Digitally Tunable MDAC based State Variable

Rahul Prakash, Eugenio Mejia
TI Designs – Precision: Verified Design
Digitally Tunable MDAC-Based State Variable Filter
Reference Design
TI Designs – Precision
Circuit Description
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modifications that help to meet alternate design goals
are also discussed.
This MDAC based state variable filter offers highly
accurate digital tuning of gain, center/cut off
frequency, and quality factor. This circuit provides
three separate filter outputs: low pass, band pass,
and high pass that can be accessed simultaneously.
Design Resources
Design Archive
TINA-TI™
DAC8812
OPA277
Ask The Analog Experts
WEBENCH® Design Center
TI Designs – Precision Library
All Design files
SPICE Simulator
Product Folder
Product Folder
RFB
IOUT
MDAC
VREF
C1
RFB
IN
VREF
MDAC
C2
RFB
RFB
IOUT
A1
A2
Gain Control – U4A
VREF
RFB
VREF
MDAC
IOUT
MDAC
VREF
A3
IOUT
MDAC
A4
LP
Frequency Control – U1A/B
IOUT
HP
RFB
IOUT
VREF
MDAC
BP
Q-Factor Control – U4B
An IMPORTANT NOTICE at the end of this TI reference design addresses authorized use, intellectual property matters and
other important disclaimers and information.
TINA-TI is a trademark of Texas Instruments
WEBENCH is a registered trademark of Texas Instruments
TIDU543-October 2014
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
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1
Design Summary
The design requirements are as follows:

Supply Voltage: 5 V

AC Input: 1 VPP

Output: Low pass, Band pass, High pass

Tunability: Gain, Center/Cut off Frequency, and Quality Factor
The design goals and performance are summarized in Table 1. Figure 1 depicts the measured transfer
function of the design.
Table 1. Comparison of Design Goals, Simulation, and Measured Performance
Cut Off Frequency
Range (3dB)
Gain Range
Quality Factor Range
Goal
Simulated
Measured
10 Hz to 30kHz
2.11 Hz to 27.98 kHz
1.99 Hz to 28.76 kHz
DAC Code Range: 0x0004 to 0xFFFF
0 dB to 6dB
-54.13 dB to 6.02 dB
-53.31 dB to 6.01 dB
DAC Code Range: 0x0040 to 0xFFFF
0 dB to 6dB
0.47 to 2.15
0.46 to 2.14
DAC Code Range: 0xFFFF to 0x4000
Figure 1: Measured Transfer Function
2
Digitally Tunable MDAC based State Variable Filter
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2
Theory of Operation
An implementation of a state variable filter using discrete resistors is shown in Figure 2
R7
R5
IN
R1
R8
C1
R6
R3
C2
R4
LP
A1
A2
A3
HP
A4
R2
BP
Figure 2: State Variable Filter
In Figure 2, amplifier A1 and A2 form summing and inverting stages followed by two op amp integrators,
A3 and A4 which act as single pole low pass filters. Amplifiers A3 and A4 are cascaded to form a secondorder filter. This configuration provides 3 filter outputs: low pass (LP), band pass (BP) and high pass (HP).
The center frequency (in the case of the band pass) or cut off frequency (in the case of the high pass and
low pass) is set by theRC circuits on both integrators. For simplicity, this RC combination is typically made
equal i.e. R3 = R4 and C1 = C2.
In order to analyze this circuit, nodal equations for each node in the circuit must be solved. The resulting
transfer function is shown in Equations 1, 2 & 3.
R5  R8
C1  C2  R1  R3  R4  R6
LP

R5  R8
R8
IN s 2 
s
C1  R2  R3  R6
C1  C2  R3  R4  R7
(1)
R5  R8
s
C1  R1  R3  R6
BP

R5  R8
R8
IN s 2 
s
C1  R2  R3  R6
C1  C2  R3  R4  R7
(2)
R5  R8
 s2
C1  R1  R3  R6
HP

R5  R8
R8
IN s 2 
s 
C1  R2  R3  R6
C1  C2  R3  R4  R7
(3)
C1  C2 , R3  R4 , R7  R8
(4)

TIDU543-October 2014
Digitally Tunable MDAC based State Variable Filter
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Equations 1, 2 & 3 are simplified by making the assumptions in Equation 4. Comparing the resulting
equations with the standard equation for a band pass filter (Equation 5), yields expressions for gain (A0),
center/cut off frequency (ω), and quality factor (Q) (Equations 6, 7 & 8).

