Chapter 19: Problem 15
Express each current or voltage in sinusoidal form. $$\mathbf{I}=7.5 \underline{/ 0^{\circ}}$$
Short Answer
Expert verified
\(I = 7.5\sqrt{2} \cos(\omega t) \text{ A}\)
Step by step solution
01
Identify Magnitude and Phase
Analyse the given complex number for the RMS value (magnitude) of the current and its phase angle. In this case, the magnitude is 7.5 A and the phase angle is 0 degrees.
02
Convert to Sinusoidal Form
Use the general form of a sinusoid, which is \(A\cos(\omega t + \phi)\). Here, \(A\) is the peak value which is \(\sqrt{2}\) times the RMS value, \(\omega\) is the angular frequency, \(t\) is the time, and \(\phi\) is the phase angle. Since the frequency is not given, assume it is \(\omega\), and the phase angle is 0 degrees as given. Calculate the peak value by multiplying the RMS value by \(\sqrt{2}\).
03
Write the Sinusoidal Form
Insert the calculated peak value and the phase angle into the sinusoidal function to express the current in sinusoidal form. The frequency term remains as \(\omega\) since it is unknown.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
RMS value
The RMS (Root Mean Square) value is a critical concept in electrical engineering, particularly when dealing with alternating current (AC). It is defined as the square root of the mean (average) of the squares of a set of values, which in the context of AC translates to representing the equivalent direct current (DC) value that would deliver the same power to a resistive load.
The RMS value is significant because it accounts for the fact that AC signals vary over time, usually in a sinusoidal fashion, meaning their instantaneous power also varies. Therefore, the RMS value provides a constant figure that can be used to calculate the power consumed. In our exercise, the RMS current is given as 7.5 A. This would be the value of a DC current that, when passed through the same resistor, would dissipate the same amount of heat as our AC current. To find the peak or maximum current value from the RMS value for sinusoidal signals, we multiply by \(\sqrt{2}\), because the peak is \(\sqrt{2}\) times higher than the RMS in a sine wave.
The RMS value is significant because it accounts for the fact that AC signals vary over time, usually in a sinusoidal fashion, meaning their instantaneous power also varies. Therefore, the RMS value provides a constant figure that can be used to calculate the power consumed. In our exercise, the RMS current is given as 7.5 A. This would be the value of a DC current that, when passed through the same resistor, would dissipate the same amount of heat as our AC current. To find the peak or maximum current value from the RMS value for sinusoidal signals, we multiply by \(\sqrt{2}\), because the peak is \(\sqrt{2}\) times higher than the RMS in a sine wave.
Angular Frequency
Angular frequency (denoted as \(\omega\)) is another important term used in AC electricity. It describes how quickly the AC waveform oscillates and is measured in radians per second. It's related to the more commonly cited frequency (f), which is measured in cycles per second or hertz (Hz), by the equation \(\omega = 2\pi f\).
In the context of our sinusoidal form conversion, angular frequency is crucial because it helps define the rate at which the current's sinusoidal wave oscillates over time. Since the frequency wasn't specified in the exercise, we represent it symbolically with \(\omega\). It's worth noting that in many real-world applications, knowing the angular frequency allows engineers to work with inductive and capacitive reactance, which are frequency-dependent, to analyze circuits more comprehensively.
In the context of our sinusoidal form conversion, angular frequency is crucial because it helps define the rate at which the current's sinusoidal wave oscillates over time. Since the frequency wasn't specified in the exercise, we represent it symbolically with \(\omega\). It's worth noting that in many real-world applications, knowing the angular frequency allows engineers to work with inductive and capacitive reactance, which are frequency-dependent, to analyze circuits more comprehensively.
Complex Numbers
The use of complex numbers in electricity paves the way for easier manipulation and analysis of alternating currents and voltages. A complex number is composed of a real part and an imaginary part, usually written in the form \( a + bi \), where \( i \)=\( \sqrt{-1} \). In electrical engineering, this is useful for phasors representation.
In our step-by-step example, the current \(\textbf{I}=7.5\underline{/ 0^{\circ}}\) firstly identifies the current's magnitude, in this case, 7.5 A, as the RMS value (the real part) and it indicates a 0-degree phase shift (the imaginary part). Complex numbers and phasors can be easily converted to a sinusoidal time function, which is a more intuitive format for understanding how the current varies over time.
In our step-by-step example, the current \(\textbf{I}=7.5\underline{/ 0^{\circ}}\) firstly identifies the current's magnitude, in this case, 7.5 A, as the RMS value (the real part) and it indicates a 0-degree phase shift (the imaginary part). Complex numbers and phasors can be easily converted to a sinusoidal time function, which is a more intuitive format for understanding how the current varies over time.
Phasor Notation
Phasor notation considerably simplifies the analysis of AC circuits. A phasor is a complex number that represents the magnitude and phase of a sinusoid, allowing simple algebraic manipulation rather than tricky differential equations. The term 'phasor' is derived from 'phase vector'.
In the given problem, the current is initially represented as a phasor \(\textbf{I}=7.5\underline{/ 0^{\circ}}\), implicating a magnitude of 7.5 A with no phase shift. To convert to sinusoidal form, we incorporate the RMS value and any phase shift into a time-varying sine or cosine function, which then becomes useful for various calculations and circuit analysis. Remember, the magnitude of the sinusoidal function is the peak value, \(\sqrt{2}\) times the RMS value, and the angle in the phasor becomes the phase (\(\phi\)) in the sinusoidal expression.
In the given problem, the current is initially represented as a phasor \(\textbf{I}=7.5\underline{/ 0^{\circ}}\), implicating a magnitude of 7.5 A with no phase shift. To convert to sinusoidal form, we incorporate the RMS value and any phase shift into a time-varying sine or cosine function, which then becomes useful for various calculations and circuit analysis. Remember, the magnitude of the sinusoidal function is the peak value, \(\sqrt{2}\) times the RMS value, and the angle in the phasor becomes the phase (\(\phi\)) in the sinusoidal expression.
