Analogue and digital electronics theory

Operational amplifiers began in the days of vaccum tubes and analog computers. They were designed such that the characterstics of the overal amplifier were largely depended on the type and amount of feedback. Thus, the complex differential amplifier itself beacme the building block that was used in many mathematical operations. They continued to evolve through the transistor era and continued to decrease in size and increase in performance.Finally in mid – 1960s a complete op-amp was integrated.Since then it has continued to improve dramatically in performance including parameters such as higher operating volatges , lower current requirents, higher current cpabilities, more tolerance to abuse, lower noise, greater stability, greater power output, higher inout impedence and higher frequencies of operations.Despite all this, the high

performance intergrated operational amplifier of today is still based on the funndamental differential ampliffier.

Differential Voltage amplilfier A differential amplifier is a type of a electronic amplifier that amplifies the difference between two input voltages but supresses any voltage common to the two inputs. In some applications, one of the differential input is connected to ground and the signal to be amplified is applied directly to the remaining input then the amplifier responds to the difference between the two inputs , but the output will be in or out of phase with the input signal depending on the which input the voltage is applied.If the input in given to (+) input then it is known as non-inverting terminal and if given to the (-) input then it is known as inverting terminal.

Figure:1 This is the schematic diagram of the op amp.

 Operation Ideally, the op amp amplifies only the difference in voltage between the two which is know as differential

input voltage and the output of the op-amp Vout is given by the equation Vout=AOL(V+-V-) Where AOL is the open loop again of the amplifier Open-loop Amplifier The term open-loop refers to the absence of a feedback look from the output to the input When a positive feedback is given to the op-amp. It acts as a regenerator and when a negative feedback is given it acts as the comparator.

 Closed loop Amplifier

The closed loop feeedback greatly reduces the gain of the circuit.If predictable opertion is desied , negative feedback is used by applying a portion of the output voltage to the inverting input.When enegative feedback is used, the circuits overall gin nd responses become determined mostly by the feedback network rathe than by the op-amp characterstics.

Figure 2: Closed loop negative feedback amplifier Inside the IC

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The below diagram emphasises on the fact that the op-amp is essentially an encapsulate circuit composed of familiar components like transistors, resistors and a single capacitor. It shows the type of circuit driving the output of the circuit. It also shows the differential inputs of the op-amp.

Figure 3: Internal circuitary of the Intergated circuit of an LM741 op-amp  Characteristics of an ideal OP-AMP

An ideal op-amp has perfect conditions to allow it to function with 100% efficiency.In practise an ideal opamp dosent exist but we try to get the op-amp as close as possible to the ideal characterstics to get a better opamp with the maximum efficiency.

 Differential voltage gain:it is the amount of amplification given to the volatge appearing between the input terminals.It is infinity for the ideal op-amp. V0=VrAv  Bandwith: It refers to the range of frequencies that can be amplified by the op-amp.The bandwidth of the ideal op-amp is infinite.  Slew Rate :The output of an ideal op amp can change as quickly as the input voltaged changes in order to reproduxe the inut waveform . it is defined as the rate of change of voltage on the output. The slew rate of an ideal an op-amp is infinite.  Input impedance:The input impedence can be represwnted by an internaal resistance between the input terminals.An ideal op-amp has infinte input immpedance.  Output impedence : the output impedence of an ideal op-amp is zero.  CMRR: The common mode rejection ratio (CMRR) of an op-amp is a metric used to quantify the ability of the device to reject common-mode signals, i.e., those that appear simultaneously and in-phase on both inputs. An ideal differential amplifier would have infinite CMRR.

