in.
 
 V out —   sw= 2 VL +( VH−VL )*{1*2^ m +( k+ 0.75)}/2^( n+m )  (22)
 
     Alternatively, if a=0.50, that is, if the period in which LSB&lt;0&gt;=1 is 50%, the output voltage Vout_SW produced as a result of switching control is given by equation (23) similarly to the case with equation (22).
 
 V out —   sw= 2 VL +( VH−VL )*{1*2^ m +( k+ 0.50)}/2^( n+m )  (23)
 
     Furthermore, if a=0.25, that is, if the period in which LSB&lt;0&gt;=1 is 25%, the output voltage Vout_SW produced as a result of switching control is given by equation (24).
 
 V out —   sw= 2 VL +( VH−VL )*{1*2^ m +( k+ 0.25)}/2^( n+m )  (24)
 
     If a=0, that is, if the period in which LSB&lt;0&gt;=1 is 0%, the output voltage Vout_SW produced as a result of switching control is given by equation (25).
 
 V out —   sw= 2 VL +( VH−VL )*(1*2^ m+k )/2^( n+m )  (25)
 
     As described in the foregoing, in the D/A converter  22 , switching control is carried out according to the least significant bit LoB&lt;0&gt; (least significant bit L 0 ). Hence, the resistances of the variable resistors  4  and  5  change according to the least significant bit LoB&lt;0&gt;, and the output voltage Vout_sw also changes according to the least significant bit LoB&lt;0&gt;. The mechanism enables production of output voltages with greater precision. 
     In addition, by giving “a” a 2^p weight, that is, a p-bit weight as indicated in equation (26), the output voltage Vout_SW is given by equation (27) and has a resolution of n+m+p bits. Note in equation (26) that q=0 to 2^p−1.
 
 a=q/ 2^ p   (26)
 
 V out —   sw= 2 VL +( VH−VL )*{1*2^( m+p )+( k+q )}/2^( n+m+p )  (27)
 
       FIG. 4  is circuit diagram of the variable resistors  4  and  5  when m=3 in the D/A converter  22 . The circuit in  FIG. 4  has the same configuration as the circuit in  FIG. 2  and additionally includes a switching control circuit  23 . Switching is controlled by means of the least significant bit L 0  to obtain Vout_L given by equation (16) and Vout_H given by equation (17). 
     Example 3 
       FIG. 5  is circuit diagram of a D/A converter  24  in accordance with example 3. The D/A converter  24  has the same configuration as the D/A converter  22  of example 2 and additionally includes a LPF (lowpass filter)  25 . The input of the LPF  25  is connected to the output of the switching circuit  3 . The LPF  25  outputs an output voltage Vout. In the D/A converter  24 , the LPF  25  is capable of removing the noise on the output voltage Vout which occurs when the switching circuit  3  carries out the switching. 
     Each reference voltage circuit of the present invention includes one of the D/A converters discussed above. The circuit has a simple configuration and is still capable of providing reference voltages with high precision. 
     Summary of Embodiments 
     Each D/A converter  1 ,  22  discussed in embodiments of the present invention is a D/A converter for converting a digital signal consisting of n upper bits HoB&lt;n−1:0&gt; and m lower bits LoB&lt;m−1:0&gt; to an output voltage Vout. The converter includes: a resistor-string circuit  2  containing (2^n)−1 resistors R of an equal resistance connected in series, a reference voltage VH and a reference voltage VL being applied across both terminals of the resistors R; a switching circuit  3  for selecting, according to the n upper bits HoB&lt;n−1:0&gt; of the digital signal, a divided voltage produced by the resistor-string circuit  2  for output as the output voltage Vout; and the variable resistors  4  and  5  connected to the terminals of the (2^n)−1 resistors R connected in series, the variable resistors  4  and  5  exhibiting resistances RH and RL determined according to the m lower bits LoB&lt;m−1:0&gt; of the digital signal. 
     According to the configuration, the output voltage Vout is produced by means of the resistances RH and RL determined in the variable resistors  4  and  5  according to the m bits of the digital signal. The configuration requires fewer wires connecting the resistor-string circuit  2  to the switching circuit  3 , fewer resistors R, and fewer switches in the switching circuit  3  than the conventional D/A converter. Circuit size is thus reduced. A D/A converter is realized which is capable of converting high bit (9 to 11 bits) digital signals. 
     Unlike the conventional D/A converter, the D/A converter  1 ,  22  needs no internal D/A converters. Therefore, current consumption does not increase due to the addition of D/A converters. Precision in D/A conversion is dictated by the relative precision of the variable resistors  4  and  5  connected to the reference voltages VH and VL. Thus, the precision in D/A conversion does not fall by as great an amount as is the case with the conventional D/A converter. A high precision D/A converter is provided. 
     In the D/A converters  1  and  22 , the resistances RH and RL of the variable resistors  4  and  5  may be given a weight by a power of 2. 
     Accordingly, the resistances RH and RL of the variable resistors  4  and  5  are readily obtainable. 
     The D/A converter  22  may further include a switching control circuit  23  for controlling switching of the resistances RH and RL of the variable resistors  4  and  5  according to a least significant bit L 0  of the digital signal containing the m lower bits LoB&lt;m−1:0&gt;. 
     Accordingly, the resistances RH and RL of the variable resistors  4  and  5  change according to the least significant bit L 0 . Vout_sw also changes according to the least significant bit L 0 . The output voltage is produced with greater precision. 
     In the D/A converter  22 , a period “a” in which the least significant bit L 0  is HIGH may be given a weight by 2^p. 
     Accordingly, the output voltage is produced with precision of n+m+p bits, that is, the number of the upper bits HoB&lt;n−1:0&gt;, plus the number of the lower bits LoB&lt;m−1:0&gt;, plus the power count of the weight given to the period “a” in which the least significant bit L 0  is HIGH. 
     The D/A converter  24  may further include a LPF  25  connected to an output of the switching circuit  3 . 
     Accordingly, the LPF  25  removes the noise on the output voltage Vout which occurs when the switching circuit  3  carries out the switching. 
     The reference voltage circuit of the present invention includes one of the D/A converters. The circuit has a simple configuration and is still capable of providing reference voltages with high precision. 
     The invention being thus described, it will be obvious that the same way may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.