Download (a) Single-Ended AC Voltage Gain

Chapter 1
Operational Amplifiers
Objectives
 Describe basic op-amp characteristics
 Discuss op-amp modes and parameters
 Explain negative feedback
 Analyze inverting, non-inverting, voltage follower, and
inverting amp configurations
 Describe the impedance characteristics of the three
op-amp configurations
 Discuss op-amp compensation
 Troubleshoot op-amps
1. INTRODUCTION
Operational amplifier (op-amp) is a differential amplifier with
characteristics as follows:
(1) very high input impedance (Zin),
(2) very low output impedance (Zout),
(3) very high voltage gain (AV).
Figure 1-1: Schematic symbol for an op-amp.
Figure 1-2: Basic op-am
The overview of a basic op-amp is as follows:
 powered by TWO (2) dc voltages, one positive (+V) and
other negative (-V),
 has TWO (2) input terminals, an inverting (-) input and a
non-inverting (+) input,
 has ONE (1) output terminal.
Internal Block Diagram of an Op-Amp
A typical op-amp is made up of three types of amplifier circuits:
(1) a differential amplifier input stage,
(2) a voltage amplifier gain stage,
(3) a push-pull amplifier output stage.
Figure 1-9 (a): Basic internal arrangement of an op-amp.
INPUT
STAGE
OUTPUT
STAGE
GAIN
STAGE
Figure 1-9 (b): Basic internal arrangement of an op-amp.
Op-amp is produced as a circuit of components integrated
into one chip.
Top View
Figure 1-3: Typical packages.
Pin 1 is indicated by a notch or
dot on dual in-line (DIP) and
surface-mount technology (SMT)
packages.
Inverting
Non-Inverting
Figure 1-4: 741 chip packaged
in an 8-pin DIP.
Differential Amp.
Voltage Amp. Output Amp.
Figure 1-5: Schematic diagram of a 741 chip.
The differential amplifier determines the input signal
modes of an op-amp. The modes are:
 Single-ended input mode
 Double-ended (differential) input mode
 Common-mode operation
 Common-mode rejection
1.1 Single-ended input mode
This mode operates when the input signal is connected to
one input and the other is grounded.
Figure 1-6: Single-ended operation.
1.2. Double-ended (differential) input mode
This mode can be operated by using only one signal or by
applying two signal at each input.
Figure 1-7: Double-ended (differential) operation.
1.3. Common-mode operation
In common mode, two signal voltages of the same phase,
frequency, and amplitude are applied to the two inputs.
When equal input signals are applied to both inputs, they
tend to cancel, resulting in a zero output voltage.
Figure 1-8: Common-mode operation.
1.4. Common-mode rejection
The measure of an op-amp’s ability to reject unwanted
signals (noise) is called the common-mode rejection
ratio (CMRR). This parameter causes the unwanted signals
do not appear on the output
Technically, CMRR is the ratio of the open-loop differential
gain, Ad, to the common-mode gain, Ac.
Ad
CMRR 
Ac
(1-1)
The CMRR is often expressed in decibels (dB) as
 Ad
CMRR (log)  20 log 10 
 Ac



(1-2)
2. DIFFERENTIAL AMPLIFIER CIRCUIT
The differential amplifier is
a circuit that has two separate
inputs and produces two
separate outputs where the
emitters are connected
together.
(I)
(NI)
It amplifies the difference
voltage between the two input
(Vdiff ).
There are three operations
can be done in a differential
amplifier circuit; dc bias, ac
operation and common
mode operation.
Figure 1-10: The basic differential
amplifier.
2.1 DC Bias
The dc bias is determined
by connecting each base
voltage to 0 V where we
obtain,
VE  VB  VBE  0.7V
.............................(1-3)
The emitter dc bias
current is
VE  (VEE )
IE 
RE
.............................(1-4)
Figure 1-11: DC bias of differential
amplifier circuit.
If both transistors have equal values of base-emitter
voltage, VBE1 = VBE2 (well matched), we obtain
I C1  I C2
IE

