Operational amplifiers (OP amps) can be used to build various electronic circuits. An ideal OP amp has infinite input impedance and zero output impedance, with infinite open-loop gain. Common configurations include inverting and non-inverting amplifiers, which provide negative and positive feedback respectively to control the closed-loop gain. Other circuits like summers, integrators, and differentiators can be built by exploiting the high input impedance and voltage amplification properties of OP amps. However, practical OP amps have non-ideal characteristics that must be accounted for, such as finite gain bandwidth, input bias currents, and output saturation levels.

1. Operational
Amplifier
Electronic Circuits
CHO, Yong Heui

2. Electronic Circuits
1. Ideal OP amp
OP amp
 OP amp 의 어떤 terminal 도 ground
에 연결되어 있지 않음 .
 3 signal terminals and 2 power
terminals (Not displayed)
Ideal OP Amp
 Differential input voltage, single-
ended output voltage
 Infinite input impedance i i = 0
 Zero output impedance
 Zero common-mode gain (v2=v1 
vo=0)
 Infinite open-loop gain A
 Infinite bandwidth : A is constant at
any frequencies.
*Differential input voltage, single-ended output voltage
2 EM Wave Lab

3. Electronic Circuits
1. Ideal OP amp
Differential and common mode
v Id = v 2 – v 1
v I cm = ½(v 1 + v 2 )
v 1 = v Icm - v Id /2
v 2 = v Icm + v Id /2
v 3 = mv d
v d = (G m v 2 – G m v 1 ) R
v 3 = mG m R (v 2 – v 1 )
Gain A = mG m R = 100·10·10
= 10 4
= 80 dB
3 EM Wave Lab

4. Electronic Circuits
2. Inverter
Closed loop gain
i2
For ideal OP amp, finite v o means v 2 -
i1 v 1 =0
v 2 – v 1 = v o /A = 0
v1
v2 v 2 ≒ v 1 : Virtual short circuit
if v 2 is grounded, v 1 is a virtual ground
i 1 = (v I – v 1 ) /R 1 = v I /R 1 terminal.
vo = v1 – i2 R2 = v1 – i1 R2 = 0 – vI R2 / R1 ∵ infinite input impedance of ideal OP
amp
vo / vI = - R2 / R1
Closed-loop gain G ≡ v o / v I = - R 2 / R 1
R2 에 의해 negative feedback 회로로 동작
(positive feedback if connected between 2 and 3)
Closed-loop gain 은 외부 수동 소자에 의해 결정 
stable and predictable, but gain loss is inevitable
4 EM Wave Lab

5. Electronic Circuits
2. Inverter
Equivalent circuit
5 EM Wave Lab

6. Electronic Circuits
2. Inverter
Finite open loop gain
= vx
6 EM Wave Lab

7. Electronic Circuits
2. Inverter
Example
R 1 should be large as a input impedance, but large R 1 causes low voltage gain
7 EM Wave Lab

8. Electronic Circuits
2. Inverter
Summer
Output is a weighted sum of input
signals
v o = v 1 (R a /R 1 )·(R c /R b ) + v 2 (R a /R 2 )·(R c /R b ) –
v 3 ·(R c /R 3 ) – v 4 ·(R c /R 4 )
Different summing coefficients are
possible
8 EM Wave Lab

9. Electronic Circuits
3. Noninverter
Closed loop gain
등가회로
9 EM Wave Lab

10. Electronic Circuits
3. Noninverter
Finite open loop gain
vo = A ( vI - vx )
i1 = - vx / R1 i 2 = ( v x – v o )/ R 2
A
1 + (R 2 / R 1 )
v o / v I = ---------------
1 + (R 2 / R 1 )
Inverting 경우
1 + ---------
A 와 분모 부분 같
음
A ≫ 1 + (R 2 / R 1 ) 이면 infinite gain 의 경우와
등가회로 같음
v I 와 v x 가 같은 값이 아니므로 (finite gain)
→ v x = R1 / (R1+ R2) · v o
v o = A ( v I - v x ) ∵ Op amp 동작 특
성
위 두식을 이용하면 같은 결과를 얻는다 .
10 EM Wave Lab

