## Chapter 4: Inductive Reactance and Impedance

## AC Circuits Table of Contents

- AC Chapter 1: Basic AC Theory
- AC Chapter 2: Complex Numbers
- AC Chapter 3: Resistors in AC Circuits
- AC Chapter 4: Inductive Reactance and Impedance
- AC Chapter 5: Capacitive Reactance and Impedance
- AC Chapter 6: Reactance and Impedance – R, L, and C
- AC Chapter 7: Resonance
- AC Chapter 8: Mixed Frequency AC Signals
- AC Chapter 9: Filters
- AC Chapter 10: Transformers

#### Videos on Page

## AC inductor circuits

Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow of electrons through them (by dropping a voltage directly proportional to the current), inductors oppose *changes* in current through them. This opposition is due to the following

- Current in a conductor creates a magnetic field around it
- Changing flux (magnetic field) induces voltage proportional to the rate of the change of the magnetic flux (Faraday’s Law)
- The polarity of the induced voltage is always such that it will try to oppose the cause of the changing current

In other words if current is increasing in magnitude, the induced voltage will “push against” the electron flow; if current is decreasing, the polarity will reverse and “push with” the electron flow to oppose the decrease. This opposition to current change is called *reactance*, rather than resistance.

Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such:

v = L\frac{di}{dt}

Where:

- v is the Voltage across the terminals of the inductance
- L is the inductance (in Henrys)
- \frac{di}{dt} is the rate of change of the current over time.

The expression *di/dt* is one from calculus, meaning the rate of change of instantaneous current (i) over time, in amps per second. The inductance (L) is in Henrys, and the instantaneous voltage (e), of course, is in volts. Sometimes you will find the rate of instantaneous voltage expressed as “v” instead of “e” (v = L di/dt), but it means the exact same thing. To show what happens with alternating current, let’s analyze a simple inductor circuit with an voltage source driving an inductor :

An oscilloscope plot of voltage and current for this purely inductive circuit shows the relationship between voltage and the rate of change of current.

We see the instantaneous voltage is zero whenever the instantaneous current is at a peak (zero change, or level slope, on the current sine wave), and the instantaneous voltage is at a peak wherever the instantaneous current is at maximum change (the points of steepest slope on the current wave, where it crosses the zero line). This results in a voltage wave that is 90^{o} 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.

### Power in Inductors

If we plot the product of current and voltage (p=iv), as a third waveform on the oscilloscope display, we see something very interesting:

Because instantaneous power is the product of the instantaneous voltage and the instantaneous current, the power equals zero whenever the instantaneous current *or* voltage is zero. Whenever the instantaneous current and voltage are both positive (above the line), the power is positive. 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^{o} out of phase, there are times when one is positive while the other is negative, resulting in equally frequent occurrences of *negative instantaneous power*.

But what does *negative* power mean? It means that the inductor is releasing power back to the circuit, while a positive power means that it is absorbing power from the circuit. Since the positive and negative power cycles are equal in magnitude and duration over time, the inductor releases just as much power back to the circuit as it absorbs over the span of a complete cycle. What this means in a practical sense is that the reactance of an inductor dissipates a net energy of zero, quite unlike the resistance of a resistor, which dissipates energy in the form of heat. Mind you, this is for perfect inductors only, which have no wire resistance.

This should make sense to us based on our understanding of inductance. While resistance is simply a form of electrical “friction” dissipating energy in the form of heat, inductance stores and releases energy by means of a magnetic field. Those periods of positive power are when the inductance stores energy and builds up its magnetic field and those periods of negative power are when the inductance releases energy, acting as a source.

Power that leaves a circuit and does not return is called true power. Power that transfers back and forth between circuit components while never leaving is called reactive power. All power in a resistive circuit is true because it leaves the circuit never to return. All power in an inductance is reactive because it merely shuttles into and out of the inductor and never leaves the circuit.

### Inductive Reactance

An inductor’s opposition to change in current is an opposition to alternating current in general, which is by definition always changing in instantaneous magnitude and direction. This opposition to alternating current is similar to resistance, but different in that it always results in a phase shift between current and voltage, and it dissipates zero power. Because of the differences, it has a different name: *reactance*. Reactance to AC is expressed in ohms, just like resistance is, except that its mathematical symbol is X instead of R. To be specific, reactance associated with an inductor is usually symbolized by the capital letter X with a letter L as a subscript, like this: X_{L}.

Since inductors drop voltage in proportion to the rate of current change, they will drop more voltage for faster-changing currents, and less voltage for slower-changing currents. What this means is that reactance in ohms for any inductor is directly proportional to the frequency of the alternating current. The exact formula for determining reactance is as follows:

X_L = 2 \pi fL

Where

- X_L is the inductive reactance
- f is the frequency in Hertz
- L is the inductance

### Video: Derive Inductor Voltage Equation (AC Sinusoid)

- Where does the equation for reactance comes from?
- Why does voltage lead current in an inductor?