2
(1-5)
resulting in a collector voltage of
VC1  VC2  VCC  I C RC  VCC
IE
 RC
2
(1-6)
2.2 AC Operation of Circuit
To operate a differential amplifier in an ac connection, two
separate ac voltage sources are connected to both bases.
There are two voltage gain
can be calculated in ac
operation:
(a) Single-Ended AC
Voltage Gain.
(b) Double-Ended AC
Voltage Gain.
Figure 1-12: AC connection of
differential amplifier circuit.
To carry out an ac analysis, each transistor is replaced by its
ac equivalent.
Figure 1-13: AC equivalent of differential amplifier circuit.
(a) Single-Ended AC Voltage Gain
Single-ended ac voltage gain is calculated by connecting one of
voltage sources to one input and the other connected to ground.
The single-ended ac voltage
gain magnitude at either
collector can be expressed
as,
Vo RC
Av 

Vi1 2re
(1-6)
where,
26 mV
re 
IE
(1-7)
re = ac emitter resistance
Figure 1-14: Connection to calculate
a single-ended ac voltage gain.
(b) Double-Ended AC Voltage Gain
By similar analysis, the differential ac voltage gain magnitude is
Vo  RC
Ad 

Vd
2ri
where, β = current gain of a transistor
ri = internal resistance of a transistor
Vd = Vi1-Vi2
(1-8)
2.3 Common-Mode Operation of Circuit
To operate a differential amplifier in a common-mode connection, the
same ac voltage source is applied to both inputs.
In most ac operation, a
differential amplifier
provides large
amplification, but in this
operation it provides small
amplification. The
voltage gain magnitude is
expressed as,
Vo
 RC
Ac 

Vi ri  2(   1) RE
..................................(1-9)
Figure 1-15: Common-mode connection.
3. DIFFERENTIAL AND COMMON-MODE OPERATION
3.1 Differential Input
Difference voltage, Vd is defined as the difference between two
input signals (Vi1 and Vi2). It is produced from two separate inputs
applied to an op-amp.
Vd  Vi1  Vi2
(1-10)
3.2 Common Input
When both input signals is same, a common voltage, Vc caused
by the two inputs can be defined as the average of the sum of the
two signals,
1
Vc  (Vi1  Vi2 )
2
(1-11)
3.3 Output Voltage
Since any signals applied to an op-amp in general have both
in-phase and out-of-phase components, the resulting
output voltage, Vo is
Vo  AdVd  AcVc
(1-12)
where Ad = differential gain of amplifier.
Ac = common-mode gain of the amplifier.
The output voltage in terms of the value of CMRR can be
expressed as,

1 Vc
Vo  AdVd 1 
 CMRR Vd



(1-13)
3.4 Opposite-Polarity Inputs
If opposite-polarity inputs are applied to an op-amp, Vi1 = -Vi2
= Vs, the resulting difference voltage is
Vd  Vi1  Vi2  Vs  (Vs )  2Vs
Vc  12 (Vi1  Vi2 )  12 [Vs  (Vs )]  0
Vo  AdVd  AcVc  Ad (2Vs )  0  2 AdVs
(1-14)
These equations illustrate that the output is the differential gain
times twice the input signal applied to one of the inputs when
the inputs are an ideal opposite signal with no common element.
3.5 Same-Polarity Inputs
If same-polarity inputs are applied to an op-amp, Vi1 = Vi2 = Vs,
the resulting difference voltage is
Vd  Vi1  Vi2  Vs  Vs  0
Vc  12 (Vi1  Vi2 )  12 (Vs  Vs )  Vs
Vo  AdVd  AcVc  Ad (0)  AcVs  AcVs
(1-15)
These equations illustrate that the output is the commonmode gain times the input signal Vs when the inputs are an
ideal in-phase signals, which shows that only common mode
operation occurs.
3.6 Common-Mode Rejection
The equations (1-14) and (1-15) provide the relationships that can
be used to measure Ad and Ac in op-amp circuits.
1. To measure Ad: Set Vi1 = -Vi2 = Vs = 0.5 V, we obtain Vd = 1 V,
Vc = 0 V and Vo = Ad
Thus, setting the input voltages Vi1 = -Vi2 = 0.5 V
results in an output voltage numerically equal to the
value of Ad.
2. To measure Ac: Set Vi1 = Vi2 = Vs = 1 V, we obtain Vd = 0 V,
Vc = 1 V and Vo = Ac
Thus, setting the input voltages Vi1 = Vi2 = 1 V results in
an output voltage numerically equal to the value of Ac.
4. OP-AMP BASICS
Ideal op-amp has the characteristics
as follows:
(1) infinite input impedance (Zin),
(2) infinite output impedance (Zout),
(3) infinite voltage gain (AV),
(4) infinite bandwidth.
Characteristics of practical op-amp
are:
(1) very high input impedance (Zin),
(2) very low output impedance (Zout),
(3) very high voltage gain (AV).
Figure 1-16: Basic op-am.
Figure 1-17: AC equivalent of op-amp circuit: (a) practical; (b) ideal.
Basic Op-Amp
A basic op-amp has the circuit characteristics as follows:
 If a voltage source is connected to the minus (-) input, the
resulting output is opposite in phase to the input signal.
 If a voltage source is applied to the plus (+) input, the output
is in phase with the input signal.
Figure 1-18: Basic op-amp.
5. PRACTICAL OP-AMP CIRCUITS
The op-amp have several circuit connections that provide
various operating characteristics. The op-amp can be
connected as:
1. an inverting amplifier
2. a non-inverting amplifier
3. an unity follower
4. a summing amplifier
5. an integrator
6. a differentiator
5.1 Inverting Amplifier
An op-amp connected as a inverting amplifier has the characteristics
as follows:
 The input signal is applied to
the inverting (-) input
through a input resistor R1.
 The non-inverting (+) input
is grounded.
 The output is obtained by
multiplying the input by a
constant gain and fed back
to the same input through a
feedback resistor Rf.
Vo  
Rf
R1
V1
Figure 1-19: Inverting amplifier.
(1-16)
5.2 Non-inverting Amplifier
The characteristics of noninverting amplifier are:
the input signal is applied to
the non-inverting (+) input
the output is applied back to
the inverting (-) input through
the feedback circuit (closed
loop) formed by R1 and Rf.
 Rf
Vo  1 
R1


