Unit 5

Linear applications of op-amp

An Operational Amplifier is basically a three-terminal device which consists of two high impedance inputs. One of the inputs is called the Inverting Input, marked with a negative or “minus” sign, ( – ). The other input is called the Non-inverting Input, marked with a positive or “plus” sign ( + ).

A third terminal represents the operational amplifiers output port which can both sink and source either a voltage or a current.

In a linear operational amplifier, the output signal is the amplification factor, known as the amplifiers gain ( A ) multiplied by the value of the input signal and depending on the nature of these input and output signals, there can be four different classifications of operational amplifier gain.

Open Loop Gain (Avo)

The main function of an operational amplifier is to amplify the input signal and the more open loop gain it has the better. Open-loop gain is the gain of the op-amp without positive or negative feedback and for such an amplifier the gain will be infinite

Input impedance (ZIN)

Input impedance is the ratio of input voltage to input current and is assumed to be infinite to prevent any current flowing from the source supply into the amplifiers input circuitry ( IIN = 0 ). Real op-amps have input leakage currents from a few pico-amps to a few milli-amps.

Output impedance (ZOUT)

The output impedance of the ideal operational amplifier is assumed to be zero acting as a perfect internal voltage source with no internal resistance so that it can supply as much current as necessary to the load.

This internal resistance is effectively in series with the load thereby reducing the output voltage available to the load. Real op-amps have output impedances in the 100-20kΩ range.

Bandwidth (BW)

An ideal operational amplifier has an infinite frequency response and can amplify any frequency signal from DC to the highest AC frequencies, so it is therefore assumed to have an infinite bandwidth.

With real op-amps, the bandwidth is limited by the Gain-Bandwidth product (GB), which is equal to the frequency where the amplifiers gain becomes unity.

Offset Voltage (VIO)

The amplifiers output will be zero when the voltage difference between the inverting and the non-inverting inputs is zero, the same or when both inputs are grounded. Real op-amps have some amount of output offset voltage.

Inverting Amplifier

The input signal vi is applied to the inverting input terminal through resistor R1 and non-inverting input terminal is grounded. The feedback from the output of the inverting terminal is provided through the feedback resistor Rf.

Since the input is applied to the inverting terminal vo and vi are opposite in polarity and hence the feedback is negative. Since the non-inverting input terminal is grounded v2=0. Due to virtual short at the input of op-amp the inverting and non-inverting input terminals are at the same potential.

Therefore, v1 = v2 = 0.

Due to high input impedance of Op-amp the current flowing into its inverting input terminal is zero. Therefore, the same current flows through R1 and Rf .

i 1 = i f

But i1 = vi – v1 / R1 = vi/R1

i f = v1 – vo / Rf = -vo/Rf

vi/R1 = - vo/Rf

Af = vo/vi = -Rf/R1

Af is the closed loop voltage gain or voltage gain with negative feedback.

Problem :

A 200mV peak to peak sine waveform voltage is applied to Op-amp inverting amplifier with Rf/R1 = 10. Sketch the output.

Solution

Peak to peak input voltage

2Vm = 200 mV

Vm = 100 mV

Vi= Vm sin wt = 100 sin wt mV

Vo = -Rf/R1 x vi = -10 x 100 sinwt mV

= -1000 sin wt mV

The input and output voltages are shown below.

Non-inverting amplifier

Non- Inverting amplifier is one in which the output is in phase with respect to input that is if you apply a positive voltage, output will be positive. The output is an non -inverted amplified version of input.

Assuming the op-amp is ideal and applying the concept of virtual short, the voltage at the inverting terminal is equal to non- inverting terminal.

Applying KCL at inverting node we get

Vi -Vo/ R2 + Vo – 0 / R1 = 0

By rearranging the terms, we will get

Voltage gain Av = Vo/ Vi = (1+ Rf/Ri)

Gain of non- inverting amplifier Av= (1+ Rf/Ri).

Problem

Design a non-inverting amplifier using Op-amp with a closed loop voltage gain of 10.

Af = 1 + Rf/R1 =10

Rf/R1 = 9

Rf = 9 R1

If R1 = 1K

Rf = 9K Ω

In an op-amp inverting amplifier R1 = 1K Ω and Rf = 100KΩ. The DC supply voltage of the op-amp is ± 15V. Calculate the output voltage if input voltage is 1V.

