Chapter Two. We are now going to look at power in RLC circuits that is circuits that contain resistance inductance and capacitance. And these quantities are often referred to as reactance and impedance. If we were to plot the current and voltage for a simple AC circuit consisting of an AC voltage source e t, and a resistor R, it would look something like this. This is sometimes referred to as the time domain because the resistors simply and directly resist the flow of electrons at all periods of time the wave form for the voltage drop across the resistor, er is exactly in phase with the wave form for the current through it, I R, which in this series circuit, and B There is only one passive element are in a circuit is equal to the circuit current AI. And the voltage drop across the resistor, er is equal to the source voltage et, we could look at any point in time along the horizontal axis of the plot and compare those values of current and voltage with each other.

Any snapshot look at a value of a wave are referred to as instantaneous values meaning the value at that instant in time. The resistor of a given size will impede AC current as defined by ohms law and is proportional to the voltage across it divided by the current through it. And when defining using a phasor notation, it is sometimes referred to as the reactance of a resistor and designated as x subscript R. xR of course, is in phase with the current IR as well as the voltage drop across it er. And in this case of only one passive element in the circuit, that is the resistor, e r is going to be equal to e T the source voltage. We can also calculate the power dissipated by the resistor and plot those values on the same graph. Note that the power is never negative in value.

When the current is positive, above the line, the voltage is also positive resulting in a power p is equal to the times I have a positive value. Conversely, when the current is a negative value below the line, the voltage is also negative, which results in a positive value for power. That is a negative number multiplied by a negative number equals a positive number. This consistent polarity of power tells us that the resistor is always dissipating power, taking it from the source and releasing it in the form of heat energy. Whether the current is positive or negative, the resistor is still dissipating energy. This time, we're going to insert a second element in the series circuit.

And that passive element is going to be an inductor. And because it's still a series circuit, the current through each of the elements the resistor and the inductor is the current in the series circuit is going to be the same current that's flowing in the inductor L. I'm going to designate the voltage drop across the inductor as e L. And I'm going to designate the current through the inductor as i l We are going to concentrate on only the voltages and currents with regard to the inductor in this case. So, the time domain graph of the current and the voltage will look like this. Remember the voltage drop across an inductor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak. In other words, the current is changing at a rate of zero or level slope on the current sine wave and the instantaneous voltage is at a peak whenever the instantaneous current is at a maximum change the points of the steepest slopes on the current one Wave where it crosses the zero line.

This results in a voltage wave that is 90 degrees out of phase with the current wave. Looking at the graph, the voltage wave seems to have a head start on the current wave, the voltage leads the current and the current lags behind the voltage. The phasers would look like this, the voltage leads the current by 90 degrees. Or you might say the current lags the voltage by 90 degrees. When dealing with AC voltages applied to an inductor, it's always confusing trying to remember whether the current leads the voltage or the current lags the voltage or vice versa. No way I try to remember it is going back to first principles of applying a DC circuit to To an inductor with a switch.

So we have the series circuit here with just the inductor connected through a switch to a battery. And when we switch the battery on, the voltage immediately rises to the source voltage. It's instantaneous. However, you can see the current take some time to build the magnetic field. So that current definitely takes longer to arrive at a maximum than the voltage. That way you can remember that the current always lags the voltage or the voltage leads the current when AC is applied to an inductor.

Let's look at an inductor in an AC circuit considering only the voltage and current in the inductor itself, because the instantaneous power delivered to the inductor is the product of the instantaneous voltage drop across That inductor and the instantaneous current in that in Dr p equals L times i L. The power equals zero whenever the instantaneous current or the voltage is zero. And whenever the instantaneous current and voltage are both positive that is above the line, the power is positive. But as with the resistor example, the power is also positive when the instantaneous current and voltage are both negative below the line. However, because the current and voltage waves are 90 degrees out of phase, there are times when one is positive and the other is negative, resulting in an equally frequent occurrence of negative instantaneous power. You will notice, however that there is on the average as many positive power flows as negative power flows resulting in a net average power flow into the inductor of zero.

So, what exactly is going on with the inductor during this positive and negative flow of energy during the positive flow of energy, the current is building a magnetic field around the inductor. This requires energy which is then stored in the magnetic field. As long as both the current through the inductor and the voltage drop across the inductor is positive this will happen. Once the current and voltage have opposite values one positive and one negative, the power flow will be negative. This is when the magnetic field around the inductor is collapsing and the stored energy will be dissipated back into the circuit. Now, both the voltage and current are negative.

So, the power is once again positive however, the current is negative which results in a building of the may have a magnetic field around the inductor but have opposite polarity to that which previously just collapsed. This too requires energy which is then stored in the reverse magnetic field. As long as both the current through the inductor in the voltage drop across the inductor are the same polarity, this will happen. Once the current and voltage have opposite values, one positive and one negative, the power flow will be negative. This is when the magnetic field around the inductor is collapsing and the stored energy will dissipate back into the circuit. Next, the current through the inductor and the voltage drop across the inductor both become positive in the cycle repeat itself over and over again 60 times a second.

This ends the chapter