EE101: Op Amp circuits (Part 6)
M. B. Patil
[email protected] www.ee.iitb.ac.in/~sequel Department of Electrical Engineering Indian Institute of Technology Bombay
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
Amplifier
x′i
β xo
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
Amplifier
x′i
β xo
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback. xo = A xi0 = A (xi + xf ) = A (xi + βxo )
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
Amplifier
x′i
β xo
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback. xo = A xi0 = A (xi + xf ) = A (xi + βxo ) xo A → Af ≡ = . xi 1 − Aβ
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback. xo = A xi0 = A (xi + xf ) = A (xi + βxo ) xo A → Af ≡ = . xi 1 − Aβ Since A and β will generally vary with ω, we re-write Af as, A (jω) → Af (jω) = . 1 − A (jω) β (jω)
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback. xo = A xi0 = A (xi + xf ) = A (xi + βxo ) xo A → Af ≡ = . xi 1 − Aβ Since A and β will generally vary with ω, we re-write Af as, A (jω) → Af (jω) = . 1 − A (jω) β (jω) As A (jω) β (jω) → 1, Af (jω) → ∞, and we get a finite xo even if xi = 0.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
Consider an amplifier with feedback. xo = A xi0 = A (xi + xf ) = A (xi + βxo ) xo A → Af ≡ = . xi 1 − Aβ Since A and β will generally vary with ω, we re-write Af as, A (jω) → Af (jω) = . 1 − A (jω) β (jω) As A (jω) β (jω) → 1, Af (jω) → ∞, and we get a finite xo even if xi = 0. In other words, we can remove xi and still get a non-zero xo . This is the basic principle behind sinusoidal oscillators.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
* The condition, A (jω) β (jω) = 1, for a circuit to oscillate spontaneously (i.e., without any input), is known as the Barkhausen criterion.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
* The condition, A (jω) β (jω) = 1, for a circuit to oscillate spontaneously (i.e., without any input), is known as the Barkhausen criterion. * For the circuit to oscillate at ω = ω0 , the β network is designed such that the Barkhausen criterion is satisfied only for ω0 , i.e., all components except ω0 get attenuated to zero.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
xi
xf
x′i
β xo
Amplifier
A x′i
xo
Frequency−sensitive network
* The condition, A (jω) β (jω) = 1, for a circuit to oscillate spontaneously (i.e., without any input), is known as the Barkhausen criterion. * For the circuit to oscillate at ω = ω0 , the β network is designed such that the Barkhausen criterion is satisfied only for ω0 , i.e., all components except ω0 get attenuated to zero. * The output xo will therefore have a frequency ω0 (ω0 /2π in Hz), but what about the amplitude?
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* A gain limiting mechanism is required to limit the amplitude of the oscillations.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* A gain limiting mechanism is required to limit the amplitude of the oscillations. * Amplifier clipping can provide a gain limiter mechanism. For example, in an Op Amp, the output voltage is limited to ±Vsat , and this serves to limit the gain as the magnitude of the output voltage increases.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* A gain limiting mechanism is required to limit the amplitude of the oscillations. * Amplifier clipping can provide a gain limiter mechanism. For example, in an Op Amp, the output voltage is limited to ±Vsat , and this serves to limit the gain as the magnitude of the output voltage increases. * For a more controlled output with low distortion, diode-resistor networks are used for gain limiting, as we shall see.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* Up to about 100 kHz, an Op Amp based amplifier and a β network of resistors and capacitors can be used.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* Up to about 100 kHz, an Op Amp based amplifier and a β network of resistors and capacitors can be used. * At higher frequencies, an Op Amp based amplifier is not suitable because of frequency response and slew rate limitations of Op Amps.
M. B. Patil, IIT Bombay
Sinusoidal oscillators
Amplifier
xo gain limiter
β xo
Frequency−sensitive network
* Up to about 100 kHz, an Op Amp based amplifier and a β network of resistors and capacitors can be used. * At higher frequencies, an Op Amp based amplifier is not suitable because of frequency response and slew rate limitations of Op Amps. * For high frequencies, transistor amplifiers are used, and LC tuned circuits or piezoelectric crystals are used in the β network.
