Low-Sensitivity Sallen-Key Filter Design with the HP-67 Programmable Calculator

December 12, 2008

This program, a part of my series of analog electronic design programs for the HP-67 programmable calculator, addresses the problem of designing second order single op-amp low- and high-pass filters using the Sallen-Key topology.

The Sallen-Key filter topology has the advantage of using a minimum of components. The simplest Sallen-Key filters use only two resistors and two capacitors. Additional resistors may be added for input attenuation (low-pass only) and gain adjustment. The following schematics illustrate these generalized Sallen-Key circuits:

The equations governing the low-pass filter are as follows,

where f is the filter’s cut-off frequency, Q is its “quality”, and H is its gain at the cut-off frequency. The corresponding equations for the high-pass filter are:

The mathematically inclined will notice that in each case, there are three equations in either nine or ten variables. Thus there is no single “right” solution. At least six or seven of the variables have to be decided arbitrarily (f, Q, H, and three or four component values), at which point the remaining variables (component values) can be solved for. An article by Texas Instruments suggests a number of simplifications to help one choose component values, but this just adds the complication of which simplification to choose.

I recently came across a pair of application notes by National Semiconductor which gives a procedure for designing Sallen-Key filters to minimize the effect of component value tolerances on the performance of the filter. A side-effect of this procedure is to reduce the number of inputs to five: f, Q, H, R_F, and R, where the latter is simply an indication of the magnitude of resistor values desired for R₁, R₂, and R₃. The procedure then dictates how all the other values are chosen, even adjusting for available “real-world” values part way through the solution.

The program presented here implements this procedure, with some minor changes:

1. Instead of asking for a desired resistor magnitude, R, the program asks for a capacitor magnitude, C, since in my experience, the capacitors drive the design.

2. The formula given for internal gain variable K in the procedure (please refer to the application note) seems to have been derived empirically, and has a jump at Q = 1.1. To make the formula simpler to implement, I modified it slightly to:
   

   The graphs of the original (pink) and revised (blue) formulae show the difference. Testing has shown that the resulting solution is generally at most one real-world capacitor value increment different (with corresponding changes in resistor values of course).

   

   Also, if H > K, then K is set equal to H, since otherwise it will not be possible to achieve the desired gain.

For the low-pass circuit, the desired gain, H, is achieved by a combination of input attenuation, α (controlled by R₁ and R₃), and the internal gain, K, of the filter (controlled by R_F and R_G). The division of this gain between the two stages depends on H and Q and is chosen to minimize the sensitivity of the circuit to component tolerances. For example, for H = 1 and Q = 2, the attenuator gain is α = 0.629 and the internal gain is K = 1.59, for a net gain of H = αK = 1.

The high-pass circuit has no attentuation stage, so H must be at least 1, and higher for Q > 0.917. If too low a value is entered for H, it is increased as necessary to make the circuit solvable.

For either circuit, to achieve a result with minimum component sensitivity without regard to gain, set H = 0. The program will automatically choose the optimal value for K (and thus H). The resulting gain will be output when determining the performance.

Using the Program

First type in the program and save it, or read it from a previously recorded magnetic card. The card should be labelled as follows:

LOW-SENSITIVITY SALLEN-KEY FILTER DESIGN
f	Q	H	C	RF
LP→RGC1C2R1R2R3		HP→RGC1C2R1R2		→f,Q,H

Filter Design from Specifications

Example: Design a 500Hz low-pass unity-gain filter with a Q of 2, using capacitors in the 10nF range, and a 47kΩ resistor for R_F:

Description	Keystrokes	Display
Use engineering notation	h  ENG   DSP 2	0.00  00
Enter f	500    f  a	500.  00
Enter Q	2    f  b	2.00  00
Enter H	1    f  c	1.00  00
Enter C	10 EEx CHS 9    f  d	10.0 -09
Enter RF	47 EEx 3    f  e	47.0  03
Compute RG	A	79.5  03
Enter real-world RG and compute C1	82 EEx 3   R/S	31.6 -09
Enter real-world C1 and compute C2	33 EEx CHS 9   R/S	3.16 -09
Enter real-world C2 and compute R1	3.3 EEx CHS 9   R/S	15.4  03
Enter real-world R1 and compute R2	15 EEx 3   R/S	95.9  03
Enter real-world R2 and compute R3	100 EEx 3   R/S	26.1  03
Enter real-world R3	27 EEx 3   R/S	27.0  03

Notes

If a resistor is to be omitted (open circuit), this program displays a value of zero for the resistance. This is different than some of my other programs, which display a “large” value representing infinity.

When the value for R_G is displayed as zero, meaning it can be omitted, the value of R_F will not matter any more, and R_F can be replaced by a direct connection.

