Calculating Common Mode and Differential Gain
SYRACUSE UNIVERSITY
ELE231 Electrical Fundamentals I
Calculating Common Mode and Differential Gain by Akhan Almagambetov, [email protected] Office Hours: Wed. 1:15-3:15 p.m., Thurs. 12:30-2:30 p.m in CST 3-204 Recitation Session # 8.2 Week of October 13th, 2008
The common mode gain and differential gain concepts are easiest to illustrate on a differential amplifier, although the same approach may be applied to other op-amp configurations1 . If we recall (from the previous handouts), the circuit diagram for an ideal differential amplifier looks similar to Figure 1 on the following page. The Operational Amplifier Basics handout2 states that the simplified equation for the output voltage, (13), is as follows: vOU T =
R4 R2 + R3
R4 R4 + 1 vIN + − vIN − R1 R1
(1)
Aside from expressing the output voltage in terms of vIN + and vIN − , we can also express it in terms of its common mode gain, Acm , and the differential gain3 , Ad . 1
R. A. Lessard, “Instrumentation Amplifier,” Feb. 16, 2007. [Microform]. A. Almagambetov, “Operational Amplifier Basics,” Oct. 6, 2008. [Online]. Available: http://home.comcast.net/~akhan.almagambetov/ele231f08/ele231_05_operational_amplifiers.pdf. 3 R. Mauro, Engineering electronics: a practical approach. Englewood Cliffs, NJ: Prentice-Hall, 1989, pp. 387-389. 2
1
Common Mode and Differential Gain IR4
vIN-vIN+
IR1
IR2
R4
v--
R1
vOUT IO
-+
v+
R2
IOUT ROUT
R3
+
vOUT
IR3
--
Figure 1: Differential amplifier
To express the common mode and differential gains, the input signals to the amplifier have to be broken down into the common-mode and differential components, respectively. The difference signal is simply defined as the difference between the two input signals: vd = vIN + − vIN −
(2)
The common-mode component is called the common-mode noise signal 4 . It is safe to assume that it is present in equal amounts on both inputs, and therefore it is expressed as the mean of the two input signals: vcm =
vIN − + vIN + 2
(3)
Equations (2) and (3) tell us that the input signals, vIN − and vIN + , can be expressed in terms of their common mode and differential components. This will make it easier to separate the two types of gain in the end:
vIN − = vcm −
4
vd 2
and
vIN + = vcm +
vd 2
(4)
Mauro, p. 387
2
10/16/2008 v.1 (JJHS) c Akhan Almagambetov Copyright May be reprinted without permission.
Common Mode and Differential Gain
What we are working toward is the following form of the output voltage equation, vOU T : vo = Acm vcm + Ad vd
(5)
Note that the common-mode gain of the amplifier circuit is denoted by Acm and the differential gain is denoted by Ad . Looking back at equation (1), what we can do to find the output voltage in terms of the common-mode and differential components, is to simply substitute the expressions that we derived in equation (4) into our output voltage equation: vOU T = vOU T =
R4 R2 + R3
R4 R2 + R3
R4 R4 vd vd − +1 vcm − vcm + R1 2 R1 2
R4 R4 1 R4 R4 R4 +1 − vcm + +1 + vd R1 R1 2 R2 + R3 R1 R1
So what do we have in the end? We can extract just the gain terms from the equation above: Acm =
Ad =
R4 R4 R4 +1 − R2 + R3 R1 R1 R4 R4 R4 1 +1 + 2 R2 + R3 R1 R1
The same procedure can be applied to any op-amp containing circuit that has a differential amplifier (or a pseudo-differential amplifier, meaning that there are no resistors on the input or feedback).