M. B. Patil, IIT Bombay
Wien bridge oscillator
Amplifier
xo
R
amplifier
β xo
Frequency−sensitive network
C
A
Z1
C R
Z2
β network
M. B. Patil, IIT Bombay
Wien bridge oscillator
Amplifier
xo
R
amplifier
β xo
Frequency−sensitive network
C
A
Z1
C R
Z2
β network
Assuming Rin → ∞ for the amplifier, we get Z2 R k (1/sC ) sRC A(s) β(s) = A =A =A . Z1 + Z2 R + (1/sC ) + R k (1/sC ) (sRC )2 + 3sRC + 1
M. B. Patil, IIT Bombay
Wien bridge oscillator
Amplifier
xo
R
amplifier
β xo
C
A
Frequency−sensitive network
Z1
C R
Z2
β network
Assuming Rin → ∞ for the amplifier, we get Z2 R k (1/sC ) sRC A(s) β(s) = A =A =A . Z1 + Z2 R + (1/sC ) + R k (1/sC ) (sRC )2 + 3sRC + 1 For A β = 1 (and with A equal to a real positive number), jωRC must be real and equal to 1/A. −ω 2 (RC )2 + 3jωRC + 1
M. B. Patil, IIT Bombay
Wien bridge oscillator
Amplifier
xo
R
amplifier
β xo
C
A
Frequency−sensitive network
Z1
C R
Z2
β network
Assuming Rin → ∞ for the amplifier, we get Z2 R k (1/sC ) sRC A(s) β(s) = A =A =A . Z1 + Z2 R + (1/sC ) + R k (1/sC ) (sRC )2 + 3sRC + 1 For A β = 1 (and with A equal to a real positive number), jωRC must be real and equal to 1/A. −ω 2 (RC )2 + 3jωRC + 1 →
ω=
1 ,A=3 RC
M. B. Patil, IIT Bombay
Wien bridge oscillator
|H|
1
V1
R
C
0.1
V2 0.01 90
C R
0
6
C = 1 nF
H
R = 158 kΩ
−90
101
102
103
104
105
f (Hz)
H(jω) =
V2 (jω) jωRC = . V1 (jω) −ω 2 (RC )2 + 3jωRC + 1
M. B. Patil, IIT Bombay
Wien bridge oscillator
|H|
1
V1
R
C
0.1
V2 0.01 90
C R
0
6
C = 1 nF
H
R = 158 kΩ
−90
101
102
103
104
105
f (Hz)
H(jω) =
V2 (jω) jωRC = . V1 (jω) −ω 2 (RC )2 + 3jωRC + 1
Note that the condition ∠H = 0 is satisfied only at one frequency, ω0 = 1/RC , i.e., f0 = 1 kHz. At this frequency, |H| = 0.33, i.e., β(jω) = 1/3.
M. B. Patil, IIT Bombay
Wien bridge oscillator
|H|
1
V1
R
C
0.1
V2 0.01 90
C R
0
6
C = 1 nF
H
R = 158 kΩ
−90
101
102
103
104
105
f (Hz)
H(jω) =
V2 (jω) jωRC = . V1 (jω) −ω 2 (RC )2 + 3jωRC + 1
Note that the condition ∠H = 0 is satisfied only at one frequency, ω0 = 1/RC , i.e., f0 = 1 kHz. At this frequency, |H| = 0.33, i.e., β(jω) = 1/3. For A β = 1 → A = 3, as derived analytically.
