📚 Sinusoidal Current | 正弦电流
In CIE A-Level Physics, sinusoidal current is the most important type of alternating current. It appears in mains electricity, signal generators, transformers, and many circuit analysis problems. Understanding its peak value, frequency, angular frequency, phase, and root-mean-square value is essential for success in the alternating currents topic.
在 CIE A-Level 物理中，正弦电流是最重要的一类交流电。它出现在市电、信号发生器、变压器以及许多电路分析问题中。理解它的峰值、频率、角频率、相位和方均根值，对于掌握交流电专题至关重要。
1. What is Sinusoidal Current? | 什么是正弦电流？
A sinusoidal current is an alternating current whose instantaneous value changes with time according to a sine or cosine function. The current increases to a positive maximum, decreases to zero, reverses direction to reach a negative maximum, and then returns to zero. This complete pattern repeats every cycle.
正弦电流是瞬时值随时间按正弦或余弦函数变化的交流电。电流先增大到正最大值，再减小到零，然后反向达到负最大值，最后回到零。这一完整过程在每个周期内重复出现。
The key quantities used to describe a sinusoidal current are the peak current I₀, the period T, the frequency f, and the angular frequency ω. These quantities are linked by simple relationships that you must be able to use confidently in calculations.
描述正弦电流的关键物理量是峰值电流 I₀、周期 T、频率 f 和角频率 ω。这些量由简单的关系联系在一起，你必须在计算中熟练运用。
2. Describing the Waveform: Period, Frequency, Angular Frequency | 描述波形：周期、频率、角频率
The period T is the time taken for one complete cycle of the sinusoidal current. Frequency f is the number of complete cycles per second, and its unit is the hertz, where 1 Hz = 1 cycle per second. The period and frequency are reciprocals of each other.
周期 T 是正弦电流完成一次完整循环所需的时间。频率 f 是每秒完整循环的次数，单位为赫兹，1 Hz = 1 周每秒。周期和频率互为倒数。
f = 1 / T
Angular frequency ω appears in the mathematical expression for sinusoidal current. It converts cycles per second into radians per second. Since one complete cycle corresponds to an angle of 2π radians, the angular frequency is given by:
角频率 ω 出现在正弦电流的数学表达式中。它将每秒的循环次数转化为每秒的弧度数。由于一个完整周期对应 2π 弧度的角度，角频率由下式给出：
ω = 2πf
3. Instantaneous Current Equation | 瞬时电流方程
If timing begins when the current passes through zero and is increasing in the positive direction, the instantaneous current i at time t can be written as:
如果计时从电流经过零点并沿正方向增大时开始，则时刻 t 的瞬时电流 i 可写为：
i = I₀ sin(ωt)
Here I₀ is the peak current, ω is the angular frequency, and t is the time. The argument ωt is measured in radians, so your calculator must be set to radian mode whenever you evaluate this expression.
这里 I₀ 是峰值电流，ω 是角频率，t 是时间。辐角 ωt 以弧度为单位，因此每当计算该表达式时，计算器必须设置为弧度模式。
If timing starts when the current is at its positive maximum value, the cosine form is used instead:
如果计时从电流处于正最大值时开始，则改用余弦形式：
i = I₀ cos(ωt)
4. Peak Value and Peak-to-Peak Value | 峰值与峰峰值
The peak value I₀ is the maximum magnitude of the current in either direction. For a sine wave centred on zero, the positive peak is +I₀ and the negative peak is −I₀.
峰值 I₀ 是电流在任一方向上的最大幅度。对于以零为中心的正弦波，正峰值为 +I₀，负峰值为 −I₀。
The peak-to-peak value is the difference between the maximum positive value and the maximum negative value. It is therefore twice the peak value:
峰峰值是最大正值与最大负值之间的差值。因此它是峰值的两倍：
peak-to-peak current = 2I₀
This distinction is important in oscilloscope questions, where the height of the trace often gives the peak-to-peak voltage before the peak voltage is found.
这一区别在示波器题目中很重要，因为示波器波形的高度通常先给出峰峰值电压，然后才能求出峰值电压。
5. Root Mean Square (RMS) Current | 方均根电流
Because a sinusoidal current changes direction and magnitude, its average value over one complete cycle is zero. This means that the simple average current is not useful for describing the heating or power effect of the alternating current.