11. Electronic Circuits
3. Noninverter
Voltage follower
Non-inverting closed-loop 에서
R1=∞, R2=0 인 경우와 동일함
Ideal Op amp 의 infinite input
impedance 를 이용하여 source
inpedance 가 큰 source 에 연결할 수 있
고 , ideal Op amp 의 zero output
impedance 를 이용하여 load inpedance
가 적은 load 에 연결할 때 사용
Buffer Amp 가 Voltage Source
와 Load impedance 에 연결
11 EM Wave Lab

12. Electronic Circuits
4. Difference
Common mode rejection ratio
For practical circuits,
Common-mode voltage gain A cm ≠
0
v o = A d ·v Id + A cm ·v Icm
현재까지 두 input voltage 의 차이만 증폭된다고
가정하였음 . 하지만 , 실제로는 공통 부분도 A cm
만큼의 gain 을 가지고 증폭된다 .
|Ad |
CMRR = 20 Log
| A cm |
12 EM Wave Lab

13. Electronic Circuits
4. Difference
Difference amplifier
지금까지는 inverting/noninverting 의 경우 input
이 없는 한쪽은 GND 였지만 , difference amp 에서
는 두 input 의 차이만을 증폭하고자 함 . (why don’t
you use Op amp itself ?)
Output port 에서 Common-mode signal 을 없애기
위해선 , inverting gain 과 noninverting gain 의
magnitude 는 같고 부호는 반대이어야 함 . (v I1 = v I2
일때 v 0 =0 이어야 함을 생각해 보면 됨 )
Inverting Noninverting
i = 0
≡
v O1 / v I 1 = - R 2 / R 1
13 EM Wave Lab

14. Electronic Circuits
4. Difference
Difference amplifier
ᅵᅵ
inverting gain ᅵᅵ ᅵᅵ
= noninverting gain ᅵᅵ
R4 R2
R2 / R1 =
R3 + R4 [1 R1
]
+
R4 R2 R2 R4
→ R3 + R4
=
R1 + R2 → R1
=
R3
R4 R2 R2
vO2 = vΙ 2
R3 + R4
[1
R1
] = vΙ 2
R1
+
R2 R2
By superposition, v O = v O1 + ( vI 2 - vI 1 ) = v Id
v O2 = R1 R1
R2
→ Ad =
R1
14 EM Wave Lab

15. Electronic Circuits
4. Difference
Difference amplifier
1 R4 1 R3
i1 =
R1
[ v Icm - v
R3 + R4 Icm ] = v Icm
R1 R3 + R4
R4 R4 R2 R3
vO = v Icm – i2R2 = v - v
R3 + R4 R3 + R4 Icm R1 R3 + R4 Icm
R4 R2 R3
=
R3 + R4
1- [ R1 R4
]v Icm
vO R4 R2 R3
For R1 = R3 , R2 = R4
Acm ≡
vΙcm = R +R 1-
3 4
[ R1 R4
]
v Id
R id ≡
iI
v Id = R 1 i I + 0 + R 1 i I ∵ vertual short
R id = 2R 1 : low input resistance for
high differential gain
15 EM Wave Lab

16. Electronic Circuits
4. Difference
Instrumentation amplifier
R2 R2
[1
R1
]
( vΙ 2 − vΙ 1 ) = 1
R1
[ ]v Ιd
+ +
R4 R2
vΟ =
R3 +
1
R1
[ ]
vΙd = Ad vΙd
R4 R2
Ad =
R3
[ 1
R1
] =
+
High input impedance and high differential
gain
Issues
 A cm is equal to 1+R 2 /R 1 at the first stage.
 Issues of imperfect match at the first two Op amps.
 Two R 1 resistors should be simultaneously varied : Not easy job
16 EM Wave Lab

17. Electronic Circuits
4. Difference
Instrumentation amplifier
2R1 2R2
(vO2 – vO1) v
2R1 + 2R2 = Ιd
→ vO2 – vO1 = [1 2R
]v Ιd
+ 1
R4 R4 R2 vO R4 R2
vO =
R3
(vO2 – vO1) =
R3
[1
R1
] vΙd → Ad ≡
vΙd =
R3
[1
R1
]
+ +
17 EM Wave Lab