M. B. Patil, IIT Bombay
Wien bridge oscillator
|H|
1
V1
R
C
0.1
V2 0.01 90
C R
0
6
C = 1 nF
H
R = 158 kΩ
−90
101
102
103
104
105
f (Hz)
H(jω) =
V2 (jω) jωRC = . V1 (jω) −ω 2 (RC )2 + 3jωRC + 1
Note that the condition ∠H = 0 is satisfied only at one frequency, ω0 = 1/RC , i.e., f0 = 1 kHz. At this frequency, |H| = 0.33, i.e., β(jω) = 1/3. For A β = 1 → A = 3, as derived analytically. SEQUEL file: ee101 osc 1.sqproj M. B. Patil, IIT Bombay
Wien bridge oscillator
gain limiter
Block diagram
Implementation
Output voltage
100 k R3 1.5 10 k Amplifier
xo gain limiter
β xo
Frequency−sensitive network
R1 amplifier
22.1 k R 2
Vo
R
Vo
C 0
158 k 1 nF C R
−1.5 0
Ref.: S. Franco, "Design with Op Amps and analog ICs"
1 t (msec)
2
β network
SEQUEL file: wien_osc_1.sqproj
M. B. Patil, IIT Bombay
Wien bridge oscillator
gain limiter
Block diagram
Implementation
Output voltage
100 k R3 1.5 10 k Amplifier
xo gain limiter
β xo
Frequency−sensitive network
22.1 k R 2
R1 amplifier
Vo
R
Vo
C 0
158 k 1 nF C R
−1.5 0
Ref.: S. Franco, "Design with Op Amps and analog ICs"
1 t (msec)
2
β network
SEQUEL file: wien_osc_1.sqproj
* ω0 =
1 1 = → f0 = 1 kHz. RC (158 k) × (1 nF)
M. B. Patil, IIT Bombay
Wien bridge oscillator
gain limiter
Block diagram
Implementation
Output voltage
100 k R3 1.5 10 k Amplifier
xo gain limiter
β xo
Frequency−sensitive network
22.1 k R 2
R1 amplifier
Vo
R
Vo
C 0
158 k 1 nF C R
−1.5 0
Ref.: S. Franco, "Design with Op Amps and analog ICs"
1 t (msec)
2
β network
SEQUEL file: wien_osc_1.sqproj
* ω0 =
1 1 = → f0 = 1 kHz. RC (158 k) × (1 nF)
* Since the amplifier gain is required to be A = 3, we must have 1 +
R2 = 3 → R2 = 2 R1 . R1
M. B. Patil, IIT Bombay
Wien bridge oscillator
gain limiter
Block diagram
Implementation
Output voltage
100 k R3 1.5 10 k Amplifier
xo gain limiter
β xo
Frequency−sensitive network
22.1 k R 2
R1 amplifier
Vo
R
Vo
C 0
158 k 1 nF C R
−1.5 0
Ref.: S. Franco, "Design with Op Amps and analog ICs"
1 t (msec)
2
β network
SEQUEL file: wien_osc_1.sqproj
* ω0 =
1 1 = → f0 = 1 kHz. RC (158 k) × (1 nF)
R2 = 3 → R2 = 2 R1 . R1 * For gain limiting, diodes have been used. With one of the two diodes conducting, R2 → R2 k R3 , and the gain reduces. * Since the amplifier gain is required to be A = 3, we must have 1 +
M. B. Patil, IIT Bombay
Wien bridge oscillator
gain limiter
Block diagram
Implementation
Output voltage
100 k R3 1.5 10 k Amplifier
xo gain limiter
β xo
Frequency−sensitive network
22.1 k R 2
R1 amplifier
Vo
R
Vo
C 0
158 k 1 nF C R
−1.5 0
Ref.: S. Franco, "Design with Op Amps and analog ICs"
1 t (msec)
2
β network
SEQUEL file: wien_osc_1.sqproj
* ω0 =
1 1 = → f0 = 1 kHz. RC (158 k) × (1 nF)
R2 = 3 → R2 = 2 R1 . R1 * For gain limiting, diodes have been used. With one of the two diodes conducting, R2 → R2 k R3 , and the gain reduces. * Since the amplifier gain is required to be A = 3, we must have 1 +
* Note that there was no need to consider loading of the β network by the amplifier because of the large input resistance of the Op Amp. That is why β could be computed independently.
M. B. Patil, IIT Bombay
Phase-shift oscillator
A
V
C1
C3
C2 R1
I
B
R2
SEQUEL file: ee101_osc_4.sqproj
Phase-shift oscillator
A
V
C1
C3
C2 R1
I
B
R2
SEQUEL file: ee101_osc_4.sqproj
Let R1 = R2 = R = 10 k, G = 1/R, and C1 = C2 = C3 = C = 16 nF .