A0  
s
Q
BP

IN s 2    s   2
Q
 R2
R1
(6)

1
C1  R3
(7)
Q
R2  R6
R5  R8
(8)
A0 
3
Component Selection
3.1
DAC Selection
(5)
To realize eight discrete resistors a minimum of six MDACs ladders must be used. By using three dual
channel MDACs with only six MDAC ladders, resistors R5 and R7 are realized by using the fixed values of
two MDAC feedback resistors.
The DAC8812 is a great option for this design. It is a dual channel MDAC with strong linearity and large
multiplying bandwidth which makes it ideal for this application.
Generally, MDACs are used in high performance applications that take full advantage of their strong dc
specifications. MDACs have a current output and are used in conjunction with an external I-V operational
amplifier. However, this design leverages the unique software-controlled output impedance property of
MDACs. An important property of MDACs for this application is their reference multiplying bandwidth. The
reference multiplying bandwidth changes with the programed DAC code and the bandwidth is decreased
at lower codes when the reference input is highly attenuated as shown in Figure 3. Plots such as these
can be found in the datasheet of any MDAC.
Figure 3: DAC8812 Reference multiplying bandwidth
4
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
TIDU543-October 2014
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MDAC
RFB
RFB
RDAC
VREF
IOUT
Figure 4: MDAC equivalent circuit
The resistors in Figure 2 are replaced by MDACs. The circuit equivalents of an MDAC are shown in Figure
4. Resistor R5 and R7 are realized using feedback resistors integrated in the multiplying DACs. The values
of these resistors are fixed and cannot be changed by writing to the MDAC. The remaining resistors are
realized by the programmable ladder impedance of the MDAC. Table 2 shows the maximum and minimum
values for the tunable resistors. Note that resistors R3/R4 and resistors R7/R8 are implemented such that
their values are tuned simultaneously (refer to Equation 4). To realize all the resistors in this filter a
minimum of six MDACs are required. Therefore three DAC8812, a 16-Bit, dual channel serial input
multiplying DAC, are used in this design. The tunable resistor range for the DAC8812 is shown in Table 2.
Keep in mind that the resistor ladders in the DAC8812 can vary by ±20 percent.
Table 2. Measured resistor values tunable range with MDAC DAC8812
3.2
Resistors
MDAC
Minimum Value
(0xFFFF Code)
Maximum Value
(0x0001 Code)
R1
U4 A
5.02 kΩ
33.00 kΩ
R2
U4 B
5.02 kΩ
25.57 kΩ
R3
U2 A
5.00 kΩ
26.91 kΩ
R4
U2 B
5.00 kΩ
22.86 kΩ
R5
U4 B RFB
5.18 kΩ
5.18 kΩ
R6
U1 B RFB
4.96 kΩ
4.96 kΩ
R7
U1 B
4.81 kΩ
22.55 kΩ
R8
U1 A
4.81 kΩ
22.48 kΩ
Amplifier Selection
This design requires four operational amplifiers. For all of these amplifiers, low input bias current is desired
so that the critical parameters of the filter have exclusive tunability via the MDAC based resistors and their
matching. The center/cut off frequency range for this design ranges from dc to 30 kHz, therefore high
speed/high unity gain bandwidth amplifiers are not required.
The OPA277 was selected for this design because it features a sufficiently low input bias current of 1 nA.
3.3
Passive Component Selection
The high side cut off frequency is determined by the combination of capacitors C1 & C2 and the
multiplying bandwidth of the MDAC with the lowest input code. The size of C1 is determined by Equation
7. The desired frequency bandwidth of the filters is 30 kHz. The DAC8812 reference input impedance is 5
kΩ paired with a 1 nF standard value for C1 will result in 32 kHz. In order to comply with Equation 4, both
C1 and C2 must be the same value for the equations to be valid.
TIDU543-October 2014
Digitally Tunable MDAC based State Variable Filter
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4
Simulation
The proposed circuit was simulated in TINA-TI™ using SPICE models of both the DAC8822 and OPA277.
The choice of using the DAC8822 as a simulation model was based on the core similarity with the
DAC8812 and the current lack of availability of a DAC8812 model. The DAC8822 and DAC8812 are very
similar devices with the minor functional difference that the DAC8822 has more feedback resistor options.
The simulation suite consists of an ac simulation test bench that included center/cut off frequency, gain
and quality factor for high pass, low pass and band pass outputs.
