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Copyright © 2016 by Wayne Stegall
Created January 16, 2016. See Document History at end for
details.
Figure
1:
Dualslope
integrator
response
superimposed
on
related
singleslope
integrator
response. 
Legend: green: first order integration. black: dualslope integration. 
Figure
2:
Dualslope
integrator
implemented
with
operational
amplifier. 
(1) 
v_{a}
v_{in} 
= –  1
sR_{1}C_{1} 
, i_{C1} = 
v_{in}
R_{1} 
(2) 
i_{C2} = i_{C1} –  v_{a}
R_{2} 
= 
v_{in}
R_{1} 
+ 
v_{in}
sR_{1}R_{2}C_{1} 
(3) 
i_{C2} =  v_{in}(sR_{2}C_{1}
+ 1)
sR_{1}R_{2}C_{1} 
(4) 
v_{out} = v_{a} –  i_{C2}
sC_{2} 
(5) 
v_{out} = –  v_{in}
sR_{1}C_{1} 
–  v_{in}(sR_{2}C_{1}+1)
s^{2}R_{1}R_{2}C_{1}C_{2} 
(6) 
v_{out} = –  v_{in}(s(R_{2}C_{2}
+R_{2}C_{1})+1)
s^{2}R_{1}R_{2}C_{1}C_{2} 
(7) 
H(s) = 
v_{out}
v_{in} 
= –  sR_{2}(C_{1}+C_{2})+1
s^{2}R_{1}R_{2}C_{1}C_{2} 
(8) 
f_{zero} = 
1
2πR_{2}(C_{1}+C_{2}) 
(9) 
H(s) = –  C_{1}+C_{2}
sR_{1}C_{1}C_{2} 
= 
1
sR_{1}(C_{1} <series> C_{2}) 

Figure
3:
Dualslope
integrator placed into context with additional openloop gain and the
application of global feedback. 
(10) 
H(s) = 
A
1+Aβ 
(11) 
H(s) = 

(12) 
H(s) =  sA_{2}R_{2}(C_{1}+C_{2})
+
A_{2}
s^{2}R_{1}R_{2}C_{1}C_{2} + sβA_{2}R_{2}(C_{1}+C_{2}) + βA_{2} 
(13) 
H(s) =  sR_{2}(C_{1}+C_{2})
+
1

In the form 
s^{2} ω_{0}^{2} 
+ 
s
ω_{0}Q 
+ 1 
(14) 
ω_{0} = 

(15) 
Q = 
R_{2}(C_{1}+C_{2}) 
= 
1
C_{1}+C_{2} 

(16) 
Q =  ω_{zero}
ω_{pole} 
Figure
4:
Circuit
used
for
SPICE
verification

Figure 5: SPICE deck configured for f_{zero} = f_{feedbackcrossover}. 
* dual slope
integrator example * Spice Opus 2.31 v1 vin 0 dc 0 ac 1 sin 0 0.1V 1kHz r1 vin vn 1.62k c1 vn vc1c2 6.19n r2 vc1c2 0 130 c2 vint vc1c2 6.19n eopa vint 0 0 vn 100k ea2 vout 0 vint 0 100 rf vout vn 51.1k .end .control set units=degrees ac dec 20 1k 1meg plot db(vout) .endc 
Figure 6: ≈ 3.5dB peak configured for f_{zero} = f_{feedbackcrossover}. 
Figure 7: SPICE deck configured for f_{zero} = ½f_{feedbackcrossover}. 
* dual slope
integrator example * Spice Opus 2.31 v1 vin 0 dc 0 ac 1 sin 0 0.1V 1kHz r1 vin vn 1.62k c1 vn vc1c2 6.19n r2 vc1c2 0 255 c2 vint vc1c2 6.19n eopa vint 0 0 vn 100k ea2 vout 0 vint 0 100 rf vout vn 51.1k .end .control set units=degrees ac dec 20 1k 1meg plot db(vout) .endc 
Figure 8: ≈ 2dB peak configured for f_{zero} = ½f_{feedbackcrossover}. 

Figure 9: SPICE deck configured back to f_{zero} = f_{feedbackcrossover} with lead compensation added at an experimentally chosen value. 
* dual slope
integrator example * Spice Opus 2.31 v1 vin 0 dc 0 ac 1 sin 0 0.1V 1kHz r1 vin vn 1.62k c1 vn vc1c2 6.19n r2 vc1c2 0 130 c2 vint vc1c2 6.19n eopa vint 0 0 vn 100k ea2 vout 0 vint 0 100 rf vout vn 51.1k cf vout vn 43p .end .control set units=degrees ac dec 20 1k 1meg plot db(vout) .endc 
Figure 10: No peak when configured back to f_{zero} = f_{feedbackcrossover} with lead compensation added at an experimentally chosen value. 

Document History
January 16, 2016 Created.