Question
A $12 \mathrm{~V}$ source is switched on at time $t=3 \mathrm{~s}$. Sketch the waveform and write the function in terms of the Heaviside step function.
Step 1
First, we need to understand what the Heaviside step function is. The Heaviside step function, also known as the unit step function, is a mathematical function that is equal to 0 for all negative values of the input (t < 0) and equal to 1 for all positive values Show more…
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Write the function $$ V(t)= \begin{cases}1 & \text { for } 0<t<a \\ 0 & \text { for } t>a\end{cases} $$ in terms of the Heaviside step function and sketch the waveform
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The Heaviside function $H$ is defined by $$H(t)=\left\{\begin{array}{ll}{0} & {\text { if } t<0} \\ {1} & {\text { if } t \geqslant 0}\end{array}\right.$$ It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t=0$ and 120 volts are applied instantaneously to the circuit. Write a formula for $V(t)$ in terms of $H(t) .$ (c) Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t=5$ seconds and 240 volts are applied instantaneously to the circuit. Write a formula for $V(t)$ in terms of $H(t) .$ (Note that starting at $t=5$ corresponds to a translation.)
FUNCTIONS AND LIMITS
A Catalog of Essential Functions
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