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We have following conclusions about the sinusoid: (1) If the input of a linear, time-invariant circuit is a sinusoid ，then the response is sinusoid of the same frequency. (2) Finding the magnitude and phase angle of a sinusoidal steady-state response can be accomplished with either real or complex sinusoids. (3) If the output of a sinusoidal circuit reaches its peak before the input, the circuit is a lead network.. Conversely, it is a lag network. (4) Using the concepts of phasor and impedance, sinusoidal circuits can be analyzed in the frequency domain in a manner analogous to resistive circuits by using the phasor versions of KCL, KVL, nodal analysis, mesh analysis and loop analysis.
apparent power(VA) reactive power VAR
real power(w)
Fig.13.3 power triangle
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Unit 13 Alternating current 交流电
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Alternating current
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Alte
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An alternative current (AC) is an electrical current whose magnitude and direction vary cyclically, as opposed to direct [1] current, whose direction remains constant . 13.1 Alternating current Alternating currents are accompanied (or caused) by alternating voltages. In English the initialism AC is commonly and somewhat confusingly used for both. Step and impulse functions are useful in determining the responses of circuits when they are first turned on or when sudden or irregular changes occur in the input; this is called transient analysis. However, to see how a circuit responds to a regular or repetitive input—the steady-state analysis—the function that is by far the most useful is the sinusoid.
0 0
90 180
270
360
ωt(s)
Fig.13.1 sine wave A sine wave, over one cycle (360。) is shown in Fig.13.1. The dashed line represents the root mean square (RMS) value at about 0.707 V peak. Where V peak is the peak voltage (unit: volt); ω is the angular frequency (unit: radians per second); T is the time to complete one cycle, and is called period (unit: second); The angular frequency ω is related to the physical frequency f which represents the number of oscillations per second (unit: hertz), by the equation: ω =2πf; t is the time (unit: second).
Square wave Triangle wave Sawtooth wave
one wave cycle
one wave cycle
one cycle
Fig.13.2 some common waveshapes (waveforms)
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They’re simply a few that are common enough to have been given distinct names. Even in circuits that are supposed to manifest “pure” sine, square, triangle, or sawtooth voltage/current waveforms, the real-life result is often a distorted version of the intended waveshape. Generally speaking, any waveshape bearing close resemblance to a perfect sine wave is termed sinusoidal, anything different being labeled as non-sinusoidal.
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Though electromechanical generators and many other physical phenomena naturally produce sine waves, this is not the only kind of alternating wave in existence. Other “waveforms” of AC are commonly produced within electronic circuitry. Here are but a few sample waveforms and their common designations in Fig.13.2.
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The sine[2] wave is the most common wave in AC and sometimes we refer to sine AC as AC in short . An AC voltage v can be described mathematically as a function of time by the following equation: v (t) =Vpeak sin (ωt)
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13.2 AC Electric power
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We have previously defined power to be the product of voltage and current in the DC circuits. For the case that voltage and current are constants, the instantaneous power is equal to the average value of the power. The voltage and current are both sinusoids in AC circuits, however, the instantaneous power, which is still the product of voltage and current, changes with time and is not equal to the average power. In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow[ 3]. The portion of power flow that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power (also referred to as active power). That portion of power flow due to stored energy that returns to the source in each cycle is known as reactive power.