V1

Figure 1-20: Non-inverting amplifier.
(1-17)
5.3 Unity Follower
An unity follower circuit is characterized as follows:
provides a voltage gain of 1 (which means there is no gain)
all of the output voltage is fed back to the inverting (-) input.
Figure 1-21: (a) Unity follower; (b) virtual-ground equivalent circuit.
5.4 Summing Amplifier
The summing amplifier is an op-amp circuit that provides an
output proportional to the sum of its inputs. Each input voltage
is multiplied by a constant-gain factor.
Figure 1-22: (a) Summing amplifier; (b) virtual-ground equivalent circuit.
The output voltage can be expressed as the sum of the
equations for each as follows:
Rf
Rf 
 Rf
Vo   V1  V2  V3 
R1
R3 
 R1
(1-18)
5.5 Integrator
If the feedback component used is a capacitor that forms an
RC circuit with the input resistor, the resulting connection is
known as an ideal integrator.
Figure 1-23: An ideal integrator.
The output voltage can be written as:
1
Vo 
V1
jRC
(1-18)
This expression can be rewritten in the time domain as:
1
vo (t )  
v1 (t )dt

RC
(1-19)
Equation (1-19) shows that the output is the integral of
the input, with an inversion and scale multiplier of 1/RC.
If more than one input voltage may be applied to an integrator,
the output voltage can be calculated as the sum of the equations
for each as follows:
 1

1
1
vo (t )  
v1 (t )dt 
v2 (t )dt 
v3 (t )dt 



R2C
R3C
 R1C

......................................................................................(1-20)
Figure 1-24: Summing-integrator circuit.
Practical integrators often have an additional resistor in
parallel with the feedback capacitor to prevent saturation.
Rf
Figure 1-25: A practical integrator.
5.6 Differentiator
If the positions of the integrator feedback capacitor (C) and
input resistor (R) are reversed, we have a differentiator.
Figure 1-26: Differentiator circuit.
The output voltage can be calculated by using the following
formula:
dv1 (t )
vo (t )   RC
dt
(1-21)
This equation illustrates that a differentiator provides an
output that is proportional to the rate of change of its
input signal.
Slew Rate
The slew rate of an op-amp is maximum rate at which the output
voltage can change in response to a change at either signal input
and written as:
Vo
SR 
t
(V / s )
(1-22)
Since frequency is related to time, the slew rate can be used to
determined the maximum signal frequency of the op-amp:
f max
SR

2 K
where, K = the peak output voltage from the op-amp.
(1-23)
The peak output voltage is half the peak-to-peak output
voltage value and expressed as:
K  Vo pp
1
2
(1-24)
While the peak-to-peak output voltage is found as
Vo pp  AdVi
(1-25)