Circuits with Feedback and Sine Wave Oscillators - Coursework Example

Circuits with Feedback and Sine Wave Oscillators Task 1Q1. Feedback is the process by which information concerning the present or past tend to influence similar phenomenon either in future or at present. Given that the process entails a chain reaction of the cause and effect forming a loop or a circuit, such an event is best described to feed back to itself. According to Sterman (2004) feedback refers to information concerning a gap existing between the reference level and the actual level of some system of parameters often used in altering a gap in a certain way. Going by this definition, it means information by itself cannot be feedback unless it is translated into some action. It is synonymous to the feedback loop, which is often take to mean a complete causal path leading from an initial detection of a gap onto a subsequent modification of such a gap (Stuart, 2007). In this regard, diagrammatically, the feedback can be represented as shown Types of Feedback There are generally two types of feedback (Ashby, 2009). These are negative feedback and positive feedback. These two terminologies can be used in two different contexts. In the first context, the terminologies refer to the gap existing between the real parameter and the reference parameter. This is based upon whether such a gap widens, in this case, it is referred to as positive, or if the gap narrows, then it is referred to as negative. Whereas, in the second context, the effect or action altering the gap basing on involvement in the non-reward (negative) and reward positive. According to Arkalgud (2005) positive feed-back lead to an increase in the gain of an amplifier, whilst the negative feed-back causes a reduction in the gain of the amplifier. However, these two contexts appear confusing. For instance, in the first case, different authors use various terms substituting negative/positive with self-correcting/self reinforcing, balancing/reinforcing, degenerative/regenerative, and discrepancy-reducing/discrepancy-enhancing among others (Arkalgud, 2005). In the second case, different authors give out the description of effect or action as a negative or positive reinforcement instead of a feedback. However, this confusion comes about because feedback could be used for either motivational or informational reasons and contains both the quantitative and the qualitative components. Qualitative feedback explains how bad, good or indifferent, whereas quantitative explains how many or otherwise how much. The negative and positive feedback reduces and increases the amplifier’s gain. In negative feedback the current or voltage feedback is applied in order to reduce the input amplifier. This is also called the inverse or degenerative feedback. In positive feedback, the current or voltage feedback is applied in order to increase the voltage input. This is also referred to as direct or regenerative feedback (Stuart, 2007). On the other hand, negative feedback can be divided into voltage feedback and current feedback. In other cases both feedbacks may exist in a circuit. In this case, both current voltage would be feedback towards the input in parallel or series. These cases would be represented as series-voltage feedback, series-current feedback, shunt current feedback, and shunt- voltage feedback (Arkalgud, 2005). Whenever positive feedback is applied in a circuit, it takes a portion of the output signal back to the input that is non-inverting. This can be useful in a comparator with the hysteresis such as the Schmitt trigger. When a circuit lacks the positive feedback, then the open loop detectors’ response is slowed down. In this case, the overall positive external feedback could be applied though somewhat different from the internal feedback which could be set in the latter purpose-designed stages. This has a far reaching influence on the zero-crossing point of detection accuracy. For instance, in a case of the general purpose op-amp, the square to sine wave converters; frequency would be below one hundred hatz. On the other hand, negative feedback will make a gain in a circuit to be stable. In this case, mathematically, it can be shown that A1= A/1+AB. If AB>> 1with the expression reducing to; A1=1/B. This means that the circuit is independent from the internal gain. In a bandwidth, negative feedback makes the gain to reduce and the bandwidth product is left constant. The bandwidth will increase in order to compensate the gain reduction. Negative feedback in a circuit makes the bandwidth to increase hence improving the frequency response of a circuit. One example of a negative feedback application in a circuit is in a non-inverting amplifier. In this amplifier, the voltage in the output is altered in the similar direction as that in the input. In this circuit the negative feedback makes the V_ be a function of Vout via the R1R2 networks. TheR2 andR2 creates a voltage divider and V_ is an input high-impedence and fails to load it appreciably. As outlined, the negative feedback is the incredibly a useful principle applying it to an operation amplifiers. It allows the creation of the practical circuits given that it can precisely set rates, gains, alongside