Phase-shift oscillator
A
V
C1
C3
C2 R1
I
B
R2
SEQUEL file: ee101_osc_4.sqproj
Let R1 = R2 = R = 10 k, G = 1/R, and C1 = C2 = C3 = C = 16 nF . Using nodal analysis, sC (VA − V ) + GVA + sC (VA − VB ) = 0
(1)
sC (VB − VA ) + GVB + sCVB = 0
(2)
Phase-shift oscillator
A
V
C1
C3
C2 R1
I
B
R2
SEQUEL file: ee101_osc_4.sqproj
Let R1 = R2 = R = 10 k, G = 1/R, and C1 = C2 = C3 = C = 16 nF . Using nodal analysis, sC (VA − V ) + GVA + sC (VA − VB ) = 0
(1)
sC (VB − VA ) + GVB + sCVB = 0
(2)
Solving (1) and (2), I =
1 (sRC )3 V. R 3 (sRC )2 + 4 sRC + 1
Phase-shift oscillator 10−2
A
V
C1
C3
C2 R1
|I(s)/V(s)| (A/V)
I
B
10−10 270
R2 6
180
(I(s)/V(s)) (deg)
SEQUEL file: ee101_osc_4.sqproj 90
102
101
103
104
105
f (Hz)
Let R1 = R2 = R = 10 k, G = 1/R, and C1 = C2 = C3 = C = 16 nF . Using nodal analysis, sC (VA − V ) + GVA + sC (VA − VB ) = 0
(1)
sC (VB − VA ) + GVB + sCVB = 0
(2)
Solving (1) and (2), I =
1 (sRC )3 V. R 3 (sRC )2 + 4 sRC + 1 M. B. Patil, IIT Bombay
Phase-shift oscillator 10−2
A
V C1
C3
C2 R1
|I(s)/V(s)| (A/V)
I
B
10−10
270
R2 6
180
(I(s)/V(s)) (deg)
SEQUEL file: ee101_osc_4.sqproj 90
101
102
103
104
105
f (Hz)
(R1 = R2 = R = 10 k, and C1 = C2 = C3 = C = 16 nF .) β(jω) =
I (jω) 1 (jωRC )3 = . V (jω) R 3(jωRC )2 + 4 jωRC + 1
M. B. Patil, IIT Bombay
Phase-shift oscillator 10−2
A
V C1
C3
C2 R1
|I(s)/V(s)| (A/V)
I
B
10−10
270
R2 6
180
(I(s)/V(s)) (deg)
SEQUEL file: ee101_osc_4.sqproj 90
101
102
103
104
105
f (Hz)
(R1 = R2 = R = 10 k, and C1 = C2 = C3 = C = 16 nF .) β(jω) =
I (jω) 1 (jωRC )3 = . V (jω) R 3(jωRC )2 + 4 jωRC + 1
For β(jω) to be a real number, the denominator must be purely imaginary. 1 1 → 3(ωRC )2 + 1 = 0, i.e., 3(ωRC )2 = 1 → ω ≡ ω0 = √ → f0 = 574 Hz . 3 RC
M. B. Patil, IIT Bombay
Phase-shift oscillator 10−2
A
V C1
C3
C2 R1
|I(s)/V(s)| (A/V)
I
B
10−10
270
R2 6
180
(I(s)/V(s)) (deg)
SEQUEL file: ee101_osc_4.sqproj 90
101
102
103
104
105
f (Hz)
(R1 = R2 = R = 10 k, and C1 = C2 = C3 = C = 16 nF .) β(jω) =
I (jω) 1 (jωRC )3 = . V (jω) R 3(jωRC )2 + 4 jωRC + 1
For β(jω) to be a real number, the denominator must be purely imaginary. 1 1 → 3(ωRC )2 + 1 = 0, i.e., 3(ωRC )2 = 1 → ω ≡ ω0 = √ → f0 = 574 Hz . 3 RC Note that, at ω = ω0 , √ 1 (j/ 3)3 1 −6 β(jω0 ) = = −8.33 × 10 . √ =− R 4 j/ 3 12 R M. B. Patil, IIT Bombay
Phase-shift oscillator
I A
V C1
B
R2
A
V C3
C2 R1
I C1
Rf
B
C3
C2 R1
β network
R2
current−to−voltage converter
Note that the functioning of the β network as a stand-alone circuit (left figure) and as a feedback block (right figure) is the same, thanks to the virtual ground provided by the Op Amp.
