AND8054 Datasheet, PDF (16/28 Pages) ON Semiconductor – Designing RC Oscillator Circuits with Low Voltage Operational Amplifiers and Comparators for Precision Sensor Applications

AND8054/D
Differentiator
C
V_IN
R
–
+
V_OUT
Figure 14. Ideal Differentiator
Listed below are the equations for the ideal differentiator
circuit shown in Figure 14.
VOUT(t)
+
*
RC
dVIN(t)
dt
VOUT(s)
VIN(s)
+
*
sRC
However, the ideal differentiator does not consider the
effect of the amplifier’s voltage offset and current bias
errors, as shown below [3].
VOUT(t)
+
*
RC
VIN(t)
dt
"
VIO
"
IBR
If the output offset and bias errors are referenced to the
input, as shown below,
dVIN(t)
dt
error
+"
VIO
RC
"
IB
C
the following observations can be made:
1. Use small R, large C.
2. V_IO ∝ 1 / RC and I_B ∝ 1 / C .
Gain (dB)
A practical differentiator circuit is shown in Figure 15. For
simplicity, this circuit will neglect the effect of resistor R₅.
There will be an error due to the operational amplifier’s
finite open loop gain as shown below:
ƪ ƫ VOUT(s)
VIN(s)
+
s2
)
s
s ^Ao
1*Ao
1
R3C4
1
R4C4
)
1
R3C3
)
1
R3R4C3C4
If A_o >> 1, the transfer equation can be simplified to:
ǒ Ǔ _VOUT(s)
ǒ Ǔǒ Ǔ ^VIN(s)
^
*s
1
R3C4
s
)
1
R3C3
s
)
1
R4C4
^
* sR4C3
ǒ_{sR3C3 ) 1}Ǔ ǒ_{sR4C4 ) 1}Ǔ
C4
R5
R4
R3
C3
V_IN
–
+
V_OUT
Figure 15. Practical Differentiator (Neglect R5)
The error due to the operational amplifier’s finite open
loop gain and bandwidth is shown graphically in Figure 16.
The oscillator’s amplifier error and bandwidth error terms
are reduced if a higher gain and increased operational
amplifier is selected. The oscillation error can be minimized
by selecting an oscillation frequency that is as low as
practical (i.e. f_oscillation ≅ 10 KHz).
The Slew Rate (SR) error of a differentiator is identical to
the equation listed for an integrator.
dVOUT(t)
dt
|
max
+
2pfpEo
+
SR
A_O
Open Loop Gain
Gain Error
ǒ Ǔ 20x log C3
C4
0
ǒ Ǔ 20x log R4
R3 ) R5
Bandwidth
Error
T_o
FP1
+
1
2pR4C4
Fz1
+
1
2pR4C3
Fp2
+
1
2pR3C3
Frequency (Hz)
Closed Loop Gain
Figure 16. Graphical Error Analysis of Ideal Differentiator
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