other significant parameters. The negative feedback can make the circuits self-correcting and stable. The fundamental principle of the negative feedback is that such output drives in such a direction creating an equilibrium condition. For an op-amp circuit without a feedback, there is lacking a corrective mechanism. The output voltage shall become saturated with a tiniest amount of the differential voltage that is applied in the inputs. Task 1Q2. A circuit with negative feedback. For one to design a circuit having negative feedback, it would be vital to consider the ideal closed gain loop and the loop gain concepts. In a negative feedback, the overall gain is always given by A1 = . In this case, the voltage gain will be reducing by factors of (). This can be represented by the voltage series feedback shown in diagram 1. Diagram 1. Without any feedback the amplifier would have a voltage gain that is given by; A=V0/VS=V0/V1. Connecting feedback signal V1 in series with the input gives; V1=VS-V1 as V0=AV1=A (VA-V1) =AVS-AV1=AVS-A(V0). This means that (1 + A)V0=AVS). The voltage gain due to the negative feedback in the circuit would given as; A1 =V0/V1=A/ (1+A), thus a negative feedback circuit. In the designed circuit, the gain in the circuit without feedback is AV. After adding an output voltage’s fraction back to the input voltage, the input to the circuit is given by the difference between signal voltage and the feedback voltage. This makes the output to be the product of voltage input and the gain in the circuit. In this respect, the negative voltage feedback in the circuit reduces the gain by a factor. In this circuit, the current gain remains unaffected. The designed circuit is a simple negative feedback amplifier. This involves an amplifier that combines an output fraction with the input fraction for the original signal to be opposed by the negative feedback. The negative feedback that is applied betters the performance of the amplifier in terms of linearity, stability gain, response of frequency, and the step response. The negative effect also reduces the parameter variations sensitivity due to the environment or the manufacturing. This advantage makes negative feedback be a useful aspect in many control systems and amplifiers. Task 1 Q3. The Effects of applying the feedback to a single and multi-stage circuits. It is worth noting that feedback amplifier are ones in which the fraction of output energy get fed back to an input of that same circuit (Skinner, 1957). It becomes positive in cases where the feedback signal becomes in phase with that of the input signal and the signal, thus becomes an additive. However, for the feedback signal that is 180 degrees out of phase with respect to that of the input signal, it becomes a negative feedback. The negative feedback amplifiers find a lot of application. In particular, in a multi-stage circuits, applying a feedback has a significant effect. First, it reduces the extremely high gain to a somewhat lower usable amount. Second, applying the feedback causes a reduction in the distortion. Third, the feedback in a multi-stage circuit causes an increase in the response of the upper frequency. It also cause an increase in the input impedance. Last but not least, applying a feedback in a multi-stage circuit leads to a reduction in the output impedance. Task 1 Q4. The circuit conditions and the methods used to achieve sinusoidal oscillation The term sinusoidal oscillator indicates that the oscillator yields sine wave output (Ashby, 2009). Depending on the variations in the output waveforms of the amplitude, there are two types of oscillations. These are damped (un sustained) and the undamped (sustained). For damped oscillations, the amplitude of the oscillations goes on increasing or decreasing continuously with time. In this case, the amplitude of oscillations that goes on decreasing is referred to as underdamped. On the other, for the amplitude of oscillations that goes on increasing continuously is referred to as the over damped. The undamped oscillations, refers to oscillations whose amplitude does remain constant over time (Stuart, 2007). The oscillator is composed of a feedback network and the amplifier. It is worthwhile noting that there are basic components needed to obtain oscillations. These are the active device, that is, the Op Amp or the Transistor, which is often used as an amplifier, and the Feedback Circuit consisting of passive components such as LC or RC combinations. An example of a simple sinusoidal circuit A circuit using a pulsed voltage source for purposes of generating a squire wave along with a filter is shown below. The 3.6864Mhz value was chosen given that it is regarded as the commonly available oscillator used for dividing nicely into some serial baud rates, and makes it excellent for purposes of serial communication. When few cycles are simulated, they provide waveforms as shown below. Simulating a few cycles of this provides the waveforms shown below, which ultimately verify a square wave alongside a resulting "sine", that tend to appear a little triangular at a glance. With a view to start the sinusoidal oscillations having constant amplitude, there is need to make sure positive feedback