Question

The Heaviside function H is defined by

\(H\left( t \right) = \left\{ \begin{array}{l}0 if t < 0\\1 if t \ge 0\end{array} \right\}\)

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\({\bf{V}}\left( {\bf{t}} \right)\)in a circuit if the switch is turned on at time\({\bf{t = 0}}\)and 120 volts are applied instantaneously to the circuit. Write a formula for\({\bf{V}}\left( {\bf{t}} \right)\)in terms of\({\bf{H}}\left( {\bf{t}} \right)\).

(c) Sketch the graph of the voltage in a circuit if the switch is turned on at time\({\bf{t = }}5\)seconds and 240 volts are applied instantaneously to the circuit. Write a formula for\({\bf{V}}\left( {\bf{t}} \right)\)in terms of\({\bf{H}}\left( {\bf{t}} \right)\). (Note that starting at\({\bf{t = }}5\)corresponds to a translation.)

Short answer

(a) The graph of the Heaviside function \({\rm{H}}\left( {\rm{t}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{0 if t < 0}}\\{\rm{1 if t}} \ge {\rm{0}}\end{array} \right\}\) is

(b) The graph of the voltage V(t) in a circuit if the switch is turned on at time \(t = 0\) and 120 volts

(c)The graph of the voltage in a circuit if the switch is turned on at time\(t = 5\) seconds and 240 volts

Step-by-step solution

Introduction

Two functions can be expressed as combination of each other with the addition, subtraction, multiplication and divide operation. One or more operation can be applied at a time.

If f and g are two functions they can be written as,

\(\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right),\left(f\right)\left(x\right),\left({f}/{g}\; \right)\left(x\right)\)

Given

The given function is,\({\rm{H}}\left( {\rm{t}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{0 if t < 0}}\\{\rm{1 if t}} \ge {\rm{0}}\end{array} \right\}\).

The given time t is 5 seconds. Voltage at \(t = 0\) and \(t = 5\) are 120 volts and 240 volts respectively.

Graph of Heaviside function

The graph of the Heaviside function \({\rm{H}}\left( {\rm{t}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{0 if t < 0}}\\{\rm{1 if t}} \ge {\rm{0}}\end{array} \right\}\)by the following the one-sided limit rule as described above,

Graph of voltage at 120 volts

The graph of the Heaviside function \({\rm{H}}\left( {\rm{t}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{0 if t < 0}}\\{\rm{1 if t}} \ge {\rm{0}}\end{array} \right\}\)by following the one-sided limit rule as described above, Voltage is 120 volts, at \(t = 0\).

Graph of voltage at 240 volts

The graph of the voltage in a circuit if the switch is turned on at time \(t = 5\) seconds and 240 volts, the Heaviside function \({\rm{H}}\left( {\rm{t}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{0 if t < 0}}\\{\rm{1 if t}} \ge {\rm{0}}\end{array} \right\}\).