Filter Performance from Chosen Components

During the calculation of the solution above, we’ve entered real-world values in response to each computed value. The real-world values of C₁ and C₂ are used when computing the values of R₁, R₂, and R₃. However, the real-world values of each of those resistors does not affect the computed value of the remaining ones. Thus, the final filter may not perform exactly as specified. To find out how it does perform, follow these steps:

Description	Keystrokes	Display
Compute resulting f	E	491.  00
Compute resulting Q	R/S	1.86  00
Compute resulting H	R/S	1.01  00

A High-Pass Example

Using the parameters already entered for the low-pass filter above, determine the components for a high-pass filter:

Description	Keystrokes	Display
Compute RG for high-pass filter	C	79.5  03
Enter real-world RG and compute C1	82 EEx 3   R/S	31.4 -09
Enter real-world C1 and compute C2	33 EEx CHS 9   R/S	3.18 -09
Enter real-world C2 and compute R1	3.3 EEx CHS 9   R/S	9.59  03
Enter real-world R1 and compute R2	10 EEx 3   R/S	97.0  03
Enter real-world R2	100 EEx 3   R/S	0.00  00

Now determine the predicted actual performance:

Description	Keystrokes	Display
Compute resulting f	E	482.  00
Compute resulting Q	R/S	1.96  00
Compute resulting H	R/S	1.57  00

Notice that H is higher than the specified unity gain. This is because the filter is not possible to construct with unity gain when Q = 2. The smallest possible gain is H = 1.59 (which due to real-world components, has become Q = 1.89 and H = 1.57). To achieve H = 1, you will need either a pre-attenuator with low output impedance, or a post-attenuator with high input impedance.

Program Listing

Line	Instruction	Comments
001♦	LBL a	Enter and store f
002	STO A
003	RTN
004♦	LBL b	Enter and store Q
005	STO B
006	RTN
007♦	LBL c	Enter and store H (gain)
008	STO C
009	RTN
010♦	LBL d	Enter and store C (capacitor scale)
011	STO 0
012	RTN
013♦	LBL e	Enter and store RF
014	STO 4
015	RTN
016♦	LBL A	Low-pass filter: RG,C1,C2,R1,R2,R3
017	CF 0
018	GTO 0
019♦	LBL C	High-pass filter: RG,C1,C2,R1,R2
020	SF 0
021♦	LBL 0	Forward solution
022	RCL B
023	2
024	.
025	2
026	×
027	.
028	9
029	−
030	RCL B
031	.
032	2
033	+
034	÷	(2.2Q-0.9)/(Q+0.2)
035	1
036	x≤y?
037	x↔y
038	STO E	K = max(1,(2.2Q-0.9)/(Q+0.2))
039	RCL C
040	x>y?
041	STO E	K = max(H,1,(2.2Q-0.9)/(Q+0.2))
042	RCL E
043	÷	H/K
044	x=0?
045	1	Use α = 1 if H/K = 0 (because H was 0)
046	F? 0
047	1	Always use α = 1 for a high-pass filter
048	STO D	α = H/K (always 1 for a high-pass filter or H = 0)
049	RCL 4
050	RCL E
051	1
052	−
053	x≠0?
054	÷	RG = RF/(K-1) if K ≠ 1, or zero if K = 1
055	R/S	Display RG and let user change it
056	STO 5
057	.	Initialize n to √0.1
058	1
059	√x
060	GSB 7	n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2))
061	RCL 8	√0.1 was stored here by subroutine 7
062	x≤y?
063	x↔y	max(n, n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2)))
064	F? 0	High-pass filter?
065	GSB 6	2nQ/(1+√(1+4Q2(K-1-n2)))
066	STO 8
067	RCL 0
068	RCL 8
069	÷	C1 = C/n
070	R/S	Display C1 and let user change it
071	STO 6
072	RCL 8
073	RCL 0
074	×	C2 = nC
075	R/S	Display C2 and let user change it
076	STO 7
077	RCL 6
078	×
079	√x	√(C1C2)
080	GSB 4	Multiply by 2πf and take reciprocal
081	STO 9	N = 1/(2πf√(C1C2))
082	RCL 7
083	RCL 6
084	÷
085	√x	n = √(C2/C1)
086	GSB 3	2nQ/(1+√(1+4Q2(K-1-n2))) or n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2))
087	STO 8
088	RCL 9
089	×
090	STO 3	(R1||R3) = nN
091	RCL D
092	÷	R1 = (R1||R3)/α
093	R/S	Display R1 and let user change it
094	STO 1
095	RCL 9
096	RCL 8
097	÷	R2 = N/n
098	R/S	Display R2 and let user change it
099	STO 2
100	RCL 3
101	1
102	RCL D
103	−	( 1-α (R1||R3) )
104	x≠0?
105	÷	R3 = (R1||R3)/(1-α) if α ≠ 1, or zero otherwise
106	R/S	Display R3 and let user change it
107	STO 3
108	RTN
109♦	LBL 4	Multiply by 2πf and take reciprocal
110	RCL A	f
111	×
112♦	LBL 1	Multiply by 2π and take reciprocal
113	2
114	×
115	π
116	×
117	1/x
118	RTN
119♦	LBL 3	Compute either 2xQ/(1+√(1+4Q2(K-1-x2))) or x(1+√(1+4Q2(1+x2)(K-1)))/(2Q(1+x2))
120	F? 0	High-pass filter?
121	GTO 7
122♦	LBL 6	Subroutine to compute 2xQ/(1+√(1+4Q2(K-1-x2)))
123	STO 8	Save x for later use
124	RCL B
125	2
126	×	( 2Q x )
127	×	( 2xQ )
128	LSTx	( 2Q 2xQ )
129	x2	( 4Q2 2xQ )
130	RCL E
131	1
132	−
133	RCL 8
134	x2
135	−	( K-1-x2 4Q2 2xQ )
136	×	( 4Q2(K-1-x2) 2xQ )
137	GSB 9	( 1+√(1+4Q2(K-1-x2)) 2xQ )
138	÷
139	RTN
140♦	LBL 7	Subroutine to compute x(1+√(1+4Q2(1+x2)(K-1)))/(2Q(1+x2))
141	STO 8	Save x in register 8 for later use both by this subroutine and the caller
142	GSB 8	( 2Q 1+x2 )
143	x2
144	×	( 4Q2(1+x2) )
145	RCL E
146	1
147	−	( K-1 4Q2(1+x2) )
148	×	( 4Q2(1+x2)(K-1) )
149	GSB 9	( 1+√(1+4Q2(1+x2)(K-1)) )
150	RCL 8
151	×	( x(1+√(1+4Q2(1+x2)(K-1))) )
152	RCL 8
153	GSB 8	( 2Q 1+x2 x(1+√(1+4Q2(1+x2)(K-1))) )
154	×	( 2Q(1+x2) x(1+√(1+4Q2(1+x2)(K-1))) )
155	÷
156	RTN
157♦	LBL 8	Subroutine to populate stack with 2Q 1+x2
158	x2
159	1
160	+
161	RCL B
162	2
163	×
164	RTN
165♦	LBL 9	Subroutine to compute 1+√(1+x)
166	1
167	+
168	√x
169	1
170	+
171	RTN
172♦	LBL E	Compute actual f, Q, and Gain
173	GSB 2	R1 or (R1||R3)
174	RCL 6
175	×
176	STO 8	Save R1C1 for use in calculating Q
177	RCL 2
178	RCL 7
179	×
180	STO 9	Save R2C2 for use in calculating Q
181	×
182	√x
183	GSB 1	Multiply by 2π and take reciprocal
184	ST I	Save actual f for use in calculating Q
185	R/S	Display actual f
186	RCL 9	( R2C2 )
187	RCL 8	( R1C1 R2C2 )
188	F? 0	High-pass filter?
189	x↔y	( R2C2 R1C1 )
190	1
191	RCL E
192	−
193	×
194	+
195	GSB 2	R1 or (R1||R3)
196	RCL 7
197	×
198	+
199	RC I	Recall actual frequency
200	×
201	GSB 1	Multiply by 2π and take reciprocal
202	R/S	Display actual Q
203	RCL 4
204	RCL 5
205	x≠0?
206	÷
207	1
208	+
209	GSB 2	R1 or (R1||R3)
210	×
211	RCL 1
212	÷
213	RTN	Return actual Gain
214♦	LBL 2	Return either R1 or (R1||R3)
215	RCL 1
216	F? 0	High-pass filter?
217	RTN	Return just R1
218	1/x
219	RCL 3
220	x≠0?	R3 exists? (0 means not)
221	1/x
222	+
223	1/x
224	RTN