R1 = 1KΩ Rf = 100KΩ

V+ = 15V and V-=-15V. vi = 1V

Af = -Rf/R1 = -100K/1K = -100

Af = vo/vi

Vo = Af vi

= -100 x 1V = -100V

The output voltage cannot exceed the DC power supply voltage . Since vo is negative and large it is limited to V-

Vo ≈ V- = -15V

The differential amplifier amplifies the difference between two voltages making this type of operational amplifier circuit a Subtractor This type of operational amplifier circuit is commonly known as a Differential Amplifier.

By connecting each input in turn to 0v ground we can use superposition to solve for the output voltage Vout. Then the transfer function for a Differential Amplifier circuit is given as:

I1 = V1-Va/R1 ; I2 = V2-Vb/R2 ; If = Va –(Vout)/R3

Summing Point Va =Vb

Vb = V2(R4/R2+R4)

If V2 =0 then Vout = -V1 (R3/R1)

If V1=0 then Vout = V2(R4/R2+R4)(R1+R3/R1)

Vout = -Vout (a) + Vout(b)

Vout = -V1(R3/R1) + V2(R4/R2+R4)(R1+R3/R1)

When resistors, R1 = R2 and R3 = R4 the above transfer function for the differential amplifier can be simplified to the following expression:

Vout = (V2 -V1)

The output is the difference between the two inputs hence, the op-amp can be used as subtractor.

The Instrumentation amplifiers consist of three op-amps. In this circuit, a non-inverting amplifier is connected to each input of the differential amplifier.

This instrumentation amplifier provides high input impedance for exact measurement of input data from transducers. The circuit diagram of an instrumentation amplifier is as shown in the figure below.

The op-amps 1 & 2 are non-inverting amplifiers and together form an input stage of the instrumentation amplifier. The op-amp 3 is a difference amplifier that forms the output stage of the instrumentation amplifier.

Working

The output stage of the instrumentation amplifier is a difference amplifier, whose output Vout is the amplified difference of the input signals applied to its input terminals.

If the outputs of op-amp 1 and op-amp 2 are Vo1 and Vo2 respectively, then the output of the difference amplifier is given by,

Vout = (R3/R2)(Vo1-Vo2)

The expressions for Vo1 and Vo2 can be found in terms of the input voltages and resistances.

Consider the input stage of the instrumentation amplifier as shown in the figure below.

The potential at node A is the input voltage V1. Hence the potential at node B is also V1, from the virtual short concept. Thus, the potential at node G is also V1.

The potential at node D is the input voltage V2. Hence the potential at node C is also V2, from the virtual short. Thus, the potential at node H is also V2.

Ideally the current to the input stage op-amps is zero. Therefore, the current I through the resistors R1, Rgain and R1 remains the same.

Applying Ohm’s law between the nodes E and F,

I = (Vo1-Vo2)/(R1+Rgain+R1) ——————— 1

I = (Vo1-Vo2)/(2R1+Rgain)

Since no current is flowing to the input of the op-amps 1 & 2, the current I between the nodes G and H can be given as,

I = (VG-VH)/Rgain = (V1-V2)/Rgain ————————- 2

Equating equations 1 and 2,

(Vo1-Vo2)/(2R1+Rgain) = (V1-V2)/Rgain

(Vo1-Vo2) = (2R1+Rgain)(V1-V2)/Rgain —————— 3

The output of the difference amplifier is given as,

Vout = (R3/R2) (Vo1-Vo2)

Therefore, (Vo1 – Vo2) = (R2/R3)Vout

Substituting (Vo1 – Vo2) value in the equation 3, we get

(R2/R3)Vout = (2R1+Rgain)(V1-V2)/Rgain

i.e. Vout = (R3/R2){(2R1+Rgain)/Rgain}(V1-V2)

The above equation gives the output voltage of an instrumentation amplifier. The overall gain of the amplifier is given by the term (R3/R2){(2R1+Rgain)/Rgain}.

Active filter

Active filters are the electronic circuits, which consist of active element like op-amp(s) along with passive elements like resistor(s) and capacitor(s).

Active filters are mainly classified into the following four types based on the band of frequencies that they are allowing and / or rejecting −

Active Low Pass Filter

If an active filter allows or passes only low frequency components and rejects or blocks all other high frequency components, then it is called as an active low pass filter.

The electric network, which is connected to the non-inverting terminal of an op-amp is a passive low pass filter. So, the input of a non-inverting terminal of an op-amp is the output of a passive low pass filter.

