Question

It is given that $v(t)=20t\cos 0.25\pi t\mathrm{V}$ over the interval $-6\le t\le 6$ s. The function then repeats itself. a) What is the fundamental frequency in rad per second? b) Is the function even? c) Is the function odd? d) Does the function have half-wave symmetry

Short answer

a) $0.25\pi$ rad/s; b) No; c) Yes; d) No.

Step-by-step solution

Understanding the Function

The given function is $v(t)=20t\cos (0.25\pi t)$ over the interval $-6\le t\le 6$ seconds and it repeats itself. It is important to recognize this function's form before solving each question component regarding periodicity and symmetry properties.

Determine the Fundamental Frequency

The function has the form $20t\cos (0.25\pi t)$. The standard cosine function $\cos (\omega t)$ has its angular frequency $\omega$, where $\omega =0.25\pi$ rad/s is given. Therefore, the fundamental frequency is $0.25\pi$ rad/s.

Check for Even Function Property

For a function to be even, $f(t)=f(-t)$ must hold. Substitute $-t$ into the function: $v(-t)=20(-t)\cos (0.25\pi (-t))=-20t\cos (0.25\pi t)$. Since $v(-t)eqv(t)$, the function is not even.

Check for Odd Function Property

For a function to be odd, $f(-t)=-f(t)$ must hold. We already found $v(-t)=-20t\cos (0.25\pi t)$. Since this equals $-v(t)$, the function is odd.

Determine Half-Wave Symmetry

A function has half-wave symmetry if it satisfies $f(t+T/2)=-f(t)$ for period $T$. The given function has $T=8$ due to $\omega =0.25\pi$ and 2$T$ cycle is 8. Compute and check: $v(t+4)=20(t+4)\cos (0.25\pi (t+4))$. This does not simplify to $-v(t)$, so it lacks half-wave symmetry.

Key concepts

Fundamental Frequency

In electric circuits, the fundamental frequency is a crucial concept. It often serves as the backbone for analyzing different waveforms. When dealing with periodic signals, identifying the fundamental frequency helps in understanding the behavior of the waveform. It is the lowest frequency of a repeating signal. In the given problem, we have a function in the form of a cosine wave: $v(t)=20t\cos (0.25\pi t)$. Here, the term $\cos (0.25\pi t)$ specifies the angular frequency, denoted by $\omega =0.25\pi$.

The fundamental frequency $f_0$ is related to the angular frequency by the equation:

• $f_0=\frac{\omega }{2\pi }$

So, substituting for $\omega =0.25\pi$ gives:

• $f_0=\frac{0.25\pi }{2\pi }=\frac{0.25}{2}=0.125\text{ Hz}$

Thus, the fundamental frequency is $0.125\text{ Hz}$, which is equivalent to the angular frequency of $0.25\pi$ rad/s.

Even and Odd Functions

A critical concept in electric circuits when examining signals is whether a function is even or odd. These terms help to classify waveforms, which can be advantageous when applying Fourier analysis. An even function is symmetric around the y-axis, described mathematically as $f(t)=f(-t)$. Conversely, an odd function satisfies $f(-t)=-f(t)$, showcasing rotational symmetry about the origin.

For the given signal $v(t)=20t\cos (0.25\pi t)$, we substituted $-t$ into the function and obtained $v(-t)=-20t\cos (0.25\pi t)$. Since this mirrors the definition of an odd function (i.e., $v(-t)=-v(t)$), the function is odd. This signifies that it does not exhibit even symmetry, which would require $v(t)=v(-t)$.

Half-Wave Symmetry

Half-wave symmetry is another way to analyze waveform characteristics. A function shows half-wave symmetry if its shape repeats every half of its period with opposite amplitude. Mathematically, this means $f(t+T/2)=-f(t)$, where $T$ is the period of the function. This property can simplify the analysis of signals, especially in harmonic assessments, because certain harmonics may be missing or reduced.

For $v(t)=20t\cos (0.25\pi t)$, we determined the period $T$ is calculated based on the angular frequency $\omega =0.25\pi$, which leads to a period of 8 seconds (since $T=\frac{2\pi }{\omega }$). To check for half-wave symmetry, examine $v(t+4)$: this does not reduce to $-v(t)$. Therefore, this particular waveform lacks half-wave symmetry, indicating all harmonics will be present.

Angular Frequency

Angular frequency $\omega$ is a fundamental part of describing circular motion or oscillations in physics and engineering. In electric circuits, it defines how quickly the waveform cycles through its periodic phases. Measured in radians per second (rad/s), it connects directly to the fundamental frequency $f_0$ via the equation $\omega =2\pi f_0$. This relationship helps in converting between angular frequency and the more commonly encountered frequency in Hertz (Hz).

In our specific function $v(t)=20t\cos (0.25\pi t)$, it is given that the angular frequency is $\omega =0.25\pi$ rad/s. This indicates that the waveform completes $0.125\text{ Hz}$ cycles per second, as established previously. Understanding angular frequency can aid in examining the temporal behavior of circuits and how different components will react to these oscillations.