sinusoidal steady state

(noun)

Indicates every voltage and current in a system is sinusoidal with the same angular frequency ω.

Related Terms

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.

  • 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.

  • Different Types of Currents

    • This solution gives the circuit voltages and currents when the circuit is in DC steady state.
    • The solution to these equations usually contain a time varying or transient part as well as constant or steady state part.
    • It is this steady state part that is the DC solution.

  • Time

    • The second is now operationally defined as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom."
    • It follows that the hyperfine splitting in the ground state of the cesium 133 atom is exactly 9,192,631,770 hertz .
    • In other words, cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted.

  • Maxwell's Equations

    • Gauss's law for magnetism states that there are no "magnetic charges (or monopoles)" analogous to electric charges, and that magnetic fields are instead generated by magnetic dipoles.
    • Ampere's law originally stated that magnetic field could be created by electrical current.
    • Electric (red) and magnetic (blue) waves propagate in phase sinusoidally, and perpendicularly to one another.

  • Detonation Waves

    • Because the flux of the flow is conserved through the transition, the state of the gas must remain on the chord .
    • There is a minimum flux that can pass through the detonation front, , and this flux also corresponds to the minimum velocity jump through the front where the final state is or the Jouguet point.
    • Furthermore, for final states above $O$ along $a'$ the gas leaves the front subsonically.
    • If the final state lies at $O$, the gas leaves the detonation front right at the speed of sound in the downstream flow.
    • At this point the postshock gas leaves the front at the sound speed so the rarefaction wave no longer overtakes the shock and the combined detonation front and rarefaction wave achieves a steady state.

  • Spherical Accretion

    • We continue looking at steady flows with two specific applications: matter flowing onto an object (accretion) and matter flowing away from an object (winds).
    • We will assume that the accretion is steady at a rate ${\dot M}$ and that the pressure $P \propto \rho^\gamma$ with $1 < \gamma < 5/3$.
    • Let's use the equation of state to eliminate $p$ from the Euler equation and use the continuity equation to eliminate $\rho$,

  • Electric Charge in the Atom

    • In the ground state, an atom will have an equal number of protons and electrons, and thus will have a net charge of 0.
    • Atoms in such a state are known as ions.
    • The steady flow of electrons is called current.