Prism Glow - Electronics - EEES - Chapter 2 - Alternating Currents and Voltages

Chapter 2-2: Sinusoidal Voltages and Currents

A Sinusoidal Current has an equation that shows the current as a function of time. It looks like this: where is the instantaneous current at any time and is the maximum value or amplitude. The quantity is expressed in radians per second and is the angular velocity.

When the instantaneous current passes through one complete set of positive and negative values, it is said to have completed one cycle. The graph above illustrates one cycle as the instantaneous current goes from the 0 degrees point to the 360 degrees point (or from 0 radians to radians).
The number of cycles that are passed through each second is known as the frequency of the wave. This frequency is expressed in cycles per second or Hertz.
The time for once cycle may be calculated by: with = Time, and = frequency.
may be plotted as a function of time or may be plotted as a function of the angle in electrical radians. One complete cycle spans radians or 360 electrical degrees.
It follows then, that .
And, by using these equations together, we can also represent the instantaneous current this way: .
In many circuits the current and voltage waves do not go through zero simultaneously, but are displaced from each other by a phase angle. We have equations to represent that current and voltage are not necessarily in phase with one another and do not necessarily cross the x-axis at 0 degrees. These equations look like this: and . and are simply padding parameters that indicate how far before the y-axis that the voltage and current curves respectively, cross the x-axis.

Graph of instantaneous current, i, voltage, e, and power, p.

The graph above shows three curves, all of them representing instantaneous values. The curves are, respectively, current, voltage and power. is the phase angle between the voltage and the current. It measures the distance between the points where each curve crosses the x-axis. In this graph, the voltage leads the current. Or we can say that since the current peaks later than the voltage, that the current lags the voltage.
The instantaneous power curve is derived from the product of the instantaneous values of and . Now, note this interesting point; even though both current and voltage are oscillating between large positive and negative values, the power is almost entirely positive! Why? The only time the power will be negative is when the sign of the voltage and current do not match. This only occurs in our example during the time covered by the phase angle where voltage has turned negative and before the current has crossed the x-axis to become negative. Were voltage and current exactly in phase, the power would be 100% positive!

Electronics > EEES > Chapter 2 - Alternating Currents and Voltages > 2-2 Sinusoidal Voltages and Currents
Chapter 2-2: Sinusoidal Voltages and Currents A Sinusoidal Current has an equation that shows the current as a function of time. It looks like this: where is the instantaneous current at any time and is the maximum value or amplitude. The quantity is expressed in radians per second and is the angular velocity. When the instantaneous current passes through one complete set of positive and negative values, it is said to have completed one cycle. The graph above illustrates one cycle as the instantaneous current goes from the 0 degrees point to the 360 degrees point (or from 0 radians to radians). The number of cycles that are passed through each second is known as the frequency of the wave. This frequency is expressed in cycles per second or Hertz. The time for once cycle may be calculated by: with = Time, and = frequency. may be plotted as a function of time or may be plotted as a function of the angle in electrical radians. One complete cycle spans radians or 360 electrical degrees. It follows then, that . And, by using these equations together, we can also represent the instantaneous current this way: . In many circuits the current and voltage waves do not go through zero simultaneously, but are displaced from each other by a phase angle. We have equations to represent that current and voltage are not necessarily in phase with one another and do not necessarily cross the x-axis at 0 degrees. These equations look like this: and . and are simply padding parameters that indicate how far before the y-axis that the voltage and current curves respectively, cross the x-axis. The graph above shows three curves, all of them representing instantaneous values. The curves are, respectively, current, voltage and power. is the phase angle between the voltage and the current. It measures the distance between the points where each curve crosses the x-axis. In this graph, the voltage leads the current. Or we can say that since the current peaks later than the voltage, that the current lags the voltage. The instantaneous power curve is derived from the product of the instantaneous values of and . Now, note this interesting point; even though both current and voltage are oscillating between large positive and negative values, the power is almost entirely positive! Why? The only time the power will be negative is when the sign of the voltage and current do not match. This only occurs in our example during the time covered by the phase angle where voltage has turned negative and before the current has crossed the x-axis to become negative. Were voltage and current exactly in phase, the power would be 100% positive!