It produces an output which is (1+Rf/R1) times the input present at the non-inverting terminal.

Active High Pass Filter

If an active filter allows or passes only high frequency components and rejects or blocks all other low frequency components, then it is called an active high pass filter.

So, the input of a non-inverting terminal of op-amp is the output of passive high pass filter.

The circuit resembles a non-inverting amplifier having the output of a passive high pass filter as an input to non-inverting terminal of op-amp. Hence, it produces an output, which is (1+Rf/R1) times the input present at its non-inverting terminal.

Active Band Pass Filter

If an active filter allows only one band of frequencies, then it is called as an active band pass filter. In general, this frequency band lies between low frequency range and high frequency range. So, active band pass filter rejects both low and high frequency components.

There are two parts in the circuit diagram of active band pass filter: The first part is an active high pass filter, while the second part is an active low pass filter.

The output of the active high pass filter is applied as an input of the active low pass filter. That means, both active high pass filter and active low pass filter are cascaded in order to obtain the output in such a way that it contains only a particular band of frequencies.

Active Band Stop filters

If an active filter rejects (blocks) a particular band of frequencies, then it is called as an active band stop filter. In general, this frequency band lies between low frequency range and high frequency range. So, active band stop filter allows passes both low and high frequency components.

The block diagram of an active band stop filter consists of two blocks in its first stage: an active low pass filter and an active high pass filter. The outputs of these two blocks are applied as inputs to the block that is present in the second stage. So, the summing amplifier produces an output, which is the amplified version of sum of the outputs of the active low pass filter and the active high pass filter.

P Control.

The Output power is directly proportional to control error. The higher the proportion coefficient, the less the output power at the same control error

Proportional control is mainly for fast-response systems with a large transmission coefficient. To adjust the proportional controller first set the maximum proportion coefficient where in the output power decreases to zero.

When the measured value is stabilized, set a specified value and gradually reduce the proportion coefficient and the control error will decrease.

If there are periodic oscillations in the system, the proportion coefficient should be increased so that control error is minimal periodic oscillations decrease to the limit.

PI Control.

The Output power equals to the sum of proportion and integration coefficients. The higher the proportion coefficient, the less the output power at the same control error. The higher the integration coefficient, the slower the accumulated integration coefficient.

PI control provides zero control error and and is insensitive to interference of the measurement channel. The PI control disadvantage is slow reaction to disturbances.

To adjust the PI controller first set the integration time equal to zero, and the maximum proportion time. Then by decreasing the coefficient of proportionality, achieve periodic oscillations in the system.

PID Control.

The Output power equals to the sum of three coefficients: proportional, integral and differential. The higher the proportion coefficient, the less the output power at the same control error.

The higher the integration coefficient, the slower the accumulated integration coefficient. The higher the differentiation coefficient, the greater the response of the system to the disturbance.

The PID controller is used in inertial systems with relatively low noise level of the measuring channel.

The advantage of PID is fast warm up time, accurate setpoint temperature control and fast reaction to disturbances.

Lag compensator

A lag compensator is one which adds a pole to the system and is used to improve the steady state response of the system. A lag compensator provides phase lag negative phase in the frequency response of system.

Using the concept of virtual ground, the op-amp lag compensator can be redrawn as shown in the figure

Using voltage divider principle, the transfer function of the system is

Vo/Vs =(1+s*C*R1)/(1+(R1+R2)*s*C),

hence the phase added to system is given as

Ω = tan-1(ω*C*R1) – tan-1(ω*C*(R1+R2))

Since R1<R1+R2 and tan-1(x) is increasing function (if a <b, tan-1(a)< tan1(b)) Ω is negative phase lag.

Lead compensator

A lead compensator is one which adds a zero the system and is used to improve the transient response of the system. A lead compensator provides phase lead positive phas) in the frequency response of system

Using the concept of virtual ground, the op-amp lag compensator can be redrawn as shown in the figure

Using voltage divider principle, the transfer function of the system is

Vo/Vs=(1+s*C*R1)/(1+(R1*R2/(R1+R2))*s*C),

hence the phase added to system is given as

Ω =tan-1(ω*C*R1) -tan-1(ω*C*R1*R2/(R1+R2)).

Since R1 > R1*R2/(R1+R2) Ω is positive(phase lead).

