Ramp Pulse

Define a ramp time pulse.

Figure 1. Define a ramp time pulse.

Time axis unit: Specify the unit to be used for the time axis.
Total signal duration ( $s_{d}$ ): The total length of the signal in the specified units.
Amplitude ( $u_{0}$ ): The amplitude of the time signal.
Pulse delay ( $t_{0}$ ): The pulse delay is the time until the peak of the time signal envelope.
Pulse width ( $p_{w}$ ): This is the half-amplitude pulse width of the signal. The pulse width is the total length of time that the signal is above 50% of its peak value ( $u_{0}$ ).
Rise time ( $τ_{1}$ ): The time required for the pulse to reach its peak value ( $u_{0}$ ) from rest.
Fall time ( $τ_{2}$ ): The time required for the pulse to reach the rest value from its peak ( $u_{0}$ ).
Note: The discharge time will be determined by the pulse width ( $p_{w}$ ).
Number of samples: The number of samples taken from the signal’s analytical equation.

(1)

u (t) = {\begin{array}{crl} 0 & for & t \leq t_{1} + t_{0} \\ u_{0} (1 - \frac{| t - t_{2} - t_{0} |}{τ_{1}}) & for & t_{1} + t_{0} \leq t \leq t_{2} + t_{0} \\ u_{0} & for & t_{2} + t_{0} \leq t \leq t_{3} + t_{0} \\ u_{0} (1 - \frac{| t - t_{3} - t_{0} |}{τ_{2}}) & for & t_{3} + t_{0} \leq t \leq t_{4} + t_{0} \\ 0 & for & t \geq t_{4} + t_{0} \end{array}

(2)

t_{1} = - \frac{p_{w} + τ_{1}}{2} t_{2} = - \frac{p_{w} - τ_{1}}{2} t_{3} = \frac{p_{w} - τ_{2}}{2} t_{4} = \frac{p_{w} + τ_{2}}{2}

The Fourier transform is as follows:

(3)

U (f) = {\begin{array}{cr} u_{0} [((p_{w} - \frac{τ_{1} + τ_{2}}{2}) s i n c ((p_{w} - \frac{τ_{1} + τ_{2}}{2}) f)) \dots \\ + \frac{1}{τ_{1} (2 π f)^{2}} (e^{- j 2 π f τ_{1}} (1 + j 2 π f τ_{1}) - 1) e^{+ j 2 π f (\frac{p_{w}}{2} + τ_{1})} \dots \\ + \frac{1}{τ_{2} (2 π f)^{2}} (e^{+ j 2 π f τ_{2}} (1 - j 2 π f τ_{2}) - 1) e^{- j 2 π f (\frac{p_{w}}{2} + τ_{2})}] e^{- j 2 π f t_{0}} & for f > 0 \\ u_{0} p_{w} & for f = 0 \end{array}