Objective:1

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a. Common mode rejection ratio When the same input voltage is applied to both input terminals of an op-amp, the op-amp is said to be operating in a common mode configuration. Since the input voltage applied is common to both the inputs, it is referred to as a common-mode voltage Vcm. A common – mode voltage Vcm can be ac, dc or a combination of ac and dc. Because ideally an op-amp amplifies only differential input voltages no common-mode output voltage Vocm should appear at the output. However, due to imperfections within an actual op-amp some common-mode voltage Vocm will appear at the output. The amplitude of this Vcm is very small and often insignificant compared to Vcm. Therefore, in practice the ratio of the output common-mode voltage Vocm to the input commonmode voltage Vcm which is called the common mode voltage gain Acm, is generally much smaller than 1. In equation form, Acm = Ideally, the common mode voltage gain Acm is zero.

CMRR can be computed by CMRR = Where Ad = differential gain Acm = common mode gain

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Figure 4: Circuit to measure CMRR

Figure: 5: CMRR vs frequency graph

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figure 6: Measurement of CMRR

b. Slew Rate The slew rate is the maximum rate of change of output voltage with respect to time, usually specified in V/µs. Generally, the slew rate is specified for unity gain and is measured by applying a step input (dc) voltage.

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Figure7: a) circuit to measure slew rate b) step input voltage and the resulting output voltage

Where Vout(t) is the output produced by the amplifier as a function of time t.

Figure 8: Measurement of slew rate

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Applications Op amps are used in a wide variety of applications in electronics. Operational amplifiers are popular building blocks in electronic circuits and they find applications in most of the consumer and industrial electronic systems. Opamps can be configured to work as different types of signal amplifiers like inverting, non-inverting, differential, summing, etc. as well as it is used to perform mathematical operations like addition, subtraction, multiplication, division, differentiation and integration. Operational amplifiers can be used in construction of active filters, providing high-pass, low-pass, band-pass, band-reject and delay functions, as comparators, oscillators and also voltage or current sources. They are also used in signal processing.  Linear applications Linear applications are those applications in which output of op-amp follows the input. Some of the applications are difference amplifier, summing amplifier, instrumentation amplifier, voltage to current convertor and, voltage amplifiers performed by linear circuits.  Non-linear applications An op amp is operating in its non-linear range when the output of the op amp is not directly proportional to the input.

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B. Objective 2 a) Integrator

The integrator is one of the fundamental circuits studied in basic electronics and its op-amp counterpart is also an important circuit for many signal processing applications. The integrator performs the mathematical operation of integration with respect to time i.e. its output voltage is proportional to the input voltage integrated over time. An integrator produces an output voltage that is proportional to both the duration and amplitude of an input signal. For example, if the input were a pulse waveform, then the output would be a voltage that was proportional to the amplitude and pulse width of the input signal. In essence the integrator computes the area( height x width) of the input signal. This corresponds to the mathematical operation in calculus called integration. The integrator may also alter the shape of the input waveform. For example, a square wave will be converted to a triangular wave in the process of being integrated. When integrated, a triangular wave will produce a parabolic waveform that very closely approximates a sinewave. In case of a sinewave input, the output will still be sinusoidal but may be shifted in phase and reduced in amplitude. For sinewave inputs, the integrator acts as a simple low-pass filter.

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 Operation If we mentally remove open capacitor C1, we will see that the circuit is a simple inverting amplifier circuits. Resistors R1 and R2 determine the voltage gain of the circuit and resistors R3 is to minimize the effects of bias current. The (-) input of the op-amp being a virtual ground point. Let us assume a step voltage applied to the input terminal and the flow of the current through R1 continues through R2 and C1. Now the change in the voltage depends on how fast the capacitor C1 can take charge on. Since the voltage across R2 does not change instantly, neither can the current, when the input voltage first makes a change in amplitude the current resulting from this voltage change is routed totally through C1. As the capacitor accumulates a charge the current through R2 begins to change. The circuit, however, is designed to ensure that the current through R2 is never allowed to be a substantial part of the capacitor current. If the input voltage is constant, the capacitor current is constant. If the input returns to its original state before the capacitor has had time to accumulate excessive voltage, this discussion is valid. With a constant current through the capacitor, we will generate a linear ramp of voltage across it (and

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therefore at the output of the op amp). With a square- wave input, the capacitor will periodically charge and discharge with equal, but opposite polarity, currents. This, of course, produces a triangle wave at the output. Resistor R 2 is included in the circuit to reduce the gain at low frequencies (DC in particular). Without R2, the bias currents (small as they are) would eventually charge C1 and cause an undesired DC offset in the output. This offset may even cause the amplifier to go into saturation. In terms of AC circuit theory, we ensure that the reactance of C1 is less than 10 percent of the value of R 2 at the lowest frequency of operation. This makes certain that most of the current will be used to charge and discharge C~.

