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The equations and their solutions are applicable from 0 Hz (i.e. direct current) to frequencies at which the transmission line structure can support higher order non-TEM modes. [2]: 282–286 The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time.
The word translinear (TL) was invented by Barrie Gilbert in 1975 [1] to describe circuits that used the exponential current-voltage relation of BJTs. [2] [3] By using this exponential relationship, this class of circuits can implement multiplication, amplification and power-law relationships. When Barrie Gilbert described this class of circuits ...
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, [citation needed] a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs ...
Let f and g be two continuously differentiable functions on , with f an even function and g an odd function. Then the second order ordinary differential equation of the form d 2 x d t 2 + f ( x ) d x d t + g ( x ) = 0 {\displaystyle {d^{2}x \over dt^{2}}+f(x){dx \over dt}+g(x)=0} is called a Liénard equation .
For example, an op amp with a GBWP of 1 MHz would have a gain of 5 at 200 kHz, and a gain of 1 at 1 MHz. This dynamic response coupled with the very high DC gain of the op amp gives it the characteristics of a first-order low-pass filter with very high DC gain and low cutoff frequency given by the GBWP divided by the DC gain.
For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order. We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps.
Given a simply connected and open subset D of and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form (,) + (,) =,is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, [1] [2] so that
Let ˙ = be a linear first order differential equation, where () is a column vector of length and () an periodic matrix with period (that is (+) = for all real values of ). Let ϕ ( t ) {\displaystyle \phi \,(t)} be a fundamental matrix solution of this differential equation.