M. B. Patil, IIT Bombay
Phase-shift oscillator
I A
V C1
B
A
V C3
C2 R1
I
R2
C1
Rf
B
C3
C2 R1
β network
R2
current−to−voltage converter
Note that the functioning of the β network as a stand-alone circuit (left figure) and as a feedback block (right figure) is the same, thanks to the virtual ground provided by the Op Amp. V (jω) = −Rf I (jω) → Aβ(jω) = −Rf
I (jω) Rf (jωRC )3 =− . V (jω) R 3(jωRC )2 + 4 jωRC + 1
M. B. Patil, IIT Bombay
Phase-shift oscillator
I A
V C1
B
A
V C3
C2 R1
I
R2
C1
Rf
B
C3
C2 R1
β network
R2
current−to−voltage converter
Note that the functioning of the β network as a stand-alone circuit (left figure) and as a feedback block (right figure) is the same, thanks to the virtual ground provided by the Op Amp. V (jω) = −Rf I (jω) → Aβ(jω) = −Rf
I (jω) Rf (jωRC )3 =− . V (jω) R 3(jωRC )2 + 4 jωRC + 1
1 1 I (jω) 1 As seen before, at → ω = ω0 = √ , we have =− . V (jω) 12 R 3 RC
M. B. Patil, IIT Bombay
Phase-shift oscillator
I A
V C1
B
A
V C3
C2 R1
I
R2
C1
Rf
B
C3
C2 R1
β network
R2
current−to−voltage converter
Note that the functioning of the β network as a stand-alone circuit (left figure) and as a feedback block (right figure) is the same, thanks to the virtual ground provided by the Op Amp. V (jω) = −Rf I (jω) → Aβ(jω) = −Rf
I (jω) Rf (jωRC )3 =− . V (jω) R 3(jωRC )2 + 4 jωRC + 1
1 1 I (jω) 1 As seen before, at → ω = ω0 = √ , we have =− . V (jω) 12 R 3 RC For the circuit to oscillate, we need Aβ = 1 → Rf (1/12 R) = 1, i.e., Rf = 12 R
M. B. Patil, IIT Bombay
Phase-shift oscillator
I A
V C1
B
A
V C3
C2 R1
I
R2
C1
Rf
B
C3
C2 R1
β network
R2
current−to−voltage converter
Note that the functioning of the β network as a stand-alone circuit (left figure) and as a feedback block (right figure) is the same, thanks to the virtual ground provided by the Op Amp. V (jω) = −Rf I (jω) → Aβ(jω) = −Rf
I (jω) Rf (jωRC )3 =− . V (jω) R 3(jωRC )2 + 4 jωRC + 1
1 1 I (jω) 1 As seen before, at → ω = ω0 = √ , we have =− . V (jω) 12 R 3 RC For the circuit to oscillate, we need Aβ = 1 → Rf (1/12 R) = 1, i.e., Rf = 12 R In addition, we employ a gain limiter circuit to complete the oscillator design.
M. B. Patil, IIT Bombay
Phase-shift oscillator
Block diagram
Implementation
3k
gain limiter 3k
V CC
6
1k
Amplifier
xo gain limiter
β xo
Output voltage
V EE
1k
Frequency−sensitive network
C1
16 nF
C2 R1
10 k Ref.: Sedra and Smith, "Microelectronic circuits"
Vo
β network 16 nF
16 nF
C3 R2
10 k
0
125 k R f
Vo amplifier (i−to−v converter)
−6 0
1
2 t (msec)
3
4
SEQUEL file: phase_shift_osc_1.sqproj
1 1 ω0 = √ → f0 = 574 Hz, T = 1.74 ms . 3 RC
M. B. Patil, IIT Bombay
Inverting amplifier, revisited
1k
Vs
R1
R2
Vo RL
SEQUEL file: inv_amp_ac.sqproj
Inverting amplifier, revisited
1k
Vs
R1
R2
Vo RL
SEQUEL file: inv_amp_ac.sqproj
* As seen earlier, AV = −R2 /R1 → |AV | should be independent of the signal frequency.
Inverting amplifier, revisited
1k
Vs
R1
R2
Vo RL
SEQUEL file: inv_amp_ac.sqproj
* As seen earlier, AV = −R2 /R1 → |AV | should be independent of the signal frequency. * However, a measurement with a real Op Amp will show that |AV | starts reducing at higher frequencies.
Inverting amplifier, revisited
40 50 k
Vs
R1
R2
25 k
Vo
AV (dB)
1k
20
10 k
R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104
105
106
f (Hz)
* As seen earlier, AV = −R2 /R1 → |AV | should be independent of the signal frequency. * However, a measurement with a real Op Amp will show that |AV | starts reducing at higher frequencies.