Registers and Flags

Register	Use
0	C – capacitor scale
1,2,3	R1, R2, R3 – filter resistors
4,5	RF, RG – feedback resistors
6,7	C1, C2 – capacitors
8	n – variable used during computation
9	N – variable used during computation
A	f – cutoff frequency
B	Q – filter quality
C	H – overall gain
D	α – input attenuator gain
E	K – internal gain
I	Temporary register

Flag	Meaning
0	High-pass filter

Revision History

2008-Dec-12 — Initial release.

References

1. Analysis of the Sallen-Key Architecture (Rev.B), Texas Instruments Application Report SLOA024B, James Karki, 2002

2. Low-Sensitivity, Lowpass Filter Design, National Semiconductor (now Texas Instruments) Application Note OA-27, Kumen Blake, 1996

3. Low-Sensitivity, Highpass Filter Design, National Semiconductor (now Texas Instruments) Application Note OA-29, Kumen Blake, 1996

                    

Related Articles

If you've found this article useful, you may also be interested in:

• Op-Amp Gain and Offset Design with the HP-41C Programmable Calculator
• Op-Amp Oscillator Design with the HP-41C Programmable Calculator
• Low-Sensitivity Sallen-Key Filter Design with the HP-41C Programmable Calculator
• Op-Amp Gain and Offset Design with the HP-67 Programmable Calculator
• Op-Amp Oscillator Design with the HP-67 Programmable Calculator
• Resistor Network Solver for the HP-67 Programmable Calculator
• A Matrix Multi-Tool for the HP 35s Programmable Calculator
• Curve Fitting for the HP 35s Programmable Calculator