AC Circuit Complex Impedance, Part 2: Reacting Nicely!

In Part 1 of this series, It’s Just a Passing Phase, we defined the sine waveform of the most common AC signals, illustrated how AC voltage and current each vary as a sine waveform in AC circuits, and we examine the concept of phase angles in describing the alignment (or lack of it) between a voltage sine wave and a current sine wave. Now we’re going to examine a bit of the dynamics of voltage and current within capacitors and inductors to get a sense for just how these components impose phase angles between voltage and current sine waves. Along the way we will describe the concept of reactance, and we will wrap up Part 2 of this Complex Impedance series with the calculation of capacitive and inductive reactances.

First, let’s harken back to the big picture points from Part 1, the roadmap of where we’re going:

  1. Voltage and current applied to AC circuits are each represented by smoothly changing sine waveforms of equal frequency, depicting the regular reversals of direction and smoothly changing magnitudes of each.
  2. The applied voltage and current sine waves often get out of step with one another so the two representations no longer oscillate together, as if one sine wave is shifted ahead or behind the other in time, or phase.
  3. The amount of deviation between the voltage and current sine wave signals in a circuit is described by a phase angle between the two signals, in units of degrees.
  4. Phase angle shifts between voltage and current are imposed by a type of opposition to current flow called reactance in AC circuit components, specifically inductive reactance and capacitive reactance, measured in units of ohms.
  5. Inductive and capacitive reactances combine in a complex way with resistance in a circuit to determine the overall impedance of the circuit.
  6. Complex impedance is described with both a magnitude in ohms and a phase angle in degrees, and there are two primary shorthand methods of representing complex impedance in writing.
  7. Impedance magnitude and phase angle impact the behavior of AC circuits, particularly with respect to power transfer and resonance, as in RF antenna circuits, oscillator circuits, matching networks, power supply circuits, and many others.

We covered points 1 through 3 in Part 1.  In Part 2 we will focus on point 4. Let’s react!

AC Circuit Components and Phase Angles:  Stating something obvious to many, an AC circuit is one designed to operate with alternating current flowing through its components, sine wave-described voltage and current surging back and forth in regular reversals. Not so obvious, perhaps, is the fact that some of the components in an AC circuit will affect the phase relationship between voltage and current in the circuit. That is, some components have the effect of causing the voltage and current to get out of synchronization with one another, and as noted in Part 1 of this series we can describe the asynchronous relationship between voltage and current sine waves with a phase angle.

Let’s consider the phase effects of three common components:  resistors (R), capacitors (C), and inductors (L).

Resistors:  Resistors impose electrical resistance in DC circuits, measured in ohms. Resistors in AC circuits do the same thing they do in DC circuits – they oppose the flow of current somewhat, depending upon their resistive value, and in AC circuits the resistance is imposed in both directions. A purely resistive circuit does not impact the voltage and current phase angle relationship. So, resistors alone in an AC circuit do not create any phase angle differences. Voltage and current will remain politely hand-in-hand. We’ll come back to the matter of resistance in combination with other effects a bit later.

A purely resistive AC circuit has no effect on voltage-current phase angle.

Capacitors:  Capacitors store electrical energy in an electric field that builds up between its two surfaces. As current flows in one direction in an AC circuit, as represented by the upper half of a current sine wave, positive charge will build on one surface of the capacitor and negative charge will build on the opposite surface until the capacitor reaches its maximum charge capacity (as measured in units of farads). As the capacitor reaches its maximum charge capacity the current must reduce over time because there is no more “room” for additional charge to be stored, even though the voltage is still pushing to pack more into the capacitor!

Thus, in a capacitor current reduces over time even as the voltage is peaking! In fact, following a surge of current that will accompany the initial application of voltage to a circuit, a capacitive circuit will settle into a sine wave rhythm of voltage and current that looks like the relationship below. Note that the energy storing cycle of the capacitor is in phase with the voltage cycle – as voltage peaks at the 90-degree and 270-degree positions of its cycle, the greatest charge is stored in the capacitor.

Capacitive Reactance Phase Angle

Ideal Capacitive Circuit Phase Angle Relationship

However, the current drops to zero as the voltage peaks at its 90-degree point in the cycle along with maximum energy storage. As described above, the voltage has pushed a maximum charge into the capacitor and no more current can flow at this point of the cycle. However, once the applied voltage begins to drop after its 90-degree peak the capacitor begins to discharge in the opposite direction. Current flows readily with the capacitor’s discharge even as the applied voltage signal passes the 0 voltage position of 180 degrees. Now in the opposite direction, current flow reduces and stops due to the capacitor reaching a maximum charge capacity with opposite (negative) polarity, even as the applied voltage tries to push more charge in the negative direction at its 270-degree peak cycle position. Rinse and repeat at various frequencies of AC cycling.

Phase Relationship:  In a purely capacitive circuit, the current (I) leads the voltage (E) by 90 degrees.

Capacitive Reactance:  Notice also that a capacitor, thanks to its limited charge-holding capacity, has an effect of opposing current flow. In spite of applied voltage, the current must reduce in magnitude, stop, and ultimately reverse. This opposition to AC current flow, coupled with the phase-shifting effect, is called capacitive reactance. We will consider the relationship of reactance and resistance in impeding AC in a moment, but first on to inductors.

Reactance couples opposition to AC current flow with a voltage-current phase angle shift.

Inductors:  An inductor stores energy in a magnetic field as current flows through it. By physical principles a magnetic field is created in the space around any conductor in which current flows. The strength of the magnetic field increases with increasing current, or it decreases with decreasing current. The direction of magnetic lines of force, or flux, depends upon the direction of current flow in the conductor. A magnetic field’s flux will oscillate around an inductor with the changing AC current direction.

