Second order equation damping factor amplifier

This invention relates generally to second order systems and, more particularly, transducers such as gyroscopes of the type employed in modern aircraft, guided missiles, and the like. It is particularly directed to the provision of electronic signal compensation for temperature caused variation of rate gyroscope damping. Precision instruments of various kinds require damping in order to perform accurately, particularly gyroscopes, accelerometers, and other transducers wherein the movement of masses is utilized to measure velocity and other functions. A good example of the problem is found in the field of gyroscopes mounted within housings filled with oil or other liquid.

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Content:

• Introduction: System Analysis
• Amplifier Output Impedance (Damping Factor) and Speakers
• Damping Factor and Why it isn’t Much of a Factor
• What is Damping Ratio : Derivation & Its Cases
• Stability - Electronic Engineering (MCQ) questions & answers
• Step response
• Earthquake Instrumentation
• Damping Control - Powersoft Audio
• Amplifiers

WATCH RELATED VIDEO: Damping factor of guitar amps: Can you hear it?

Introduction: System Analysis

A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. First, focus on the variables that cannot change abruptly; capacitor voltage and inductor current.

Example: The switch in Fig. We are going to find:. Thus, the inductor acts like a short circuit, while the capacitor acts like an open circuit. The inductor acts like a short circuit and the capacitor like an open circuit. Thus, we have. A series RLC circuit is shown in Fig. The circuit is being excited by the energy initially stored in the capacitor and inductor.

A more compact way of expressing the roots is. The three elements in parallel have the same voltage across. According to the passive sign convention, the current through each element is leaving the top node. Applying KCL at the top node, taking the derivative with respect to t and dividing by C results in. The solution has two components: the transient response v t t and the steady-state response v ss t ; that is,.

In the circuit in Fig. Hence the steady-state response is. Again, the complete solution consists of the transient response i t t and the steady-state response i ss t ; that is,. Study Guides. Second-Order Circuits A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements Finding Initial and Final Values First, focus on the variables that cannot change abruptly; capacitor voltage and inductor current.

There are two key points to keep in mind in determining the initial conditions. Carefully handle the polarity of voltage across the capacitor and the direction of the current through the inductor; and are defined strictly according to the passive sign convention. The capacitor voltage is always continuous so that and the inductor current is always continuous so that , where denotes the time just before switching and is the time just after, assuming that the switching takes place at.

The result is Thus, c. Figure 3: A source-free series RLC circuit The energy is represented by the initial capacitor voltage and initial inductor current. Applying KVL around the loop and differentiating with respect to t , This is a second-order differential equation. The solution is of the form and substituting this to the DE, the characteristic equation is where are the two roots of the characteristic equation of the differential. Since there are two possible solutions from the two values of , A complete or total solution would therefore require a linear combination of and.

Thus the natural response of the series RLC circuit is , where the constants and are determined from initial values. If , the underdamped case; roots are complex. When this happens, both roots and are negative and real. The response is , which decays and approaches zero as t increases. The second-order differential equation becomes Solving the DE gives the natural response of the critically damped circuit: a sum of a negative exponential and a negative exponential multiplied by a linear term,.

A typical critically damped response is shown in Fig. It is a sketch of , which reaches a maximum value of at , one time constant, and then decays all the way to zero. The roots may be written as , where and which is called the damped frequency. Both and are natural frequencies because they help determine the natural response. Using Euler's identities, and replacing constants with constants , the natural response is.

The natural response for this case is exponentially damped and oscillatory in nature. It has a time constant of and a period of. Figure 6: Underdamped response. Assume initial inductor current and initial capacitor voltage , and. Figure 7: A source-free parallel RLC circuit The three elements in parallel have the same voltage across. In this case the roots are complex and may be expressed as.

The response is The constants in each case can be determined from the initial conditions: and. To find ,. Hence the steady-state response is Complete solutions for the three cases: Overdamped: Critically damped: Underdamped: and are, respectively, the voltage across C and the current through L. Once the capacitor voltage is known, can be obtained which is the same current through C, L, and R. Hence, the voltage across the resistor is , while the inductor voltage is.

In the circuit, the final value of the current through L is the same as the source current. Complete solutions for the three cases Overdamped: Critically damped: Underdamped: These equations only apply for finding the inductor current. Once the inductor current is known, can be obtained which is the same voltage across C, L, and R. Hence, the current through the resistor is , while the capacitor current is.

Amplifier Output Impedance (Damping Factor) and Speakers

This program will show the effect that the speaker wire has on the effective damping factor. Input Section Amplifier Power Output? Email Home Page. Use F11 to go to full screen viewing if using Google Translate.

Damping Factor and Why it isn’t Much of a Factor

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What is Damping Ratio : Derivation & Its Cases

A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. First, focus on the variables that cannot change abruptly; capacitor voltage and inductor current. Example: The switch in Fig.

Stability - Electronic Engineering (MCQ) questions & answers

Apparatus for measuring the damping ratio of a second order system. The apparatus takes the form of a mass rate flow meter comprising an impact plate adapted to be inserted in a fluid flow. The plate and its mounting means forms a second order system which has a damping ratio proportional to the rate of mass flow. The plate is connected in a loop with an integrator, or a differentiator, and a multiplier. The integrator of differentiator provides, among other things, phase shifting. The loop gain is increased until the loop oscillates.

Step response

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I was wondering what would you classify the damping state under damped, over-damped, critically damped a second-order ode system with a negative damping ratio? To me it doesn't make much sense since a negative damping ratio results in an unstable system. I assume the G s which you give is the Laplace transform of the Green's function please indicate next time what your symbols mean otherwise, a question can be hard to understand.

Earthquake Instrumentation

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory , step response is the time behaviour of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.

Damping Control - Powersoft Audio

RELATED VIDEO: What is the damping factor of an amplifier?

Damping is the power on or to prevent or reduce its oscillation in an oscillatory system. So, in a physical system, the generation of damping can be done through the process that dissolves the stored energy within the oscillation. The best examples are resistance within electronic oscillators , viscous drag within mechanical systems, light absorption as well as scattering in optical oscillators. This article discusses an overview of the damping ratio and its derivation. A damping ratio definition is a dimensionless measure used to describe how oscillations within a system can decompose once a disturbance occurs is known as the damping ratio. The behavior of oscillatory can be exhibited by many systems once they are worried about their location of stationary equilibrium.

Amplifiers

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Damping of the oscillatory system is the effect of preventing or restraining or reducing its oscillations gradually with time. The damping ratio in physical systems is produced by the dissipation of stored energy in the oscillation. It is the restraining or decaying of vibratory motions like mechanical oscillations, noise, and alternating currents in electrical and electronic systems by dissipating energy.