Inverting amplifier, revisited
40 50 k
Vs
R1
R2
25 k
Vo
AV (dB)
1k
20
10 k
R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104
105
106
f (Hz)
* As seen earlier, AV = −R2 /R1 → |AV | should be independent of the signal frequency. * However, a measurement with a real Op Amp will show that |AV | starts reducing at higher frequencies. * If |AV | is increased, the gain “roll-off” starts at lower frequencies.
Inverting amplifier, revisited
40 50 k
Vs
R1
R2
25 k
Vo
AV (dB)
1k
20
10 k
R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104
105
106
f (Hz)
* As seen earlier, AV = −R2 /R1 → |AV | should be independent of the signal frequency. * However, a measurement with a real Op Amp will show that |AV | starts reducing at higher frequencies. * If |AV | is increased, the gain “roll-off” starts at lower frequencies. * This behaviour has to do with the frequency response of the Op Amp which we have not considered so far.
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vi
Vo
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106
The gain of the 741 Op Amp starts falling at rather low frequencies, with fc ' 10 Hz!
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vi
Vo
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106
The gain of the 741 Op Amp starts falling at rather low frequencies, with fc ' 10 Hz! The 741 Op Amp (and many others) are designed with this feature to ensure that, in typical amplifier applications, the overall circuit is stable (and not oscillatory).
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vi
Vo
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106
The gain of the 741 Op Amp starts falling at rather low frequencies, with fc ' 10 Hz! The 741 Op Amp (and many others) are designed with this feature to ensure that, in typical amplifier applications, the overall circuit is stable (and not oscillatory). In other words, the Op Amp has been internally compensated for stability.
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vi
Vo
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106
The gain of the 741 Op Amp starts falling at rather low frequencies, with fc ' 10 Hz! The 741 Op Amp (and many others) are designed with this feature to ensure that, in typical amplifier applications, the overall circuit is stable (and not oscillatory). In other words, the Op Amp has been internally compensated for stability. The gain of the 741 Op Amp can be represented by, A0 A(s) = , 1 + s/ωc with A0 ≈ 105 (i.e., 100 dB), ωc ≈ 2π × 10 rad/s.
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vo
Vi
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106 ωt
A(jω) =
A0 . 1 + jω/ωc
For ω ωc , we have A(jω) ≈
A0 . jω/ωc
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vo
Vi
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106 ωt
A(jω) =
A0 . 1 + jω/ωc
For ω ωc , we have A(jω) ≈
A0 . jω/ωc
|A(jω)| becomes 1 when A0 = ω/ωc , i.e., ω = A0 ωc .
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vo
Vi
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106 ωt
A(jω) =
A0 . 1 + jω/ωc
For ω ωc , we have A(jω) ≈
A0 . jω/ωc
|A(jω)| becomes 1 when A0 = ω/ωc , i.e., ω = A0 ωc . This frequency, ωt = A0 ωc , is called the unity-gain frequency. For the 741 Op Amp, ft = A0 fc ≈ 105 × 10 = 106 Hz.
