Found myself curious as to whether time travel was possible and if so would this mean there are more than one constants of time, basically mean this time we measure by day and night is not the only one of its kind? I checked it online and found this \/ I feel so much better now ~ Differential equation Main article: LTI system theory First order LTI systems are characterized by the differential equation {dV \over dt} + \frac{1}{\tau} V = f(t) where τ represents the exponential decay constant and V is a function of time t V = V(t). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output. Classical examples for f(t) are: The Heaviside step function, often denoted by u(t): u(t)=\begin{cases} 0, & t < 0 \\ 1, & t \ge 0 \end{cases} the impulse function, often denoted by δ(t), and also the sinusoidal input function: f(t) = A \sin(2 \pi f t) or f(t) = A e^{j \omega t }, where A is the amplitude of the forcing function, f is the frequency in Hertz, and ω = 2π f is the frequency in radians per second. Example solution An example solution to the differential equation with initial value V0 and no forcing function is V(t) = V_o e^{-t / \tau} where V_o = V(t=0) is the initial value of V. Thus, the response is an exponential decay with time constant τ. Discussion Suppose V(t) = V_0 e^{-{t \over \tau}}. This behavior is referred to as a decaying exponential function. The time \tau (tau) is referred to as the time constant and can be used (as in this case) to indicate how rapidly an exponential function decays. Here: t = time (generally t>0 in control engineering) V0 = initial value (see specific cases below). Specific cases 1) Let t=0; then V=V_0 e^0, and so V=V_0 2) Let t= \tau; then V=V_0 e^{-1} \approx 0.37 V_0 3) Let V=f(t)=V_0 e^{-{t \over \tau}}, and so \lim_{t \to \infty}f(t) = 0 4) Let t=5 \tau; then V=V_0 e^{-5} \approx 0.0067V_0 After a period of one time constant the function reaches e−1 = approximately 37% of its initial value. In case 4, after five time constants the function reaches a value less than 1% of its original. In most cases this 1% threshold is considered sufficient to assume that the function has decayed to zero - as a rule of thumb, in control engineering a stable system is one that exhibits such an overall damped behavior. Relation of time constant to bandwidth
Posted on: Mon, 02 Dec 2013 19:51:09 +0000
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