 Derivation of the Input and output relation of the integrator

RI in

Cf

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Figure 9 : Integrator circuit By KCL, Iin IC = IC IC

Q = CV IC

So, the output voltage is the integration of the input voltage. And the terms represents the integration times of this integrator.

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B. Output waveforms for different input waveforms shown using multisim

Fig 1.2 The performance of the circuit when the input is triangular wave.

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Fig 1.3 The performance of the circuit when the input is square wave

Fig 1.3 The performance of the circuit when the input is sine wave

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c) Integrator as a Low pass filter The name low-pass circuit is because it passes low frequencies readily attenuates high frequencies. The attenuation of high frequency is due to the reactance of the capacitor which decreases with the increasing frequency. As output is taken across the capacitor and reactance of a capacitor is inversely proportional to the frequency. At very high frequencies the capacitor acts as a short circuit and therefore the out drops to zero. When a sinusoidal input Vi is applied to it then the output Vo is given by:

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Output is a frequency dependent quantity. At frequencies, -JXc<<R, under this condition, expression becomes almost zero i.e. Vo = 0

At low frequencies, when -JXc >> R Vo = Vi Thus, the integrator circuit attenuated signals of high frequencies and do not affect signals of low frequencies making it a low pass filter. By changing the input signal to that of a sine wave of varying frequency the Op-amp Integrator begins to behave more like an active Low Pass Filter , passing low frequency signals while attenuating the high frequencies. When the input frequency is 50Hz

Fig 1.4 When the input frequency is 100 Hz

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Fig 1.5

When the input frequency is 150Hz

When the input frequency is 200Hz

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Input frequency (Hz) Gain (dB) Output voltage 50Hz -9.617 100Hz -15.395 150Hz -18.843 200Hz -21.453

4. Objective 3

A. Oscillator

Oscillator is to generate alternating current or voltage waveforms. More precisely, an oscillator is a circuit that generates a repetitive waveform of fixed amplitude and frequency without any external input signal. They are used in radio, television, computers, and communications.

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Oscillator principle An oscillator is a type of feedback amplifier in which part of the output is fed back to the input via feedback circuit. If the signal feedback is of proper magnitude and phase the circuit produces alternating currents or voltages.

Figure 10: Schematic diagram of the basic oscillator

This diagram looks identical to that of the feedback amplifiers but here the input voltage is zero (vin=0) and the feedback is positive because most oscillators use positive feedback. Finally, the closedloop gain of the amplifier is denoted by Av rather than Af. From the block diagram Vi = Vf + Vs Vo=AvVi Vf=BVo Vs = 1-AvB Using the above relationships, the following equation is obtained: =

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However, Vs = 0 and Vo ≠ 0 implies that AvB = 1 Expressed in polar form. AvB = 1

The above equation gives the two requirements for oscillation:  The magnitude of the loop gain should be equal to 1  The total phase shift of the loop gain must be equal to 0 or 360 . This is also known as barkhausen criteria.

Operation

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Figure 10

Oscillator Feedback let us consider the feedback circuit shown above. In the figure Vin is the voltage of ac input driving the input terminals B-C of an amplifier having voltage gain A.

The amplified output voltage is

Vout = A Vin

This voltage drives a feedback circuit that is usually a resonant circuit, as we get maximum feedback at one frequency. The feedback voltage returning to point a is given by equation

Vf = A β Vin

where β is the gain of feedback network

If the phase shift through the amplifier and feedback circuit is zero, then A β Vin is in phase with the input signal Vin that drives the input terminals of the amplifier.