For a deeper dive into the derivation of the equation for the inductive reactance as well as the maths that explains the relationship between voltage and current in an inductor, check out this video. The first 5 minutes goes through the maths and then the rest consists of examples

**Example**: If we apply sinusoidal AC voltages of 60, 120 and 2500 Hz to a 10 mH inductor it will manifest the reactances in table below.

Frequency (Hertz) | Reactance (Ohms) |
---|---|

60 | 3.7699 |

120 | 7.5398 |

2500 | 157.0796 |

*Reactance of a 10 mH inductor:*

In the reactance equation, the term “2\pi f” (everything on the right-hand side except the L) has a special meaning unto itself. It is the number of radians per second that the alternating current is “rotating” at, if you imagine one cycle of AC to represent a full circle’s rotation. A *radian* is a unit of angular measurement: there are 2\pi radians in one full circle, just as there are 360^{o} in a full circle. If the alternator producing the AC is a double-pole unit, it will produce one cycle for every full turn of shaft rotation, which is every 2\pi radians, or 360^{o}. If this constant of 2\pi is multiplied by frequency in Hertz (cycles per second), the result will be a figure in radians per second, known as the *angular velocity* of the AC system.

Angular velocity may be represented by the expression 2\pi f , or it may be represented by its own symbol, the lower-case Greek letter Omega, which appears similar to our Roman lower-case “w”: ω. Thus, the reactance formula X_L=2\pi f XL = 2πfL could also be written as X_L=\omega L .

It must be understood that this “angular velocity” is an expression of how rapidly the AC waveforms are cycling, a full cycle being equal to 2\pi radians. It is not necessarily representative of the actual shaft speed of the alternator producing the AC. If the alternator has more than two poles, the angular velocity will be a multiple of the shaft speed. For this reason, ω is sometimes expressed in units of *electrical* radians per second rather than (plain) radians per second, so as to distinguish it from mechanical motion.

Any way we express the angular velocity of the system, it is apparent that it is directly proportional to reactance in an inductor. As the frequency (or alternator shaft speed) is increased in an AC system, an inductor will offer greater opposition to the passage of current, and vice versa. Alternating current in a simple inductive circuit is equal to the voltage (in volts) divided by the inductive reactance (in ohms), just as either alternating or direct current in a simple resistive circuit is equal to the voltage (in volts) divided by the resistance (in ohms). An example circuit is shown here:

*Inductive reactance*

I = \frac{V}{X_L}=\frac{10V}{3.7699\Omega}=2.6526A

### Inductive Impedance

We need to keep in mind that voltage and current are not in phase for an inductor. As was shown earlier, the voltage has a phase shift of +90^{o} with respect to the current (voltage leads). If we represent these phase angles of voltage and current mathematically in the form of complex numbers, we find that an inductor’s opposition to current has a phase angle, too:

Opposition = \frac{10V\angle90^o}{2.6526A\angle 0^o}

Opposition = 3.7699\Omega \angle 90^o or Opposition = 0+j3.7699\Omega

*Current lags voltage by 90 ^{o} in an inductor.*

Mathematically, we say that the phase angle of an inductor’s opposition to current is 90^{o}, meaning that an inductor’s opposition to current is a positive imaginary quantity. Combining the magnitude of opposition with the phase angle of the opposition, creates a complex number which is called the ** impedance**.

- Reactance of an inductor: X_L
- Impedance of an inductor: Z_L=X_L\angle90^o=0+jX_L

The idea of impedance becomes critically important in circuit analysis, especially for complex AC circuits where reactance and resistance interact. It will prove beneficial to represent *any* component’s opposition to current in terms of complex numbers rather than scalar quantities of resistance and reactance.

## If a 10uH inductor is in a circuit that operates at 100kHz, what is it’s impedance?

X_L = 2\pi fL = 2\pi (100000Hz)(0.00001F) = 6.283\Omega

Z_L = X_L\angle90^o = 6.283\Omega\angle90^o

**REVIEW:**

*Inductive reactance*is the opposition that an inductor offers to alternating current due to its phase-shifted storage and release of energy in its magnetic field. Reactance is symbolized by the capital letter “X” and is measured in ohms just like resistance (R).- Inductive reactance can be calculated using this formula: X_L = 2 \pi f L
- The
*angular velocity*of an AC circuit is another way of expressing its frequency, in units of electrical radians per second instead of cycles per second. It is symbolized by the lower-case Greek letter “omega,” or ω. - Inductive reactance
*increases*with increasing frequency. In other words, the higher the frequency, the more it opposes the AC flow of electrons. - Inductive impedance ( is a complex number which combines the reactance of the inductor with a phase angle of 90
^{o}. The phase angle of 90^{o}is the phase angle difference between voltage across and current through the inductor

