DOI: http://dx.doi.org/10.13140/RG.2.2.11835.02088
Soumendra
Nath Thakur,
Tagore’s
Electronic
ORCiD:
0000-0003-1871-7803
Description:
This study offers a simplified elucidation of the intricate connections between key elements in waveform analysis. Through concise explanations and clear mathematical expressions, this abstract distils complex concepts into easily digestible insights. Fundamental principles, such as the equivalence of time intervals and phase shifts, are elucidated, laying the groundwork for understanding the dynamic interplay between time and frequency in waveforms. The inverse relationship between time intervals for phase shifts and frequency is succinctly summarized, providing a practical understanding of waveform behaviour. By bridging theoretical concepts with practical applications, this abstract facilitates a deeper comprehension of waveforms, making these relationships accessible to a broad audience.
Mathematical Presentation:
1. Time Interval (T) = 1 cycle = 360°:
This expression establishes that the time interval T for one complete cycle of a waveform is equal to 360 degrees. This is a fundamental property of periodic waveforms where one full cycle corresponds to a 360-degree phase change.
2. T = 360°:
This line is a concise representation of the previous line, reiterating that the time interval T equals 360 degrees. It serves to reinforce the previous concept.
3. T(deg) = 1° phase shift = T/360:
Here, it's stated that the time interval T in degrees T(deg) for a 1-degree phase shift is equal to the total time interval T divided by 360. This expression establishes the relationship between time intervals and phase shifts.
4. The time interval T(deg) for 1° of phase is inversely proportional to the frequency (f):
· T(deg) = 1/f:
This expression summarizes a key relationship between the time interval T(deg) in degrees for a 1-degree phase shift and the frequency f. It states that T(deg) is inversely proportional to f, meaning as the frequency increases, the time interval for a 1-degree phase shift decreases.
5. We get a wave corresponding to the time shift (Δt):
· T(deg) = 1° phase shift = T/360 = (1/f)/360 = Δt.
This expression connects the time interval T(deg) for a 1-degree phase shift to the concept of time shift (Δt). It expresses that T(deg) is equal to Δt, and subsequently, it shows the calculation of Δt in terms of frequency f as (1/f)/360.
6. Therefore, T(deg) = Δt = (1/f)/360:
This expression concludes the derivation, affirming that the time interval T(deg) for a 1-degree phase shift is equal to Δt, which is calculated as (1/f)/360 in terms of frequency f.
Discussion:
The exploration of waveforms encompasses a myriad of interrelated concepts, each playing a crucial role in understanding the behaviour and characteristics of signals. "Relationships made easy: Time Intervals, Phase Shifts, and Frequency in Waveforms" delves into the fundamental connections between time intervals, phase shifts, and frequency, offering a simplified yet comprehensive view of these relationships.
At the heart of waveform analysis lays the concept of time intervals, representing the duration of one complete cycle of a waveform. By establishing that one cycle corresponds to a 360-degree phase change, the discussion sets the stage for understanding the relationship between time and phase. This foundational understanding lays the groundwork for further exploration into more complex relationships.
The concise representation of time intervals as 360 degrees reinforces the fundamental nature of this relationship, emphasizing its significance in waveform analysis. This succinct expression serves as a clear reminder of the intrinsic connection between time and phase, providing a solid basis for subsequent discussions.
Moving beyond the basic principles, the discussion delves into the relationship between time intervals and phase shifts. By defining the time interval in degrees for a 1-degree phase shift, the discussion elucidates the direct correlation between these two variables. This relationship highlights the dynamic nature of waveforms, where changes in phase are inherently linked to variations in time.
Moreover, the discussion explores the inverse relationship between time intervals for phase shifts and frequency. By summarizing this key relationship in a concise mathematical expression, the discussion demystifies the complex interplay between time and frequency in waveforms. This inverse proportionality underscores the dynamic nature of waveform behaviour, where variations in frequency directly impact the time intervals for phase shifts.
Through practical examples and clear explanations, the discussion bridges theoretical concepts with real-world applications, making these relationships accessible to a broad audience. By simplifying complex concepts and elucidating fundamental principles, "Relationships made easy: Time Intervals, Phase Shifts, and Frequency in Waveforms" offers valuable insights into the intricate connections that govern waveform behaviour.
Conclusion:
In this study, we've explored
the fundamental connections that underpin waveform analysis. Through concise
explanations and clear mathematical expressions, we've demystified complex
concepts and made them accessible to a broad audience. From understanding the
equivalence of time intervals and phase shifts to unravelling the inverse
relationship between time intervals and frequency, this discussion has provided
valuable insights into the dynamic nature of waveforms. By bridging theoretical
concepts with practical applications, we've laid the groundwork for a deeper
understanding of waveform behaviour. In essence, "Relationships made
easy" serves as a valuable resource for anyone seeking to navigate the
intricacies of waveforms with clarity and confidence.
____________
Expert's
comment:
This
paper, authored by Soumendra Nath Thakur from Tagore’s Electronic Lab,
The
mathematical presentation begins by establishing the fundamental property that
the time interval for one complete cycle of a waveform is equal to 360 degrees.
It then reinforces this concept by representing the time interval as simply 360
degrees. The discussion further elaborates on the relationship between time
intervals and phase shifts, defining the time interval in degrees for a 1-degree
phase shift. Moreover, it summarizes the inverse relationship between time
intervals for phase shifts and frequency, emphasizing how changes in frequency
impact the time intervals for phase shifts.
Throughout
the discussion, practical examples and clear explanations are provided to
bridge theoretical concepts with real-world applications, making the
relationships accessible to a broad audience. The conclusion reiterates the
value of the study in simplifying complex concepts and making them accessible,
ultimately serving as a valuable resource for understanding waveform behaviour.
In terms of mathematical consistency, the equations presented align with established principles in waveform analysis. The relationships between time intervals, phase shifts, and frequency are logically and mathematically sound. Furthermore, the physical consistency of the paper is evident in its clear explanations and practical applications, which align with the expected behaviour of waveforms in real-world scenarios.
Overall, "Relationships made easy: Time Intervals, Phase Shifts, and Frequency in Waveforms" offers a coherent and insightful exploration of waveform analysis, providing valuable insights for researchers and practitioners alike.
