E1C06 09/14/2010
11:55:6 Page 234
Using Kirchhoff’s law about the noninverting input loop yields
I i t
ð ÞR 1 þ E o t
ð Þ ¼ E i t
ð Þ
Then the circuit input resistance R i is
R i ¼
E i t
ð Þ
I i t
ð Þ
¼
E i t
ð ÞR 1
E i t
ð Þ À E o t
ð Þ
ð6:50Þ
Likewise the output resistance is found to be
R o ¼
R 2
1 þ A o
ð6:51Þ
Because A o is large, Equations 6.49 to 6.51 show that the input impedance, developed as a resistance,
of the voltage follower can be large, its output impedance can be small, and the circuit will have near
unity gain. Acceptable values for R 1 and R 2 range from 10 to 100 kV to maintain stable operation.
Input signal integration and differentiation can be performed with the operational amplifier
circuits. For the integration circuit in Figure 6.20e, the currents through R 2 and C are given by
I R 2 t
ð Þ ¼
E i t
ð Þ À E A t
ð Þ
R 2
I C t
ð Þ ¼ C
d
dt
E o t
ð Þ À E A t
ð Þ
ð
Þ
Summing currents at node A yields the integrator circuit operation:
E o t
ð Þ ¼ À
1
R 2 C
Z
E i t
ð Þdt
ð6:52Þ
As a special note, if the input voltage is a constant positive DC voltage, then the output signal will be
a negative linear ramp, E 0 ¼ ÀE i t=RC, with t in seconds, a feature used in ramp converters and
integrating digital voltmeters (see Section 7.5 in Chapter 7). Integrator circuits are relatively
unaffected by high-frequency noise as integration averages out noise.
Following a similar analysis, the differentiator circuit shown in Figure 6.20f performs the
operation
E o t
ð Þ ¼ ÀR 2 C
dE i t
ð Þ
dt
ð6:53Þ
Differentiator circuits amplify high-frequency noise in signals to the point of masking the true
signal. Adding resistor R i limits high-frequency gain to a À3 dB cutoff frequency of f c ¼ 1=2pR i C.
In noisy environments, a passive RC differentiator may simply perform better.
6.7 ANALOG SIGNAL CONDITIONING: SPECIAL-PURPOSE CIRCUITS
Analog Voltage Comparator
A voltage comparator provides an output that is proportional to the difference between two input
voltages. As shown in Figure 6.21a, a basic comparator consists of an operational amplifier
operating in a high-gain differential mode. In this case, any difference between inputs E i 1 and E i 2 is
amplified at the large open-loop gain A o . Accordingly, the output from the comparator saturates
234 Chapter 6 Analog Electrical Devices and Measurements
11:55:6 Page 234
Using Kirchhoff’s law about the noninverting input loop yields
I i t
ð ÞR 1 þ E o t
ð Þ ¼ E i t
ð Þ
Then the circuit input resistance R i is
R i ¼
E i t
ð Þ
I i t
ð Þ
¼
E i t
ð ÞR 1
E i t
ð Þ À E o t
ð Þ
ð6:50Þ
Likewise the output resistance is found to be
R o ¼
R 2
1 þ A o
ð6:51Þ
Because A o is large, Equations 6.49 to 6.51 show that the input impedance, developed as a resistance,
of the voltage follower can be large, its output impedance can be small, and the circuit will have near
unity gain. Acceptable values for R 1 and R 2 range from 10 to 100 kV to maintain stable operation.
Input signal integration and differentiation can be performed with the operational amplifier
circuits. For the integration circuit in Figure 6.20e, the currents through R 2 and C are given by
I R 2 t
ð Þ ¼
E i t
ð Þ À E A t
ð Þ
R 2
I C t
ð Þ ¼ C
d
dt
E o t
ð Þ À E A t
ð Þ
ð
Þ
Summing currents at node A yields the integrator circuit operation:
E o t
ð Þ ¼ À
1
R 2 C
Z
E i t
ð Þdt
ð6:52Þ
As a special note, if the input voltage is a constant positive DC voltage, then the output signal will be
a negative linear ramp, E 0 ¼ ÀE i t=RC, with t in seconds, a feature used in ramp converters and
integrating digital voltmeters (see Section 7.5 in Chapter 7). Integrator circuits are relatively
unaffected by high-frequency noise as integration averages out noise.
Following a similar analysis, the differentiator circuit shown in Figure 6.20f performs the
operation
E o t
ð Þ ¼ ÀR 2 C
dE i t
ð Þ
dt
ð6:53Þ
Differentiator circuits amplify high-frequency noise in signals to the point of masking the true
signal. Adding resistor R i limits high-frequency gain to a À3 dB cutoff frequency of f c ¼ 1=2pR i C.
In noisy environments, a passive RC differentiator may simply perform better.
6.7 ANALOG SIGNAL CONDITIONING: SPECIAL-PURPOSE CIRCUITS
Analog Voltage Comparator
A voltage comparator provides an output that is proportional to the difference between two input
voltages. As shown in Figure 6.21a, a basic comparator consists of an operational amplifier
operating in a high-gain differential mode. In this case, any difference between inputs E i 1 and E i 2 is
amplified at the large open-loop gain A o . Accordingly, the output from the comparator saturates
234 Chapter 6 Analog Electrical Devices and Measurements