## Series resistor-inductor circuits

A series network is one where all components are connected in-line with each other such that they share a single path for electric charges to flow. The following illustration shows four components (each one represented as a nondescript rectangle) connecte in series with each other, with lettered points for reference:

Any series network provides only one path for current, so there three fundamental properties that arise:

- Series connected components experience the same current at any given time (Conservation of Electric Charge: every charge entering one portion of a series network must eventually exit. I_A=I_B=I_C=I_D=I_E)
- Total voltage across a string of series components is equal to the phasor sum of the voltages across each individual component. (Conservation of Energy: the sum of all energy gains and losses must equal the total gain/loss). V_{total} = V_A + V_B + V_C + V_D + V_E . This is graphically equivalent to stacking voltage phasors tip-to-tail in a phasor diagram.
- Impedances add in series – Total impedance for a string of series-connected impedances is equal to the vector sum of those impedance values. Z_{total} = Z_A + Z_B + Z_C + Z_D + Z_E . This is graphically equivalent to stacking impedance vectors tip-to-tail in a vector diagram

These properties hold true for any series network because they are rooted in fundamental conservation laws. They apply for AC as well as DC circuits, and they apply for any types of series connected components. The major difference between these properties as applied to AC versus DC is that all calculations must be done using phasors in AC rather than simple numbers as in DC.

Take this circuit as an example to work with:

*Series resistor inductor circuit: Current lags applied voltage by 0 ^{o} to 90^{o}.*

You can follow the analysis of the circuit in this video, or in the text below

The resistor will offer 5 Ω of resistance to AC current regardless of frequency, while the inductor will offer 3.7699 Ω of reactance to AC current at 60 Hz. Because the resistor’s resistance is real, so it’s impedance is 5 Ω ∠ 0^{o}, or 5 + j0 Ω. The inductor’s reactance is imaginary, so it’s impedance is 3.7699 Ω ∠ 90^{o}, or 0 + j3.7699 Ω. The combined effect of the two components will be an opposition to current equal to the complex sum of the two numbers. This combined opposition will be a vector combination of the resistor’s impedance and the inductor’s impedance.

Z_{total} = 5\Omega(resistance)+3.7699\Omega(inductive reactance)

Z_{total} = (5\Omega\angle0^o) + (3.7688\Omega\angle90^o)

orZ_{total} = (5+j0 \Omega) + (0 + j3.7699\Omega)

Z_{total} = 6.262\Omega\angle37.016 = 5 + j3.7688\Omega

Impedance is related to voltage and current just as you might expect, in a manner similar to resistance in Ohm’s Law:

v=i\times Z, \large{i=\frac{v}{Z}}, \large{Z=\frac{v}{i}}

In fact, this is a far more comprehensive form of Ohm’s Law than what was taught in DC electronics (E=IR), just as impedance is a far more comprehensive expression of opposition to the flow of electrons than resistance is. *Any* resistance and any reactance, separately or in combination (series/parallel), can be and should be represented as a single impedance in an AC circuit.

To calculate current in the above circuit, we first need to give a phase angle reference for the voltage source, which is generally assumed to be zero. (The phase angles of resistive and inductive impedance are *always* 0^{o} and +90^{o}, respectively, regardless of the given phase angles for voltage or current).

\large{i=\frac{v}{Z} }

\large{i=\frac{10V/angle^o}{6.262\Omega\angle37.016^o}}>/p>

i=1.597A\angle-37.016^o

As with the purely inductive circuit, the current wave lags behind the voltage wave (of the source), although this time the lag is not as great: only 37.016^{o} as opposed to a full 90^{o} as was the case in the purely inductive circuit.

*Current lags voltage in a series L-R circuit.*

For the resistor and the inductor, the phase relationships between voltage and current haven’t changed. Voltage across the resistor is in phase (0^{o} shift) with the current through it; and the voltage across the inductor is +90^{o} out of phase with the current going through it. We can verify this mathematically:

v_R = i_R \times Z_R

(1.597A\angle-37.016^o)(5\Omega\angle0^o) = 7.9847V\angle-37.016^o

Notice that the phase angle of v_R is equal to the phase angle of the current.

The voltage across the resistor has the exact same phase angle as the current through it, telling us that E and I are in phase (for the resistor only).

v_L = i_L \times Z_L

(1.597A\angle-37.016^o)(3.7699\Omega\angle90^o) = 6.0203V\angle52.984^o

Notice that the phase angle of v_L is exactly 90^{o} more than the phase angle of the current

The voltage across the inductor has a phase angle of 52.984^{o}, while the current through the inductor has a phase angle of -37.016^{o}, a difference of exactly 90^{o} between the two. This tells us that E and I are still 90^{o} out of phase (for the inductor only).

