sinusoidal steady state

Examples of sinusoidal steady state in the following topics:

• Phasors

  • Phasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
  • The sinusoidal function $A cos( \omega t + \phi)$ can be written as a phasor: $Ae^{j \theta}$ .
  • This can be particularly useful because the frequency factor (which includes the time-dependence of the sinusoid) is often common to all the components of a linear combination of sinusoids.
  • Sinusoidal Steady State and the Series RLC CircuitPhasors may be used to analyze the behavior of electrical and mechanical systems that have reached a kind of equilibrium called sinusoidal steady state.
  • In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency ω.

• Driven Oscillations and Resonance

  • This equation can be solved exactly for any driving force, using the solutions z(t) which satisfy the unforced equation: $\frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0$, and which can be expressed as damped sinusoidal oscillations $z(t) = A \mathrm{e}^{-\zeta \omega_0 t} \ \sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \varphi \right)$in the case where ζ ≤ 1.
  • In the case of a sinusoidal driving force: $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t)$, where $\!
  • \omega$ is the driving frequency for a sinusoidal driving mechanism.
  • The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude $\!
  • Steady state variation of amplitude with frequency and damping of a driven simple harmonic oscillator.

• Resistors in AC Circuits

  • It is the steady state of a constant-voltage circuit.
  • If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating-current circuit.
  • The voltage and current are sinusoidal and are in phase for a simple resistance circuit.

• Steady-State Approximation

  • The steady state approximation can be used to determine the overall rate law when the rate-determining step is unknown.
  • Both cases can be addressed by using what is known as the steady state approximation.
  • With the steady state assumption, we can write the following:
  • We had no knowledge of the rate-determining step, so we used the steady state approximation for our reaction intermediate, N2O2.
  • Simplify overall rate laws using the steady state approximation for reactions with various or unknown rate-limiting steps, explainting the steady state approximation and when it is valid

• Managing the Business Cycle

  • When the economy is not at a steady state, the government and monetary authorities have policy mechanisms to move the economy back to consistent growth.
  • When the economy is not at a steady state and instead is at a point of either overheating (growing to fast) or slowing, the government and monetary authorities have policy mechanisms, fiscal and monetary, respectively, at their disposal to help move the economy back to a steady state growth trajectory.

• Position, Velocity, and Acceleration as a Function of Time

  • This equation simply states that the acceleration of the waveform (Left: second derivative with respect to time) is proportional to the Laplacian (Right: second spatial derivative) of the same waveform.
  • Consider one of the most common waveforms, the sinusoid.
  • A general form of a sinusoidal wave is $y(x,t) = A sin(kx-\omega t + \phi)$, where A is the amplitude of the wave, $\omega$ is the wave's angular frequency, k is the wavenumber, and $\phi$ is the phase of the sine wave given in radians.
  • We looked closely into the sinusoidal wave.
  • Since a wave with an arbitrary shape can be represented by a sum of many sinusoidal waves (this is called Fourier analysis), we can generate a great variety of solutions of the wave equation by translating and summing sine waves that we just looked closely into.

• Absorptive State

  • When the gastrointestinal tract is full, anabolism exceeds catabolism; this is the absorptive state.
  • Absorptive state is the period in which the gastrointestinal tract is full and the anabolic processes exceed catabolism.
  • The glycogen and fat will be stored in the liver and adipose tissue, respectively, as reserves for the post-absorptive state.
  • Some are used to make plasma proteins, but most leave through liver sinusoids to be used by body cells to construct proteins.

• The Representation Function

  • The two-chamber structure had functioned well in state governments.
  • Two senators were chosen by state governments which benefited smaller states.
  • When the Constitution was ratified in 1787, the ratio of the populations of large states to small states was roughly 12 to one.
  • Since each state has two senators, residents of smaller states have more clout in the Senate than residents of larger states.
  • Critics, such as constitutional scholar Sanford Levinson, have suggested that the population disparity works against residents of large states and causes a steady redistribution of resources from large states to small states.

• Immigration and Border Security

  • Immigration and border security are two important issues for United States policy.
  • Illegal immigrants are those non-citizens who enter the United States without government permission and are in violation of United States nationality law or stay beyond the termination date of a visa, also in violation of the law.
  • The illegal immigrant population in the United States in 2008 was estimated by the Center for Immigration Studies to be about 11 million people, down from 12.5 million people in 2007.
  • Illegal immigrants who come generally for economic opportunities or to escape political oppression, continue to outpace the number of legal immigrants - a trend that has held steady since the 1990s.
  • Rate of immigration to the United States relative to sending countries' population size, 2001–2005

• Catabolic-Anabolic Steady State