M. B. Patil, IIT Bombay
Frequency response of Op Amp 741
ideal
Vo
Vi
Gain (dB)
100
Op Amp 741
−20 dB/decade
0
10−1
f (Hz)
106 ωt
A(jω) =
A0 . 1 + jω/ωc
For ω ωc , we have A(jω) ≈
A0 . jω/ωc
|A(jω)| becomes 1 when A0 = ω/ωc , i.e., ω = A0 ωc . This frequency, ωt = A0 ωc , is called the unity-gain frequency. For the 741 Op Amp, ft = A0 fc ≈ 105 × 10 = 106 Hz. Let us see how the frequency response of the 741 Op Amp affects the gain of an inverting amplifier. M. B. Patil, IIT Bombay
Inverting amplifier, revisited
R2
R2 Vs
R1
Vs
Vo
R1
R2 Vs
Ro Vi Ri
AV (s) Vi
Vo
R1 Vi
AV (s) Vi
Vo
M. B. Patil, IIT Bombay
Inverting amplifier, revisited
R2
R2 Vs
R1
Vs
Vo
R1
R2 Vs
Ro Vi Ri
AV (s) Vi
Vo
R1 Vi
AV (s) Vi
Vo
Assuming Ri to be large and Ro to be small, we get R2 R1 −Vi (s) = Vs (s) + Vo (s) . R1 + R2 R1 + R2
M. B. Patil, IIT Bombay
Inverting amplifier, revisited
R2
R2 Vs
R1
Vs
Vo
R1
R2 Vs
Ro Vi Ri
AV (s) Vi
Vo
R1 Vi
AV (s) Vi
Vo
Assuming Ri to be large and Ro to be small, we get R2 R1 −Vi (s) = Vs (s) + Vo (s) . R1 + R2 R1 + R2 Using Vo (s) = AV (s) Vi (s), Vo (s) R2 1 » „ « – „ « =− R1 + R2 1 R1 + R2 s Vs (s) R1 1+ + R1 A0 R1 A0 ωc
M. B. Patil, IIT Bombay
Inverting amplifier, revisited
R2
R2 Vs
R1
Vs
Vo
R1
R2 Vs
Ro Vi Ri
AV (s) Vi
Vo
R1 Vi
AV (s) Vi
Vo
Assuming Ri to be large and Ro to be small, we get R2 R1 −Vi (s) = Vs (s) + Vo (s) . R1 + R2 R1 + R2 Using Vo (s) = AV (s) Vi (s), Vo (s) R2 1 » „ « – „ « =− R1 + R2 1 R1 + R2 s Vs (s) R1 1+ + R1 A0 R1 A0 ωc R2 1 ωc A0 ωt ≈− , with ωc0 = = . R1 1 + s/ωc0 1 + R2 /R1 1 + R2 /R1
M. B. Patil, IIT Bombay
Inverting amplifier, revisited
1k
Vs
R1
R2
Vo RL
SEQUEL file: inv_amp_ac.sqproj
Inverting amplifier, revisited
1k
Vs
R1
R2
Vo RL
SEQUEL file: inv_amp_ac.sqproj
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
Inverting amplifier, revisited
1k
Vs
R1
R2
R2
5k
gain (dB) 14
fc ′ (kHz) 167
Vo RL
SEQUEL file: inv_amp_ac.sqproj
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
Inverting amplifier, revisited
40
Vs
R1
R2
R2
5k
gain (dB) 14
fc ′ (kHz) 167
Vo
AV (dB)
1k
20
R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
Vs
R1
R2
R2
Vo
gain (dB)
fc ′ (kHz)
5k
14
167
10 k
20
91
AV (dB)
1k
20
R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
Vs
R1
R2
R2
Vo
gain (dB)
fc ′ (kHz)
5k
14
167
10 k
20
91
AV (dB)
1k
20
10 k R2 = 5 k
RL SEQUEL file: inv_amp_ac.sqproj
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
Vs
R1
R2
R2
Vo RL
gain (dB)
fc ′ (kHz)
5k
14
167
10 k
20
91
25 k
28
38
SEQUEL file: inv_amp_ac.sqproj
AV (dB)
1k
20
10 k R2 = 5 k
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
Vs
R1
R2
R2
5k
Vo RL
gain (dB) 14
fc ′ (kHz) 167
10 k
20
91
25 k
28
38
SEQUEL file: inv_amp_ac.sqproj
25 k AV (dB)
1k
20
10 k R2 = 5 k
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
Vs
R1
R2
R2
5k
Vo RL
gain (dB) 14
fc ′ (kHz) 167
10 k
20
91
25 k
28
38
50 k
34
19.6
SEQUEL file: inv_amp_ac.sqproj
25 k AV (dB)
1k
20
10 k R2 = 5 k
0
101
102
103
104 f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
105
106
Inverting amplifier, revisited
40
50 k
R1
5k
Vo RL
gain (dB) 14
fc ′ (kHz) 167
10 k
20
91
25 k
28
38
50 k
34
19.6
SEQUEL file: inv_amp_ac.sqproj
25 k AV (dB)
1k
Vs
R2
R2
20
10 k R2 = 5 k
0
101
102
103
104
105
106
f (Hz)
R2 1 ωt Vo (s) =− ωc0 = , (ft = 1 MHz). Vs (s) R1 1 + s/ωc0 1 + R2 /R1
M. B. Patil, IIT Bombay