As long as we are on-topic, there is another concept that is used as a metric in measuring the quality of the amplifier circuit. This metric is called the common-mode rejection ratio (CMRR) and its units are decibels (dB). You can calculate the CMRR using the formula below5 :
CM RR = 20 log10
5
Mauro, p. 388
3
Ad Acm
(6)
ELE231 Electrical Fundamentals I
Calculating Common Mode and Differential Gain by Akhan Almagambetov, [email protected] Office Hours: Wed. 1:15-3:15 p.m., Thurs. 12:30-2:30 p.m in CST 3-204 Recitation Session # 8.2 Week of October 13th, 2008
The common mode gain and differential gain concepts are easiest to illustrate on a differential amplifier, although the same approach may be applied to other op-amp configurations1 . If we recall (from the previous handouts), the circuit diagram for an ideal differential amplifier looks similar to Figure 1 on the following page. The Operational Amplifier Basics handout2 states that the simplified equation for the output voltage, (13), is as follows: vOU T =
R4 R2 + R3
R4 R4 + 1 vIN + − vIN − R1 R1
(1)
Aside from expressing the output voltage in terms of vIN + and vIN − , we can also express it in terms of its common mode gain, Acm , and the differential gain3 , Ad . 1
R. A. Lessard, “Instrumentation Amplifier,” Feb. 16, 2007. [Microform]. A. Almagambetov, “Operational Amplifier Basics,” Oct. 6, 2008. [Online]. Available: http://home.comcast.net/~akhan.almagambetov/ele231f08/ele231_05_operational_amplifiers.pdf. 3 R. Mauro, Engineering electronics: a practical approach. Englewood Cliffs, NJ: Prentice-Hall, 1989, pp. 387-389. 2
1
Common Mode and Differential Gain IR4
vIN-vIN+
IR1
IR2
R4
v--
R1
vOUT IO
-+
v+
R2
IOUT ROUT
R3
+
vOUT
IR3
--
Figure 1: Differential amplifier
To express the common mode and differential gains, the input signals to the amplifier have to be broken down into the common-mode and differential components, respectively. The difference signal is simply defined as the difference between the two input signals: vd = vIN + − vIN −
(2)
The common-mode component is called the common-mode noise signal 4 . It is safe to assume that it is present in equal amounts on both inputs, and therefore it is expressed as the mean of the two input signals: vcm =
vIN − + vIN + 2
(3)
Equations (2) and (3) tell us that the input signals, vIN − and vIN + , can be expressed in terms of their common mode and differential components. This will make it easier to separate the two types of gain in the end:
vIN − = vcm −
4
vd 2
and
vIN + = vcm +
vd 2
(4)
Mauro, p. 387
2
10/16/2008 v.1 (JJHS) c Akhan Almagambetov Copyright May be reprinted without permission.
Common Mode and Differential Gain
What we are working toward is the following form of the output voltage equation, vOU T : vo = Acm vcm + Ad vd
(5)
Note that the common-mode gain of the amplifier circuit is denoted by Acm and the differential gain is denoted by Ad . Looking back at equation (1), what we can do to find the output voltage in terms of the common-mode and differential components, is to simply substitute the expressions that we derived in equation (4) into our output voltage equation: vOU T = vOU T =
R4 R2 + R3
R4 R2 + R3
R4 R4 vd vd − +1 vcm − vcm + R1 2 R1 2
R4 R4 1 R4 R4 R4 +1 − vcm + +1 + vd R1 R1 2 R2 + R3 R1 R1
So what do we have in the end? We can extract just the gain terms from the equation above: Acm =
Ad =
R4 R4 R4 +1 − R2 + R3 R1 R1 R4 R4 R4 1 +1 + 2 R2 + R3 R1 R1
The same procedure can be applied to any op-amp containing circuit that has a differential amplifier (or a pseudo-differential amplifier, meaning that there are no resistors on the input or feedback).
As long as we are on-topic, there is another concept that is used as a metric in measuring the quality of the amplifier circuit. This metric is called the common-mode rejection ratio (CMRR) and its units are decibels (dB). You can calculate the CMRR using the formula below5 :
CM RR = 20 log10
5
Mauro, p. 388
3
Ad Acm
(6)