Conversely, a changing magnetic field about any conductor induces an electromotive force (a voltage) in the conductor. This is called electromagnetic induction. Interestingly, as current is pushed through an inductor component by an applied voltage, the inductor builds up a magnetic field over time. But since the magnetic field is changing in strength as it builds up, it also has the effect of inducing another voltage within the inductor component. But, this induced voltage will be of the opposite polarity of the applied voltage that is building up the magnetic field with current flow in the first place!

The induced opposite-polarity voltage is called the back EMF, and it is the back EMF against which the applied voltage to the circuit must do work to create current and store energy as a magnetic field. In an AC circuit containing an inductor component, you can imagine the regular reversals of current building (inflating) and releasing (deflating) a magnetic field about the inductor with each sine wave cycle of current, causing a regular flip-flopping field of magnetic flux polarity (“north-south” magnetic field reversals), and an associated back EMF being induced and flip-flopping in cycle with it all.

The voltage applied to a circuit must overcome the back EMF induced by changing magnetic flux.

Tracking the phase effects of the inductor is a little more challenging than for the capacitor. Keep in mind that the magnitude of the back EMF voltage is greatest when the current change (and resulting magnetic field change) is greatest. It is the change in current flow (and resulting magnetic field change) that induces the back EMF. And the back EMF opposes the applied EMF, although not at quite equivalent magnitude.

Consider this phase relationship diagram depicting applied voltage, back EMF voltage, and the current in an inductive circuit.

Ideal Inductive Circuit Phase Angle Relationship

Ideal Inductive Circuit Phase Angle Relationship

At the 90-degree position along the time axis, the applied voltage is zero and the current flow is at its positive peak. As the applied voltage begins to go negative the current is reduced from its peak value and, therefore, changes in magnitude. This change induces a back EMF opposite in polarity from the applied EMF, but perhaps of somewhat lesser magnitude (voltage), so there is still a small resultant voltage of the polarity of the applied EMF signal to produce current flow.

As the current changes with the sum voltage of EMF and back EMF, its change induces back EMF that is in phase with the applied EMF. The change in current is greatest as it reverses direction at the zero line, so back EMF is greatest at these positions.

Following a negative peak value the applied EMF voltage returns toward zero, and the current levels off to its negative peak maximum. Since the current magnitude is no longer changing rapidly at a current peak position, the back EMF falls to zero. Again, cycle after cycle repeats this steady state phase relationship in inductors.

In the inductor, the greatest storage of energy in the magnetic field occurs when the current magnitude peaks, since current flow creates the magnetic field. So, in a way counter to the capacitor, the energy storage of the inductor is in phase with the current rather than with the voltage waveform.

Further, the resultant waveform cycles of this complex dance among EMF, back EMF, and current settles into one in which the applied voltage leads the current, or exactly opposite of the capacitive relationship.

Phase Relationship:  In a purely inductive circuit, the voltage (E) leads the current (I) by 90 degrees.

Inductive Reactance:  As noted, the back EMF opposes the applied EMF, with the sum voltage in the inductor being of reduced magnitude. A reduced voltage results in reduced current flow, so the inductor also has a reactance, or an opposition to AC current flow through the inductor.

ELI the ICE man:  You may want to remember reactance phase shifting effects using the mnemonic ELI the ICE man. With ELI, the voltage or EMF (E) in an inductor (L) leads the current (I). With ICE, the current (I) in a capacitor (C) leads the voltage (E).

Reactance and Frequency:  Summarizing a few important points, reactance in AC circuits opposes the flow of AC current and has the unit ohms. Reactance may be inductive or capacitive in nature, and the phase relationship that results between voltage and current is different in each pure case. Let’s now consider the computation of reactances along with the effect of the applied signal frequency.

Capacitive and inductive reactance changes with the applied frequency.

Capacitive Reactance:  Capacitive reactance increases as the applied signal frequency decreases. Lower frequencies mean that more current will flow in each individual sine wave cycle. More current flowing generally results in greater capacitor charge accumulating with each cycle, and thus, more energy must be exchanged each cycle taxing the voltage source. Additionally, a lower capacitance value increases capacitive reactance, as a capacitor may achieve maximum charge storage more readily and reduce current flow commensurately. [Read more about the dynamics of capacitive reactance with this Question of the Week.]

Capacitive reactance (XC) is calculated as:

X_{c}=\frac{1}{2\pi fC}

Where:

XC is capacitive reactance in ohms
f is the frequency in hertz
C is the capacitance in farads

Inductive Reactance: Inductive reactance increases with increasing frequency. With increased frequency, the rate of change of the current and resulting magnetic field changes also increases. Since changing current and magnetic fields results in greater back EMF, the opposition to current flow is increased.

Inductive Reactance (XL) is calculated as:

X_{L}={2\pi fL}

Where:

XL is inductive reactance in ohms
f is the frequency in hertz
L is the inductance in henries

So, you may calculate the magnitude of reactance in a capacitor or an inductor easily if you know the applied AC frequency and the value of the capacitor or inductor.

But, what about circuits that contain both inductors and capacitors, and that have resistance to boot? What is to be done with these combinations? How do you merge capacitive reactance, inductive reactance, and resistance to obtain a circuit’s impedance? And what about that phase angle — how does it figure into this whole situation?

We will dive deeply into those questions in Part 3 of this series on Complex Impedance, coming soon!

Go to Part 3, Putting It All Together
Back to Part 1, It’s Just a Passing Phase

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