Since the non-inverting input terminal of op-amp is grounded v2=0. Due to virtual short at the input of op-amp the inverting and non-inverting input terminals are at the same potential

Therefore v1=v2=0

Due to high impedance of op-amp the current flowing into its input terminal is zero. Therefore the same current flows through R and C

That is

i1 = if -------------------(1)

where i1 = vi – v1 /R = vi -/R---------------------(2)

and if = C d/dt[v1 – v0] = -C dv0/dt --------------------(3)

Substituting (2) and (3) in (1) we get

vi = - C/R dv0/dt ------------------------(4)

dvo/dt = (-1/RC) vi ---------------------------- (5)

Integrating both sides of eq(5) we get

v0 = - 1/RC + v0(0) --------------------------(6)

where v0(0) is the initial voltage on the capacitor at t=0.

Note: v0(o) represents the constant of integration.

From eq(6) we find that the output voltage is proportional to the integral of the input voltage.

Therefore v0 = - 1/RC

Problem:

The integrator circuit as shown in the figure has R = 500K Ω and C=1µF. Find and plot the output voltage for the inputs as shown in the figure.

c)vi= 2 sin 4tV d) vi= 4tV

Solution:

We know that

v0 = - 1/RC + v0(0)

Here R= 500K Ω and C = 1µF

1/RC = 1/ 500 x 1000 x 1 x 10-6

= 1/RC = 2

= -2 ; vi =- 5V

= -2

= 10t V

Which means it is a ramp voltage with positive slope.

When vi=5V

-2 ; vi = 5V

= -2

= -10t V

Which means it is a ramp voltage with negative slope.

When vi= 2 sin4t V

Vo = -2

= -4 [-cos 4t] /4

= cos 4t

When vi=4t V

Vo =

= -8 [t2/2]

= - 4V

By utilizing an op-amp and few other external components, linear voltage regulator is built The same circuit is also a voltage stabilizer, able to stabilize voltage at a grade better than The transistor is used to supply the load with much more current than the op-amp itself could possibly supply.

The D1 diode is a Zener -type diode and it is used for voltage reference.

Basic op-amp voltage regulator

D1 is biased through Rz. When correctly reserve biased, the Zener diode keeps the voltage across its leads close to the Zener breakdown voltage. The op-amp is used as a linear voltage amplifier.

Due to the high open loop voltage gain of the op-amp the op-amp remains in its linear region, the voltage difference between its inverting (V-) and non-inverting input (V+) is almost equal to zero.

In other words, the voltage at its non-inverting input, in respect to the ground, equals the voltage at its inverting input:

V-=V+ -------(1)

Equation (1) holds true for any op-amp working at its linear region (as an amplifier).

R1 and R2 form a voltage divider, and the voltage (V-) at their connection point is also given by the well- known voltage-divider formula:

V-=VL·R1/(R1+R2) ------ (2)

However, V+ is also equal to the Zener breakdown voltage (Vz), because the non-inverting input of the op-amp is directly connected to the cathode of the Zener diode.

V+=VZ -----------------(3)

After solving (1),(2) and (3), we get:

VL=VZ·(1+R2/R1) --------------(4)

From equation (4), we conclude that VL voltage which is the voltage applied to the load is directly proportional to the Zener voltage.

As long as Zener voltage remains stable, VL also remains stable. Additionally, the voltage applied to the load, can be easily adjusted by adjusting R1, R2 or both. For continues voltage adjustment, R1 and R2 can be replaced by a potentiometer, having its wiper at the non-inverting input of the op-amp, and its other leads at the ground and the VL line, respectively.

Wien bridge oscillator is an audio frequency sine wave oscillator of high stability and simplicity. The feedback signal in this circuit is connected to the non-inverting input terminal so that the op-amp is working as a non-inverting amplifier.

The feedback network does not provide any phase shift. The circuit can be viewed as a Wien bridge with a series combination of R1 and C1 in one arm and parallel combination of R2 and C2 in the adjoining arm. Resistors R3 and R4 are connected in the remaining two arms.

The condition of zero phase shift around the circuit is achieved by balancing the bridge. The series and parallel combination of RC network form a lead-lag circuit.

At high frequencies, the reactance of capacitor C1 and C2 approaches zero. This causes C1 and C2 appears short. Here, capacitor C2 shorts the resistor R2. Hence, the output voltage Vo will be zero since output is taken across R2 and C2 combination. So, at high frequencies, circuit acts as a 'lag circuit'.