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Figure 11

Now we connect point ‘a’ to point ‘b’ and simultaneously remove voltage source Vin, then feedback voltage A β Vin drives the input terminals b c of the amplifier, as shown in fig. (b) above. CASES:  Aβ < 1 (or) Aβ < Vin - The signal will die out.  Aβ > 1 – The output signal will build up.  Aβ = 1 – The output signal is a steady sine wave.

Figure 12

There is no need of an input signal for the initiation of oscillations. To obtain a selfsustained oscillation, the condition β A = 1 must be satisfied. The value of β A is made greater than unity. As a result, the system starts oscillating by amplifying noise voltage which is always present. An average value of β A of 1 can be obtained by the saturation factors in the practical circuits. The waveforms that are obtained will not

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be exactly sinusoidal. If the value of β A is closer to the value 1, the waveform becomes more sinusoidal. The figure above shows how the noise voltage results in a build-up of a steady state oscillation condition.

At the resonant frequency, the phase shift around the loop is made 0° by deliberate design. The phase shift is different from 0° above and below the resonant frequency. Thus, the resonant frequency of the feedback circuit will be the only frequency where the oscillations will be obtained.

B. COLPITTS Oscillator Oscillator is an amplifier with the positive feedback and it converts DC input signal into AC output waveform with certain variable frequency drive and certain shape of output waveform (like sine wave or square wave, etc) by using the positive feedback instead of input signal. Oscillators which utilizes the inductor L and capacitor C in their circuit are called as LC oscillator which is a type of linear oscillator. Colpitts Oscillator Theory It consists of a tank circuit which is an LC resonance sub circuit made of two series capacitors connected in parallel to an inductor and frequency of oscillations can be determined by using the values of these capacitors and inductor of the tank circuit.

This oscillator is almost similar to Hartley oscillator in all aspects; hence, it is termed as electrical dual of Hartley oscillator and is designed for the generation of high frequency sinusoidal oscillations with the radio frequencies typically ranging from 10 KHz to 300MHz. The major difference between these two oscillators is that it uses tapped capacitance, whereas the Hartley oscillator uses tapped inductance.

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Colpitts Oscillator Working Whenever power supply is switched on, the capacitors C1 and C2 shown in the above circuit start charging and after the capacitors get fully charged, the capacitors starts discharging through the inductor L1 in the circuit causing damped harmonic oscillations in the tank circuit.

Figure 13: Tank Circuit with Capacitors and Inductors Thus, an AC voltage is produced across C1 & C2 by the oscillatory current in the tank circuit. While these capacitors get fully discharged, the electrostatic energy stored in the capacitors get transferred in the form of magnetic flux to the inductor and thus inductor gets charged.

Similarly, when the inductor starts discharging, the capacitors start charging again and this process of energy charging and discharging capacitors and inductor continues causing the generation of oscillations and the frequency of these oscillations can be determined by using the resonant frequency of the tank circuit consisting of inductor and capacitors. This tank circuit is considered as the energy reservoir or energy storage. This is because of frequent energy charging and discharging of the inductor, capacitors that part of LC network forming the tank circuit.

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The continuous undamped oscillations can be obtained from the Barkhausen criterion. For sustained oscillations, the total phase shift must be 360 or 0 . In the above circuit as two capacitors C1 & C2 are centre tapped and grounded, the voltage across capacitor C2 (feedback voltage) is 1800 with the voltage across capacitor C1 (output voltage). The common emitter transistor produces 180 phase shift between the input and output voltage. Thus, from the Barkhausen criterion we can get undamped continuous oscillations. The resonant frequency is given by

ƒr=1/(2П√(L1*C)) Where ƒr is the resonant frequency

C is the equivalent capacitance of series combination of C1 and C2 of the tank circuit

It is given as

C=(C1*C2)/((C1+C2)) L1 represents the self-inductance of the coil.