Question

Find an expression involving \(u_{c}(t)\) for a function \(g\) that ramps up from zero at \(t=t_{0}\) to the value \(h\) at \(t=t_{0}+k\) and then ramps back down to zero at \(t=t_{0}+2 k\)

Short answer

Question: Write down the expression for the function \(g(t)\) that starts at zero at \(t=t_0\), reaches a value of \(h\) at \(t=t_0+k\), and goes back to zero at \(t=t_0+2k\). Answer: The expression for \(g(t)\) is given by: \(g(t) = \frac{h}{k}(t-t_0)u_c(t-t_0) + \frac{-h}{k}(t-t_0-2k)u_c(t-t_0-k)\).

Step-by-step solution

Create the rising part of the ramp

To create the rising part of the ramp, we will use a linear function that starts at 0 and increases to \(h\) at \(t=k\). We can write this linear function as \(f_r(t) = \frac{h}{k}(t-t_0)f_1(t)\), where \(f_1(t) = u_c(t-t_0)\) is the unit step function shifted by \(t_0\). Now, \(f_r(t)\) will be equal to 0 for \(t < t_0\) and will increase linearly to reach the value of \(h\) for \(t = t_0 + k\).

Create the falling part of the ramp

To create the falling part of the ramp, we will use another linear function that starts at \(h\) and decreases to 0 at \(t=2k\). We can write this linear function as \(f_f(t) = \frac{-h}{k}(t-t_0-2k)f_2(t)\), where \(f_2(t) = u_c(t-t_0-k)\) is the unit step function shifted by \(t_0+k\). Now, \(f_f(t)\) will be equal to 0 for \(t < t_0+k\) and will decrease linearly to reach the value of 0 for \(t = t_0 +2k\).

Combine the rising and falling parts of the ramp

The complete function \(g(t)\) consists of the sum of the rising and falling parts of the ramp, \(f_r(t)\) and \(f_f(t)\). We can combine them by adding them as follows: \(g(t) = f_r(t) + f_f(t)\).

Simplify the expression for g(t)

Finally, we will simplify this expression for \(g(t)\) by substituting the expressions of \(f_r(t)\) and \(f_f(t)\) from Steps 1 and 2: \(g(t) = \frac{h}{k}(t-t_0)u_c(t-t_0) + \frac{-h}{k}(t-t_0-2k)u_c(t-t_0-k)\) This expression represents the function \(g(t)\) that ramps up from 0 at \(t=t_0\) to the value \(h\) at \(t=t_0+k\) and then ramps back down to 0 at \(t=t_0+2k\), with the help of unit step functions \(u_c(t)\).

Key concepts

Linear Functions

Linear functions are at the heart of understanding ramp functions. A linear function is characterized by a constant rate of change, which creates a straight line when plotted on a graph. In the context of ramp functions, these straight lines are used to make the 'ramps' up and down.
To construct the ramping up part of a function, we use a linear equation of the form \( f(t) = mt + b \), where \( m \) represents the slope and \( b \) is the intercept. The slope \( m \) determines how steeply the function increases or decreases.
A positive slope means that the function is increasing, whereas a negative slope means it is decreasing. These linear functions are modified using unit step functions to ensure they start and end at the correct times.

Unit Step Function

The unit step function, denoted as \( u_c(t) \), is a powerful tool for controlling when a linear function is active. It acts as a switch, turning a function on at a specified time and off when it is no longer needed. This function is defined as follows:

• \( u_c(t) = 0 \) for \( t < 0 \)
• \( u_c(t) = 1 \) for \( t \geq 0 \)

This simple representation allows us to build piecewise functions, which are functions made up of multiple sub-functions each defined over different intervals. In the context of the ramp function, the unit step function is used to precisely control the part of the linear function in effect. By shifting the unit step, we decide when the linear part starts and ends, creating a ramp that rises and then falls back to zero.

Ramp Functions

Ramp functions are a special type of piecewise function, typically used to model signals or processes that gradually increase and decrease over time. They are useful in engineering and physics for simulating real-world systems, like the rise and fall of temperature or voltage.

A complete ramp function, as described in the solution, is built from two linear segments: a rising and a falling part. \( f_r(t) \), the rising segment, increases linearly from zero to a certain height \( h \), controlled by the linear equation and unit step function. Similarly, \( f_f(t) \) decreases back down to zero.
By combining these segments, using the function \( g(t) = f_r(t) + f_f(t) \), we get a piecewise function that model the whole ramp behavior. Using mathematical operations from linear algebra and unit step concepts, constructing these ramp functions becomes a systematic process, easily tailored to suit any ramping needs.