Figure 5 shows the realization of the resistors in this circuit using MDAC DAC8822.
Figure 5: State Variable Filter using DAC8822
4.1
Center/Cut Off Frequency Simulation
The center/cut off frequency simulations were carried out for all the outputs (low pass, band pass and high
pass). The MDAC ladders U2A/B can be tuned, thereby changing the frequency from 2.11 Hz (code
0x0004) to 27.98 kHz (code 0xFFFF). See Equation (7). Note that 0x0000 is not used in this design
because at this code all the ladder switches are open and the ladder resistance is Hi-Z.
Figure 6 shows center frequency simulations for the high pass output. For frequency simulation data for
band pass and low pass filters refer to Appendix B.
Figure 6: Cut off frequency simulation – High pass output
6
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
TIDU543-October 2014
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4.2
Gain Simulation
The gain simulations were carried out for all the outputs (low pass, band pass and high pass). The MDAC
ladder U4A can be tuned, thereby changing the gain from -54.13 dB (code 0x0040) to 6.02 dB (code
0xFFFF). See Equation 6.
Figure 7 shows gain simulations for the low pass output. For gain simulation data for band pass and high
pass filters refer to Appendix B.
Figure 7: Gain simulation – Low pass output
4.3
Quality Factor Simulation
The quality factor simulations were carried out for all the outputs (low pass, band pass and high pass). The
MDAC ladder U4B can be tuned, thereby changing the quality factor from 0.47 (code 0xFFFF) to 2.15
(code 0x4000). See Equation (8).
Figure 8 shows gain simulations for the band pass output.
Figure 8: Quality Factor simulation – Band pass output
TIDU543-October 2014
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
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5
PCB Design
The PCB schematic and bill of materials can be found in the Appendix A.
5.1
PCB Layout
General PCB layout best-practices should be followed for this design. Analog and digital lines must not be
traced out parallel to each other in order to reduce the coupling of digital signals onto analog signal paths.
Figure 9: PCB Layout
8
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
TIDU543-October 2014
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6
Measurement
The circuit was tested using a bode plot analyzer that provided the input stimulus and measured the output
response.
6.1
Center/Cut-off Frequency Measurement
The cut-off frequency setting is directly proportional to the DAC code of U2A/B. Each LSB change will
adjust the frequency by approximately 0.5 Hz. For frequency measurement data for band pass and low
pass filters refer to Appendix B.
Figure 10: Cut-off frequency measurement – High pass output
Table 3. High Pass Frequency Results @ 2 kHz
Code
0x0004
0x8000
0xFFFF
TIDU543-October 2014
Simulated Freq.
2.1100 Hz
14690 Hz
27980 Hz
Measured Freq.
1.9900 Hz
14962 Hz
28757 Hz
Error
-5.69 %
-1.85 %
-2.78 %
Digitally Tunable MDAC based State Variable Filter
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6.2
Gain Measurement
The gain setting is directly proportional to the DAC code of U4A. Each LSB change will adjust the gain by
approximately 1 mdB. Measurements at very low codes are susceptible to noise. For gain measurement
data for band pass and high pass filters refer to Appendix B.
Figure 11: Gain measurement – Low pass output
Table 4. Low Pass Gain Results
Code
0x0040
0x8000
0xFFFF
10
Simulated Gain
-54.1276 dB
0.0012 dB
6.0200 dB
Measured Gain
-53.3100 dB
0.0007 dB
6.0140 dB
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
Error
-0.8176 dB
0.0004 dB
0.0060 dB
TIDU543-October 2014
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6.3
Quality Factor Measurement
The quality factor measurement is inversely proportional to the DAC code of U4B. Each LSB change will
adjust the quality factor by approximately 52µ. The quality factor can be controlled independently of gain
by using another MDAC to control resistors R5 & R6. For information about how to calculate quality factor,
please refer to Appendix B.
Figure 12: Quality factor measurement – Band pass output
Table 5. Band Pass Quality Factor Results
Code
0x4000
0x8000
0xFFFF
TIDU543-October 2014
Simulated Q-Factor
2.1524
0.9461
0.4691
Measured Q-Factor Error
2.1424
-0.46 %
1.0077
6.52 %
0.4660
-0.65 %
Digitally Tunable MDAC based State Variable Filter
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7
Modifications
