5.2 IDEAL OPERATIONAL AMPLIFIER 229
Solution
We have
P=
∑^6
n= 1
VnIn
whereVnis the voltage at terminalnandInis the current flowingintoterminaln. Note that currents
I 1 andI 2 are zero sinceRi=∞,vout=Avin=10(1)=10 V. The power supply maintains terminal
5atV 5 =Vps=20 V and terminal 6 atV 6 =0;I 3 =−I 4 andI 5 =−I 6. Then
P=V 3 I 3 +V 4 I 4 +V 5 I 5 +V 6 I 6 =V 3 (−IL)+V 4 IL+V 5 Ips−V 6 Ips
=(V 4 −V 3 )IL+
(
VpsIps
)
=−VoutIL+VpsIps
=
(
Vps−Vout
)
Ips=
(
Vps−Vout
)
IL
Noting thatIL=Vout/RL, one has
P=( 20 − 10 )
(
10
100
)
=1W
Since many amplifiers are designed with large values ofRiin order to keep the input power
low, the approximation ofRi∼=∞is often justified. However, the assumption thatIps=ILis not
as justified because a certain amount of additional current (though kept small in order to minimize
the waste) will pass directly through the amplifier from terminal 5 to terminal 6 without going
through the load, makingIpsslightly larger thanIL.
The behavior of an amplifier always depends on the frequency of the sinusoidal signal
in question. In general, the parameters of the amplifier model vary with the signal frequency.
The variation of
∣
∣A ̄
∣
∣, as well as that of the phase angleθA, with frequency for an ampli-
fier is known as itsfrequency response.The value of
∣
∣A(f ) ̄
∣
∣always drops off at sufficiently
high frequencies. SinceA(ω) ̄ = V ̄out/V ̄in, the phase differenceθAbetween output and in-
put sinusoids is also a function of frequency. However, at low frequenciesθAis often zero,
and thereforeA ̄ can be regarded as real. Figure 5.1.3 shows variations of
∣
∣A ̄
∣
∣andθAtyp-
ical of audio, video, bandpass, and operational amplifiers. The input and output resistances
should be generalized to impedancesZi(ω) andZo(ω) when frequency effects are important.
However, most commercial amplifiers are designed to makeZiandZoreal and constant over
the useful frequency range of the amplifier block. If the frequency response of a block it-
self is known, the frequency response of a larger circuit containing the amplifier block can
be found.
5.2 Ideal Operational Amplifier
The operational amplifier, known also asop amp, consists of several transistors, diodes, capacitors,
and resistors. It is available in integrated-circuit form for less than one U.S. dollar. Being
inexpensive, compact, and versatile, operational amplifiers are used in a variety of simple circuits.
Op-amp circuits, as we will see later, usually contain negative feedback.
Op-amp circuits themselves can be regarded as building blocks. These blocks are charac-
terized by their input resistance, output resistance, and open-circuit voltage amplification. The
symbol for the op amp is shown in Figure 5.2.1. Two terminals labeled+and−are available
for inputs. The voltages of these terminals are labeled with respect to the common terminal,
