General Properties of Electronic Single-Loop Feedback Systems |
181 |
noting that: |
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Vo = −Vi′ Kvo/(sτa + 1) |
(4.28) |
Equation 4.28 is solved for Vi′ and substituted into Equation 4.27, and the closed-loop transfer function is found:
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= A (s) = |
−KvoRF [RF + R1(1+ Kv )] |
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sτa(RF |
+ R1) [RF + R1(1+ Kv )] |
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The low and mid-frequency closed-loop gain is:
Av (0) = −KvoRF
[RF + R1(1+ Kv )]
and the closed-loop amplifier’s BW is its −3-dB frequency whereAv(0) / 2. This is simply:
fb = [RF + R1(1+ Kv )]
[2πτa (RF + R1)] hertz
Thus, its gain*bandwidth product is:
= KvoRF[RF + R1(1+ Kvo )] = GBWPamp [RF + R1(1+ Kvo )][2πτa (RF + R1)]
= GBWPoa α
Kvo RF
2πτa (RF + R1)
(4.29)
(4.30)
Av(fb) =
(4.31)
(4.32)
where α is the feed-forward attenuation of the op amp’s feedback circuit. Thus the inverting op amp circuit with feedback has a GBWP that is lower
than the GBWP of the op amp. As RF is increased, Av(0) increases and
GBWPamp GBWPoa. In all cases, the use of NFB extends the high-frequency half-power frequency at the expense of gain.
4.3.4Decrease in Gain Sensitivity
Gain sensitivity is the fractional change in closed-loop gain resulting from changes in amplifier circuit component values. In the case of the basic SISO feedback system of Figure 4.3(A), the closed-loop gain is:
Vo |
= A = |
αKv |
(4.33) |
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Vs |
1+ βKv |
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© 2004 by CRC Press LLC
182 |
Analysis and Application of Analog Electronic Circuits |
The sensitivity of Av with respect to the gain Kv is defined by:
SAv ∫ Av
Av
Kv Kv
Kv
so now |
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∂A ˆ |
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Av |
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˜ Kv |
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∂Kv ↓ |
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and the partial derivative is: |
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∂A |
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α[(1+ βKv )− βKv ] |
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αK K −1 |
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∂Kv |
(1+ βKv )2 |
= (1+ βKv ) RD |
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Thus,
(4.34)
(4.35)
(4.36)
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Av |
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Av |
Kv |
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(4.37) |
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Kv |
[RD] |
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and the sensitivity is: |
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SAv ∫ |
Av |
Av |
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Kv Av |
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1 |
(4.38) |
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Kv[RD] Kv Kv |
[RD] |
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Kv |
Kv |
Kv |
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In op amps, 1/[RD] is on the order of 10−4 to 10−6, so a circuit with NFB is generally quite insensitive to variations in its dc open-loop gain, Kvo.
Also of interest is how sensitivity varies as a function of frequency in NFB circuits. According to Northrop (1990),
Complex algebra must be used in the calculation of such sensitivities, because the closed-loop gain [frequency response] is complex. Here we express the closed-loop gain in polar form for convenience:
A (jω) = ΩA (jω)Ω ejθ(ω) |
(4.39) |
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The gain sensitivity, given by Equation [4.38], can also be written as
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A |
dA |
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d(ln Av ) |
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v = |
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(4.40) |
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d(ln x) |
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© 2004 by CRC Press LLC