由于正弦电流的方向和大小都在变化，它在一个完整周期内的平均值为零。这意味着简单的平均电流无法用来描述交流电的热效应或功率效应。
To compare an alternating current with an equivalent direct current, we use the root mean square value. The rms current is the value of direct current that would deliver the same average power to a resistor as the alternating current.
为了将交流电与等效直流电进行比较，我们使用方均根值。方均根电流是能够在电阻中提供与交流电相同平均功率的直流电流值。
For a sinusoidal current, the rms value is related to the peak value by:
对于正弦电流，方均根值与峰值的关系为：
Irms = I₀ / √2 ≈ 0.707 I₀
6. Deriving the RMS Value | 方均根值的推导
To derive the rms value, we square the instantaneous current, average i² over one complete cycle, and then take the square root. For i = I₀ sin(ωt), the square is:
为了推导方均根值，我们先将瞬时电流平方，在一个完整周期内对 i² 取平均，然后开平方。对于 i = I₀ sin(ωt)，其平方为：
i² = I₀² sin²(ωt)
Using the trigonometric identity sin²(ωt) = (1 − cos(2ωt)) / 2, the average value of cos(2ωt) over a full period is zero. Therefore the average of sin²(ωt) over one complete cycle is 1/2.
利用三角恒等式 sin²(ωt) = (1 − cos(2ωt)) / 2，cos(2ωt) 在一个完整周期内的平均值为零。因此 sin²(ωt) 在一个完整周期内的平均值为 1/2。
The mean square current is therefore:
因此均方电流为：
⟨i²⟩ = I₀² / 2
Taking the square root gives the root mean square current:
开平方后得到方均根电流：
Irms = √(I₀² / 2) = I₀ / √2
7. Average Power in a Resistor | 电阻中的平均功率
For a resistor, the instantaneous power is given by P = i²R. Because the current changes with time, the power also fluctuates. In fact, the power varies at twice the frequency of the current, and it is always positive for a resistor because i² is always positive.
对于电阻，瞬时功率由 P = i²R 给出。由于电流随时间变化，功率也会波动。实际上，功率的变化频率是电流频率的两倍，而且对于电阻来说功率始终为正，因为 i² 恒为正。
The average power delivered to the resistor can be calculated directly using the rms current:
传递给电阻的平均功率可以直接使用方均根电流计算：
Pavg = Irms² R
This is why rms values are so useful: they allow us to use the direct-current power formula for an alternating current in a resistive circuit.
这就是方均根值如此有用的原因：它使我们可以对纯电阻电路中的交流电直接使用直流功率公式。
Alternatively, using the peak current, the average power is:
或者，使用峰值电流时，平均功率为：
Pavg = ½ I₀² R
8. Phase and Phase Difference | 相位与相位差
In AC circuits, two sinusoidal quantities of the same frequency can reach their peaks at different times. The phase difference is the angle by which one wave leads or lags the other, and it is measured in radians or degrees.
在交流电路中，两个同频率的正弦量可能在不同时刻达到峰值。相位差是一个波领先或滞后另一个波的角度，单位为弧度或度。
For example, if the current is i = I₀ sin(ωt) and the voltage is v = V₀ sin(ωt + π/2), then the voltage reaches its peak earlier than the current. The voltage leads the current by π/2 rad, or 90°.
例如，如果电流为 i = I₀ sin(ωt)，电压为 v = V₀ sin(ωt + π/2)，那么电压比电流更早达到峰值。电压领先电流 π/2 弧度，即 90°。
Δφ = π/2 rad = 90°
Phase differences become especially important when studying capacitors and inductors, where voltage and current are not in phase.
相位差在研究电容器和电感器时尤为重要，因为它们的电压和电流不同相。
9. Sinusoidal Voltage and Resistive Circuits | 正弦电压与纯电阻电路
In a purely resistive AC circuit, the sinusoidal voltage and current are in phase. If the voltage across the resistor is v = V₀ sin(ωt), then the current is i = I₀ sin(ω
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