A) The Phasor Form Of The Sinusoid 20 Cos(4t + 139) Is 20
Understanding the phasor representation of sinusoidal signals is fundamental in electrical engineering, especially in the analysis of AC circuits. Consider the sinusoid 20 Cos(4t + 139°). Its amplitude is 20, its angular frequency is 4 rad/sec, and its phase shift is 139°. When converted into its phasor form, this sinusoid can be represented as a complex number that encapsulates both magnitude and phase, simplifying the analysis of signals especially when dealing with superposition or impedance calculations. The key point here is that the magnitude of the phasor directly corresponds to the amplitude of the sinusoid, and the phase corresponds to the phase shift of the wave.
Understanding the Sinusoid
- Amplitude: 20
- Angular frequency: 4 rad/sec
- Phase shift: 139°
Converting to Phasor Form
The general form of a sinusoid in the time domain is:
v(t) = A cos(ωt + φ)
where:
- A = amplitude
- ω = angular frequency
- φ = phase angle in degrees or radians
The corresponding phasor is a complex number expressed as:
V = A∠φ
or in rectangular form as:
V = A cos(φ) + j A sin(φ)
Phasor Representation of 20 Cos(4t + 139°)
- The magnitude is 20
- The phase angle is 139°
Therefore, the phasor can be written as:
V = 20∠139°
This form is particularly useful for circuit analysis, as it enables straightforward addition of sinusoidal voltages and currents by simply adding their phasors algebraically.
Using Phasors, The Value Of 1 Cos(20t + ...)
In the second part of the discussion, the focus is on determining the value of a sinusoid of the form 1 Cos(20t + ...). Using phasors, this process simplifies to representing the sinusoid as a complex number and analyzing its properties. Though the phase angle is not specified in the given fragment, the method remains the same regardless of the phase value. We will explore the general approach for calculating the instantaneous value of such a sinusoid using phasor concepts.
General Approach to Phasor Analysis
- Identify the amplitude (A), angular frequency (ω), and phase angle (φ) of the sinusoid.
- Express the sinusoid as a phasor: V = A∠φ.
- Use the phasor to perform desired operations or calculations.
- Convert the phasor back to the time domain if needed, using the inverse transformation:
v(t) = Re{V e^{jωt}} = A cos(ωt + φ)
Calculating the Instantaneous Value
Suppose the sinusoid is:
v(t) = 1 Cos(20t + φ)
with an unknown phase φ. The steps are:
- Express the phasor as:
V = 1∠φ
- If φ is known, the instantaneous value at time t is:
v(t) = Re{V e^{j20t}} = A cos(20t + φ)
where:
- A = magnitude of the phasor (here, 1)
- φ = phase angle in radians or degrees
In the absence of a specified phase angle, it's common to assume φ = 0°, which simplifies the sinusoid to:
v(t) = cos(20t)
Practical Application: Example Calculation
- Given: v(t) = 1 cos(20t + φ)
- Assuming φ = 30°, then the phasor is:
V = 1∠30°
The instantaneous value at time t is:
v(t) = cos(20t + 30°)
Summary of Phasor Methodology
- Convert sinusoidal signals into their phasor form for simplified analysis.
- Phasors allow for easy addition, subtraction, and multiplication, which correspond to superposition and impedance calculations in circuit theory.
- The inverse process involves converting the phasor back into the time domain to find the instantaneous value.
Conclusion
The use of phasors significantly streamlines the analysis of sinusoidal signals in electrical systems. For the specific sinusoid 20 Cos(4t + 139°), the phasor form is simply 20∠139°, capturing both magnitude and phase in a compact form. When dealing with other sinusoidal signals like 1 Cos(20t + φ), the same principles apply: identify the amplitude and phase, represent the signal as a phasor, perform calculations in the complex domain, and convert back to the time domain to find instantaneous values. Mastery of phasor analysis is essential for engineers working with AC circuits, signal processing, and communications, providing a powerful tool for simplifying complex sinusoidal interactions.