At low frequencies, both capacitors act as open because capacitor offers very high reactance. Again, output voltage will be zero because the input signal is dropped across the R1 and C1 combination. Here, the circuit acts like a 'lead circuit'.

But at one particular frequency between the two extremes, the output voltage reaches to the maximum value. At this frequency only, resistance value becomes equal to capacitive reactance and gives maximum output. Hence, this frequency is known as oscillating frequency (f).

Consider the feedback circuit, on applying voltage divider rule

Vf(s) = Vo(s) x Zp(s)/ Zp(s) + Zs(s)

Zs(s) = R1 + 1/sC1 and Zp(s) = R2|| 1/sC2

Let R1=R2=R and C1=C2=C . On solving

β = Vf(s)/ Vo(s) = RsC /(RsC) 2 + 3RsC + 1 ---------------------------(1)

Since the op-amp is operated in the non-inverting configuration the voltage gain

Av = Vo(s)/ Vf(s) = 1 + R3/R4 -------------------(2)

Applying the condition for sustained oscillations, = Av β =1

RsC /(RsC) 2 + 3RsC + 1. 1 + R3/R4

S=jw

(1 + R3/R4) ( jwRC/ - R2 C2 w2 + 3 jwRC + 1) =1

jw RC (1 + R3/R4)= (- R2 C2 w2 + 3 jwRC + 1)

jw[(1 + R3/R4)RC – 3RC] = 1- R2 C2 w2

To obtain the frequency of oscillation equate the real part to zero

1- R2 C2 w2 = 0

w = 1/RC

f = 1/ 2 π RC

To obtain the condition for gain at the frequency of oscillation equate the imaginary part to zero.

jw[(1 + R3/R4)RC – 3RC] = 0

jw[(1 + R3/R4)RC= jw3RC

[(1 + R3/R4) =3

R3/R4 =2

Therefore R3 = 2 R4 is the required condition.

Phase Shift oscillator

In an RC Oscillator circuit, the input is shifted 180o through the feedback circuit returning the signal out-of-phase and 180o again through an inverting amplifier stage to produces the required positive feedback.

This then gives us “180o + 180o = 360o” of phase shift which is effectively the same as 0o, thereby giving us the required positive feedback.

In other words, the total phase shift of the feedback loop should be “0” or any multiple of 360o to obtain the same effect.

The circuit on the left shows a single resistor-capacitor network whose output voltage “leads” the input voltage by some angle less than 90o.

In a pure or ideal single-pole RC network. it would produce a maximum phase shift of exactly 90o, and because 180o of phase shift is required for oscillation, at least two single-poles networks must be used within an RC oscillator design.

However, in reality it is difficult to obtain exactly 90o of phase shift for each RC stage so we must therefore use more RC stages cascaded together to obtain the required value at the oscillation frequency.

The amount of actual phase shift in the circuit depends upon the values of the resistor (R) and the capacitor (C), at the chosen frequency of oscillations with the phase angle ( φ ) being given as:

Xc = 1/2π fc R=R

Z = [ R 2 + Xc 2 ] ½

Ø = tan -1 Xc /R

Analog to digital converter samples the analog signal on each falling or rising edge of sample clock. In each cycle, the ADC gets of the analog signal, measures and converts it into a digital value. The ADC converts the output data into a series of digital values by approximates the signal with fixed precision.

In ADCs, two factors determine the accuracy of the digital value that captures the original analog signal. These are quantization level or bit rate and sampling rate.

The figure depicts how analog to digital conversion takes place. Bit rate decides the resolution of digitized output

The figure displays 3-bit ADC is used for converting analog signal.

Assume that one volt signal has to be converted from digital by using 3-bit ADC as shown below. Therefore, a total of 2^3=8 divisions are available for producing 1V output. This results 1/8=0.125V is called as minimum change or quantization level represented for each division as 000 for 0V, 001 for 0.125, and likewise upto 111 for 1V. If we increase the bit rates like 6, 8, 12, 14, 16, etc. we will get a better precision of the signal. Thus, bit rate or quantization gives the smallest output change in the analog signal value that results from a change in the digital representation.

References:

2. Electronic Devices and Circuits by VENKATA RAO, McGraw Hill

3. Microelectronic circuits by Smith

4. Analog Electronic circuits by A.P Godse

5. Analog Electronic circuits by J.B Gupta.