Problem 2
Question
\(\cdot\) The plate on the back of a certain computer scanner says that the unit draws 0.34 A of current from a \(120 \mathrm{V}, 60\) Hz line. Find (a) the root-mean-square current, (b) the current ampli- tude, (c) the average current, and (d) the average square of the current.
Step-by-Step Solution
Verified Answer
(a) 0.34 A, (b) 0.48 A, (c) 0 A, (d) 0.1156 A².
1Step 1: Understanding Root Mean Square (RMS) Current
The root-mean-square (RMS) current for an alternating current (AC) circuit is usually given directly. For this scanner, the RMS current is clearly specified as 0.34 A. Typically, when current is quoted in specifications, it is the RMS value.
2Step 2: Finding the Current Amplitude
In AC circuits, the relationship between RMS current \( I_{rms} \) and current amplitude \( I_0 \) is given by the formula \( I_{rms} = \frac{I_0}{\sqrt{2}} \). Therefore, to find \( I_0 \), rearrange the formula to get \( I_0 = I_{rms} \times \sqrt{2} \). Substituting \( I_{rms} = 0.34 \) A, we have \( I_0 = 0.34 \times \sqrt{2} \approx 0.48 \) A.
3Step 3: Calculating the Average Current
For AC circuits, the average current over a complete cycle is zero because the current is alternating and has equal positive and negative halves. Therefore, the average current is 0 A.
4Step 4: Calculating the Average Square of the Current
The average square of the current is given by \( (I_{rms})^2 \). Given \( I_{rms} = 0.34 \) A, the average square of the current is \( (0.34)^2 = 0.1156 \) A².
Key Concepts
AC circuitscurrent amplitudeaverage currentaverage square of the current
AC circuits
Alternating Current (AC) circuits are an essential part of our daily lives, powering numerous devices in our homes and workplaces. Unlike direct current (DC), where the flow of electric charge is unidirectional, AC circuits have a current that reverses its direction periodically. This periodic change allows AC circuits to efficiently deliver electricity over long distances, making it ideal for household and commercial use.
In an AC circuit, the current and voltage follow a sinusoidal wave pattern. This wave-like nature is key to understanding how various electrical measurements, like the RMS current and current amplitude, are derived. The frequency of the AC signal, often measured in Hertz (Hz), tells us how many cycles of this wave occur per second. For example, a standard household AC power supply is often 60 Hz, meaning it goes through 60 complete cycles each second.
Understanding AC circuits helps us comprehend how power is delivered and utilized, making sense of technical specifications like those found on the back of electrical devices such as a computer scanner.
In an AC circuit, the current and voltage follow a sinusoidal wave pattern. This wave-like nature is key to understanding how various electrical measurements, like the RMS current and current amplitude, are derived. The frequency of the AC signal, often measured in Hertz (Hz), tells us how many cycles of this wave occur per second. For example, a standard household AC power supply is often 60 Hz, meaning it goes through 60 complete cycles each second.
Understanding AC circuits helps us comprehend how power is delivered and utilized, making sense of technical specifications like those found on the back of electrical devices such as a computer scanner.
current amplitude
The current amplitude, often denoted as \( I_0 \), is the peak value of the current in an AC circuit. It represents the maximum extent of the current in either the positive or negative direction during each cycle of the AC waveform. In practical terms, it's the highest current that flows through the circuit at any point within a single oscillation.
To find the current amplitude from the RMS current \( I_{rms} \), we use the relationship:
Understanding the concept of current amplitude is crucial in scenarios where the peak current might impact the performance or safety of a circuit, such as in designing electrical components that can withstand surges.
To find the current amplitude from the RMS current \( I_{rms} \), we use the relationship:
- \( I_{0} = I_{rms} \times \sqrt{2} \)
Understanding the concept of current amplitude is crucial in scenarios where the peak current might impact the performance or safety of a circuit, such as in designing electrical components that can withstand surges.
average current
The average current in an AC circuit is an interesting concept because, unlike DC, it measures the average value of a current that changes direction. In a complete cycle of an ideal sinusoidal AC waveform, the current spends equal time in the positive and negative regions of the cycle. This symmetry results in an average current of zero over one full cycle.
Why is this significant? Knowing that the average current is zero underscores the balanced nature of AC systems. Even though there are momentary peaks, the net movement of charge over time is balanced. This has consequences for the stability of the power systems and how devices manage intermittent power without accumulating excess charge.
When dealing with AC systems, emphasis is placed on RMS values and not the average current, since RMS provides a meaningful measure of power usage.
Why is this significant? Knowing that the average current is zero underscores the balanced nature of AC systems. Even though there are momentary peaks, the net movement of charge over time is balanced. This has consequences for the stability of the power systems and how devices manage intermittent power without accumulating excess charge.
When dealing with AC systems, emphasis is placed on RMS values and not the average current, since RMS provides a meaningful measure of power usage.
average square of the current
The average square of the current, calculated as \((I_{rms})^2\), plays an important role when discussing power calculations in AC circuits. This value represents the mean value of the current squared over one cycle of the AC waveform, an essential aspect of understanding power.
In electrical engineering, calculating the average square of the current is fundamental for determining the power dissipated by resistive elements in the circuit. Since power is related to the square of current, RMS current is preferred over peak or average current in power calculations. This allows for more accurate assessments of energy consumed.
For instance, in our example, where RMS current \( I_{rms} = 0.34 \) A, the average square of the current becomes \((0.34)^2 = 0.1156\) A². This measurement is critical in assessing how much heat or energy is transferred by the AC current over time.
In electrical engineering, calculating the average square of the current is fundamental for determining the power dissipated by resistive elements in the circuit. Since power is related to the square of current, RMS current is preferred over peak or average current in power calculations. This allows for more accurate assessments of energy consumed.
For instance, in our example, where RMS current \( I_{rms} = 0.34 \) A, the average square of the current becomes \((0.34)^2 = 0.1156\) A². This measurement is critical in assessing how much heat or energy is transferred by the AC current over time.
Other exercises in this chapter
Problem 1
\(\bullet\) You have a special lightbulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant.
View solution Problem 3
A capacitance \(C\) and an inductance \(L\) are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If
View solution Problem 4
\(\bullet\) (a) Compute the reactance of a 0.450 \(\mathrm{H}\) inductor at frequencies of 60.0 \(\mathrm{Hz}\) and 600 \(\mathrm{Hz}\) . (b) Compute the reacta
View solution Problem 5
\(\bullet\) A radio inductor. You want the current amplitude through a \(0.450-\mathrm{mH}\) inductor (part of the circuitry for a radio receiver) to be 2.60 \(
View solution