Steady-state sinusoidal (AC) current
- Many important applications in electrical engineering involves AC currents and voltages.Electric power is distributed worldwide using AC signals with a frequency of either 50 or 60 HZ.
- The transmission and the reception of electromagnetic waves necessary in any wireless applications involves AC signals.
- when AC signals are initially applied to electric circuits, transient responses are produced in the circuit that decay to zero over time.
- It covers analysis of circuits when the source is sinusoidal.
The analysis techniques are exactly the same as those used when the source was DC.
- What makes AC analysis more difficult is the mathematics, as will be seen in the next section. Fortunately there are mathematical tools and short cuts, such as using Phasors.
Review of Sinusoidal Sources
A sinusoidal source has the following attributes:
- An amplitude
- A frequency
- A Phase Angle
Steady-state sinusoidal time-varying voltage and current waveforms can be given be
where v and i are the time-varying voltage and current, and Vm and Im are the peak values (magnitudes or amplitudes) of the voltage and current waveforms. In equation 3.2, θ is known as the phase angle, which is normally defined with reference to the voltage waveform.
The term cos θ is called a power factor. Remember that we assumed a voltage having a zero phase. In general, the phase of the voltage may have a value other than zero. Then θ should be taken as the phase of the voltage minus the phase of the current.
In a linear circuit excited by sinusoidal sources, in the steady-state, all voltages and currents are sinusoidal and have the same frequency. However, there may be a phase difference between the voltage and current depending on the type of load used.
The three basic passive circuit elements, the resistor (R), the inductor (L), and the capacitor (C), are considered in this chapter. An ac load may be a combination of these passive elements, such as R + L or R + C.
Note that the current and voltage waveforms in the resistor are in phase, while inductances and capacitors both have a 90° phase shift between voltage and current. The inductor current waveform lags the inductor voltage waveform by 90°, while in the capacitor, the current leads the voltage by 90°.
The peak-to-peak value is also used in the analysis of ac circuits; it is the difference between the highest and lowest values of the waveform over one cycle. This can easily be visualized in the ac waveforms generated.
It might seem difficult to describe an ac signal in terms of a specific value, since an ac signal is not constant. However, as shown in Chapter 2, these sinusoidal signals are periodic, repeating the same pattern of values in each period. Therefore, when a voltage or a current is described simply as ac, we will refer to its rms or effective value, not its maximum value, which simplifies the description of ac signal.
In general, for nonsinusoidal systems (distorted waveforms), the power factor PF is equal to
where Stotal is the total apparent (or complex) power in VA, which is equal to Vrms · Irms.
If the voltage and current waveforms of an ac system are measured in real time, it is much easier to calculate the power factor simply by using the definitions of average and rms values (given in equations 2.3 and 2.4).
However, in many utility applications, the distortion in ac voltage is usually small, hence the voltage waveform can be assumed as an ideal sine wave at fundamental frequency. This assumption simplifies the analysis, which results in an analytical solution of power factor for the nonsinusoidal systems as