Depending on the design requirement other multiplying DACs can be used in this design. DAC8812 was
selected for its serial interface, high resolution, and superior multiplying bandwidth. Table 6 shows other
MDAC options for application that may not require high resolution or that may require a parallel interface.
Table 7 shows alternative amplifiers that can be used in this design for larger bandwidth or minimal input
bias current.
Table 6. Alternative MDACs
MDAC
Resolution
Channel Count
Interface
Reference multiplying bandwidth
DAC8812
16 bits
2
Serial
10 MHz
DAC8822
16 bits
2
Parallel
10 MHz
DAC8802
14 bits
2
Serial
10 MHz
DAC8805
14 bits
2
Parallel
10 MHz
DAC7822
12 bits
2
Parallel
10 MHz
Table 7. Alternative operational amplifiers
8
Amplifier
Supply
Bandwidth
Input bias current (Typ.)
OPA277
±18 V
1 MHz
±500 nA
OPA211
±18 V
80 MHz
±50 nA
OPA188
±18 V
2 MHz
±160 pA
OPA170
±18 V
1.2 MHz
±8 pA
About the Author
Rahul Prakash is a design and systems engineer in the precision digital to analog converters group at
Texas Instruments. Rahul received his BTech in Electrical and Electronics Engineering from the Netaji
Subhas Institute of Technology, India, and MS in Electrical Engineering from University of Texas at Dallas.
Eugenio Mejia is an applications engineer in the precision digital to analog converters group at Texas
Instruments. Eugenio received his Bachelors of Science in Electrical Engineering from Texas A&M
University.
12
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
TIDU543-October 2014
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9
Acknowledgements & References
1.
Engineer It, What is a multiplying DAC (MDAC)? (Video)
2.
Build a three phase sine wave generator with the UAF421 (SBFA013)
3.
Design a 60 Hz notch filter with the UAF42 (SBFA012)
4.
TLC7528, Digitally-controlled state-variable filter application information (SLAS062E)
5.
TIPD137, ±10V 4-Quadrant Multiplying DAC (TIDU031)
TIDU543-October 2014
Digitally Tunable MDAC based State Variable Filter
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Appendix A.
A.1 Electrical Schematic
Figure A-1: Electrical Schematic
A.2 Bill of Materials
Figure A-2: Bill of Materials
14
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
TIDU543-October 2014
www.ti.com
Appendix B.
B.1 Center/Cut Off Frequency Simulation & Measurements
Figure B-1: Cut-off frequency simulation – Low pass output
Figure B-2: Cut-off frequency measurement – Low pass output
Table 8. Low Pass Frequency Results
Code
0x0004
0x8000
0xFFFF
TIDU543-October 2014
Simulated Freq.
2.2138 Hz
17.1738 kHz
36.81 kHz
Measured Freq.
1.9387 Hz
16.8585 kHz
35.6480 kHz
Error
-12.45 %
-1.84 %
-3.16 %
Digitally Tunable MDAC based State Variable Filter
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Figure B-3: Cut-off frequency simulation – Band pass output
Figure B-4: Cut-off frequency measurement – Band pass output
Table 9. Band Pass Frequency Results
Code
0x0004
0x8000
0xFFFF
16
Simulated Freq.
2.2514 Hz
17.7834 kHz
35.4852 kHz
Measured Freq.
1.9387 Hz
16.7400 kHz
35.7890 kHz
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
Error
-13.89 %
-5.87 %
0.86 %
TIDU543-October 2014
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B.2 Gain Simulation & Measurements
Figure B-5: Gain simulation – High pass output
Figure B-6: Gain measurement – High pass output
Table 10. High Pass Gain Results @ 50 kHz
Code
0x0004
0x8000
0xFFFF
TIDU543-October 2014
Simulated Gain
-52.8254 dB
1.2612 dB
7.3731 dB
Measured Gain
-45.7880 dB
1.1631
7.3293
Error
-7.04 dB
0.10 dB
0.04 dB
Digitally Tunable MDAC based State Variable Filter
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Figure B-7: Gain simulation– Band pass output
Figure B-8: Gain measurement – Band pass output
Table 11. Band Pass Gain Results
Code
0x0004
0x8000
0xFFFF
18
Simulated Gain
3.7835 dB
-2.2214 dB
-56.3345 dB
Measured Gain
3.5845 dB
-2.4316 dB
-53.9780 dB
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
Error
0.20 dB
0.21 dB
-2.36 dB
TIDU543-October 2014
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B.3 Quality Factor Simulation & Measurements
The quality factor is calculated using Equation (9), the center frequency (fc) and the -3 dB bandwidth (Δf3dB) of the bandpass filter.
Q
TIDU543-October 2014
fc
f 3dB
Digitally Tunable MDAC based State Variable Filter
Copyright © 2014, Texas Instruments Incorporated
(9)
19
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