18. Electronic Circuits
5. Nonideal OP amp
Nonideal OP amp
Differential open-loop gain  Noninfinite CMRR, noninfinite input
resistance, nonzero output resistance :
Closed-loop circuits 에서 Not critical
A0
A(jw) =
1+ jw/w b
For w = 0
For w ≫ w b
18 EM Wave Lab

19. Electronic Circuits
5. Nonideal OP amp
Frequency response
←
= vx
− R2 / R 1
Vo(s)/Vi(s) ≈ s
1
+ ωT / (1 + R2 / R1 )
For A0 ≫ 1+R2/R1
ωT
ω3dB = −−−−−−−−
− + R2 /
1
R1
For noninverting closed-loop case,
only DC gain (1 + R 2 /R 1 ) is different
19 EM Wave Lab

20. Electronic Circuits
5. Nonideal OP amp
Frequency response
Closed-loop gain R2/R1 f3dB=ft/(1+R2/R1)
+1000 999 1kHz
1 + R2 / +100 99 10kHz Constant gain-BW
product
R1 +10 9 100kHz
+1 0 1MHz
-1 1 0.5MHz
− R 2 / R1 -10 10 90.9kHz
-100 100 9.9kHz
-1000 1000 0.99kHz
ωT = A0 · ωb
20 EM Wave Lab

21. Electronic Circuits
6. Large signal
Output voltage saturation
21 EM Wave Lab

22. Electronic Circuits
6. Large signal
Slew rate
22 EM Wave Lab

23. Electronic Circuits
6. Large signal
Slew rate
23 EM Wave Lab

24. Electronic Circuits
6. Large signal
Full power bandwidth
Unity-gain voltage follower 의 input 에 sine wave 를 인가하고 출력 전압의 진폭이 최대
가 되도록 할 경우 , slew 현상이 발생하지 않는 입력신호의 최대 주파수
24 EM Wave Lab

25. Electronic Circuits
7. DC effect
Offset voltage
25 EM Wave Lab

26. Electronic Circuits
7. DC effect
Equivalent model
DC biasing issue
DC signal issue
Inverting Noninverting
26 EM Wave Lab

27. Electronic Circuits
7. DC effect
Capacitive coupling
Only for DC Only for AC
A ≠ 1 + R 2 /R 1 STC HPF
27 EM Wave Lab

28. Electronic Circuits
7. DC effect
Input bias current
두 전류 I B1 , I B2 는 거의 같은 값을 가지지만 OP amp 내부의
mismatch 로 인해 약간 차이가 남
Only for
DC
B2
28 EM Wave Lab

29. Electronic Circuits
7. DC effect
Input bias current
V O = I B1 R 2 ≈ I B R 2  limits on R 2
V O = -I B2 R 3 + R 2 (I B1 – I B2 R 3 /R 1 )
(for I B1 = I B2 = I B3 )
두 input 단자에서 본 저항 값이 같다 .
29 EM Wave Lab

30. Electronic Circuits
7. DC effect
Input bias current
For R 3 = ( R 1 ∥R 2 ) and I B1 ≠ I B2 ≠ I B3
I B1 = I B + I OS /2 I B2 = I B - I OS /2 →
V O = I OS R 2 (compare with V O = I B1 R 2 in case of without R 3 )
AC coupled inverting amp AC coupled non-inverting amp
30 EM Wave Lab

31. Electronic Circuits
8. Integrator
Impedance characteristics
31 EM Wave Lab

32. Electronic Circuits
8. Integrator
Example
32 EM Wave Lab

33. Electronic Circuits
8. Integrator
Miller integrator
Integrator Frequency : w int = 1/RC
Infinite DC gain : weak at DC imperfection
33 EM Wave Lab

34. Electronic Circuits
8. Integrator
DC imperfection
34 EM Wave Lab

35. Electronic Circuits
8. Integrator
Differentiator
HPF with infinite corner frequency.
Noise magnifier at High Frequency
35 EM Wave Lab