Question

An alternating current-direct current (AC-DC) voltage signal is made up of the following two components, each measured in volts \((\mathrm{V}): V_{\mathrm{AC}}(t)=10 \sin t\) and \(V_{\mathrm{DC}}(t)=15\). a) Sketch the graphs of these two functions on the same set of axes. Work in radians. b) Graph the combined function \(V_{A C}(t)+V_{\mathrm{DC}}(t)\) c) Identify the domain and range of \(V_{\mathrm{AC}}(t)+V_{\mathrm{DC}}(t)\) d) Use the range of the combined function to determine the following values of this voltage signal. i) minimum ii) maximum

Short answer

Domain: \((-\touchstone, \touchstone)\), Range: [5, 25], Min: 5, Max: 25.

Step-by-step solution

Understand the Components

Identify the two components of the voltage signal. The AC component is given by the function \(V_{AC}(t)=10 \sin t\), and the DC component is a constant \(V_{DC}(t)=15\).

Sketch the Individual Graphs

Graph \(V_{AC}(t)=10 \sin t\) first. This is a sine wave with amplitude 10 and period \(2\pi\). Then graph \(V_{DC}(t)=15\), which is a horizontal line at \(y=15\).

Combine the Functions

Graph the combined function \(V(t)=V_{AC}(t)+V_{DC}(t)=10 \sin t + 15\). This function will be the sine wave shifted upwards by 15 units.

Identify the Domain

Since both functions are defined for all real numbers, their domain is \((-\touchstone,\touchstone)\). Therefore, the domain of the combined function is also \((-\touchstone,\touchstone)\).

Identify the Range

The range of the sine function \(10 \sin t\) is from -10 to 10. When shifted up by 15, the range is from 5 to 25. Therefore, the range of \(V(t)=10 \sin t + 15\) is \([5, 25]\).

Determine Minimum Value

The minimum value of the combined function \(V(t)=10 \sin(t) + 15\) occurs when \(\sin(t) = -1\). Therefore, \(V_{\min} = 10(-1) + 15 = 5\).

Determine Maximum Value

The maximum value of the combined function \(V(t)=10 \sin(t) + 15\) occurs when \(\sin(t) = 1\). Therefore, \(V_{\max} = 10(1) + 15 = 25\).

Key concepts

Sine Wave

A sine wave is a smooth and periodic oscillation. It is a fundamental waveform that occurs in many applications, particularly in electrical engineering.
The formula for a sine wave can be written as \(y = A \sin(Bt + C) + D\). Here, \(A\) is the amplitude, which indicates the peak value of the wave (how high and low it goes). For \(V_{\text{AC}}(t)=10 \sin t\), the amplitude is 10 volts.
The variable \(t\) is the input, often representing time. The term inside the sine function (\(Bt+C\)) affects the frequency and phase of the wave. Since \(B=1\) and \(C=0\) in \(V_{\text{AC}}(t)\), the wave completes one cycle every \(2\pi\) units of time, known as the period. Our sine wave starts at \(t=0\) and has no phase shift.
Sine waves are essential because they form the foundational building block of more complex signals in areas such as electronics, acoustics, and signal processing.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods.
A common example of a periodic function is the sine function. It repeats every \(2\pi\) units.
For our sine wave formula, \(V_{\text{AC}}(t)=10 \sin t\), it means that the function values repeat every \(2\pi\) units of time. This repeating nature of periodic functions makes them useful for representing phenomena that cycle over time, such as alternating current in electrical signals.
Because periodic functions are predictable and repeatable, they are valuable for analysis and signal processing. Understanding the periods and characteristics of these functions allow us to foresee and manipulate waves in technology and nature.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.
For \(V_{\text{AC}}(t)=10 \sin t\), the domain is \((-\infty, \infty)\) since sine is defined for all real numbers. The range of \(10 \sin t\) is \([-10, 10]\), as sine fluctuates between -1 and 1, scaling by the amplitude of 10.
For the DC voltage \(V_{\text{DC}}(t)=15\), the domain is also \((-\infty, \infty)\), yielding a constant output of 15.When combining these two as \(V(t)=10 \sin t + 15\), the domain remains \((-\infty, \infty)\), covering all real inputs. The range translates as follows: add 15 to each output of the \(10 \sin t\) function. Thus, the combined function ranges from 5 to 25. Identifying domain and range is crucial for understanding the behavior and limits of functions used in various scientific and engineering fields.

Function Combination

Function combination involves creating a new function by performing operations on two or more existing functions.
In our example, we combined two voltage functions: an AC component \(V_{\text{AC}}(t)=10 \sin t\) and a DC component \(V_{\text{DC}}(t)=15\). By adding these functions, we created a new function \(V(t)=10 \sin t + 15\).
This results in a sine wave that oscillates around a new centerline, shifted up by 15 units, representing the DC component.
Function combination techniques are essential in various subjects like algebra, calculus, and signal processing, as they allow us to analyze and construct new functions that describe more complex behavior from simpler components. This method is widely used in signal processing to understand the effects of different signals working together.