is achieved as a primary conditions. The other oscillator circuit conditions must be satisfied. These are often referred to as Barkhausen conditions. They include, first, the magnitude of loop gain (Aβ) has to be unity implying that both the gain of the feedback network (β) and the product of the gain of the amplifier (A) have to be unity. Secondly, the phase shift that is around a loop must be either 0 degrees or 360 degrees. This implies that the feedback network, as well as the phase shift across the amplifier must be 0 or 360 degrees. Considering practical Oscillator, it is worth noting that practically, obtaining sustained oscillations at certain desired frequency of oscillation, there are basic requirements that an oscillator circuit has to satisfy. As mentioned above, the circuit has to have a positive feedback and whenever a positive feedback is applied in the circuit, the circuit gain is often given by Af = A/(1- Aβ) In this case, the equation shows that where the value of ‘Aβ’ equals to 1, the overall gain goes to infinity implying that there is an output without an external input. In practice, getting sustained oscillations, during when the circuit gets turned on, loop gain has to be somewhat slightly greater than 1. In this case, it ensures the oscillations build up within the circuit. But, the moment the suitable level of the output voltage is approached, then the loop gain has to t decrease automatically to 1. It is only then that the circuit can maintains a sustained oscillation. If not, the circuit would still operate as over-damped. It can be achieved in a circuit either through decreasing amplifier gain (A) or by decreasing the feedback gain (β). Task 2 Q 2. Building and evaluating the sine wave oscillator to the given specification Oscillators are those circuits that often yield specific and periodic waveforms which are sinusoidal, squire, sawtooth or triangular. Generally, they use a form of active device, a crystal or a lamp surrounded by some passive devices such as inductors, resistors, and capacitors in generating the output. Oscillators fall in two classes: these are sinusoidal and the relaxation. Sinusoidal oscillators are composed of amplifiers along with the external components used in generating the oscillation, often created by an operational amplifier op amps. Whereas, the relaxation oscillators do generate sawtooth, nonsinuoidal triangular waveforms. The focus in this discussion is on a sine wave oscillators often made from an operational amplifier op amps. More often than not, sine wave oscillators, serve as test waveforms or references in many circuits. It is worth noting that a pure sine wave is characterised by a single frequency with no harmonic. This imply that a sine wave might be a an input to any circuit or device along with the output harmonics being measured for purposes of determining the extend of distortion. In order to provide a certain shape, waveforms in a relaxation oscillator, can be generated from a given sine wave. Requirement for Oscillation The simplest or Canonical form of the negative feedback system is useful in demonstrating requirements for the occurrence of an oscillation (Mayr, 2009). Fig 1 is a block diagram showing the system by which Vout indicate the output voltage, while Vin indicate the input voltage from an amplifier gain block ‘A’, whereas β is the signal referred to as the feedback factor fed back into the summing junction. The letter E, stands for the error term, which is always equal to the input voltage and the feedback factor summation. The classical expression for the feedback system can be derived mathematically as follows with equation 1 as a defining equation for output voltage while equation 2 as the error. Vout = E * A……………………………………………………………………….(1). E = Vin + βVout.........................................................................................................(2). When the error term, is eliminated from the equation, the equation becomes Vout/A = Vin – βVout..................................................................................................(3). After collecting the like terms, the equation becomes Vin = Vout(1/A + β)....................................................................................................(4). When the equation is rearranged, the terms yield equation 5, which is a classical form of the feedback expression. Vout/Vin= A/1+Aβ...................................................................................................(5). From this, it is clear that oscillator do not need the externally applied input signal but they use a fraction of an output signal often created by a feedback network as an input signal. Additionally, oscillation occur whenever some feedback system cannot find the stable steady state following the failure of a transfer function to be satisfied and the system becomes unstable whenever the denominator part of equation 5 reduces to zero. This implies that the key to design the oscillator is to make sure Aβ = -1. Satisfying such a condition demand that magnitude of a loop gain becomes a unity and the corresponding phase shift has to be 180 degrees. For the negative feedback system, the expression becomes Aβ = 1 Read More