We can also mathematically prove that these complex values add together to make the total voltage, just as Kirchhoff’s Voltage Law would predict:

v_{total} = v_R + v_L

v_{total} = 7.9847V\angle-37.016^o + 6.0203V\angle52.984^o

v_{total} = 10V\angle0^o

Let’s check the validity of our calculations with SPICE. I’ve done the simulation with both the KiCAD Spice simulator and LTSpice, so you can choose your favourite SPICE and follow along.

The simulated results came out as:

- V_R = 7.985V\angle-37.02^o
- V_L = 6.02V\angle52.98^o
- I=1.597\angle-37.02^o

With all these figures to keep track of for even such a simple circuit as this, it would be beneficial for us to use the “table” method. Applying a table to this simple series resistor-inductor circuit would proceed as such. First, draw up a table for E/I/Z figures and insert all component values in these terms (in other words, don’t insert actual resistance or inductance values in Ohms and Henrys, respectively, into the table; rather, convert them into complex figures of impedance and write those in):

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | Volts | ||

I | Amps | |||

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | Ohms |

Although it isn’t necessary, I find it helpful to write *both* the rectangular and polar forms of each quantity in the table. If you are using a calculator that has the ability to perform complex arithmetic without the need for conversion between rectangular and polar forms, then this extra documentation is completely unnecessary. However, if you are forced to perform complex arithmetic “longhand” (addition and subtraction in rectangular form, and multiplication and division in polar form), writing each quantity in both forms will be useful indeed.

Now that our “given” figures are inserted into their respective locations in the table, we can proceed just as with DC: determine the total impedance from the individual impedances. Since this is a series circuit, we know that opposition to electron flow (resistance *or* impedance) adds to form the total opposition:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | Volts | ||

I | Amps | |||

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | 5+j3.7699 6.262\angle37.016^o | Ohms |

Now that we know total voltage and total impedance, we can apply Ohm’s Law (I=V/Z) to determine total current:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | Volts | ||

I | 1.2751-j0.9614 1.597\angle-37.016^o | Amps | ||

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | 5+j3.7699 6.262\angle37.016^o | Ohms |

Just as with DC, the total current in a series AC circuit is shared equally by all components. This is still true because in a series circuit there is only a single path for electrons to flow, therefore the rate of their flow must uniform throughout. Consequently, we can transfer the figures for current into the columns for the resistor and inductor alike:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | Volts | ||

I | 1.2751-j0.9614 1.597\angle-37.016^o | 1.2751-j0.9614 1.597\angle-37.016^o | 1.2751-j0.9614 1.597\angle-37.016^o | Amps |

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | 5+j3.7699 6.262\angle37.016^o | Ohms |

Now all that’s left to figure is the voltage drop across the resistor and inductor, respectively. This is done through the use of Ohm’s Law (V=IZ), applied vertically in each column of the table:

R | L | Total | ||
---|---|---|---|---|

V | 6.3756-j4.8071 7.9847\angle-37.016^o | 3.6244+j4.8071 6.0203\angle52.984^o | 10+j0 10\angle0^o | Volts |

I | 1.2751-j0.9614 1.597\angle-37.016^o | 1.2751-j0.9614 1.597\angle-37.016^o | 1.2751-j0.9614 1.597\angle-37.016^o | Amps |

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | 5+j3.7699 6.262\angle37.016^o | Ohms |

And with that, our table is complete. The exact same rules we applied in the analysis of DC circuits apply to AC circuits as well, with the caveat that all quantities must be represented and calculated in complex rather than scalar form. So long as phase shift is properly represented in our calculations, there is no fundamental difference in how we approach basic AC circuit analysis versus DC.

Now is a good time to review the relationship between these calculated figures and readings given by actual instrument measurements of voltage and current. The figures here that directly relate to real-life measurements are those in *polar notation*, not rectangular! In other words, if you were to connect a voltmeter across the resistor in this circuit, it would indicate **7.9847** volts, not 6.3756 (real rectangular) or 4.8071 (imaginary rectangular) volts. To describe this in graphical terms, measurement instruments simply tell you how long the vector is for that particular quantity (voltage or current).

Rectangular notation, while convenient for arithmetical addition and subtraction, is a more abstract form of notation than polar in relation to real-world measurements. As I stated before, I will indicate both polar and rectangular forms of each quantity in my AC circuit tables simply for convenience of mathematical calculation. This is not absolutely necessary, but may be helpful for those following along without the benefit of an advanced calculator. If we were to restrict ourselves to the use of only one form of notation, the best choice would be polar, because it is the only one that can be directly correlated to real measurements.

Impedance (Z) of a series R-L circuit may be calculated, given the resistance (R) and the inductive reactance (X_{L}). Since V=IR, V=IX_L, and V=IZ, resistance, reactance, and impedance are proportional to voltage, respectively. Thus, the voltage phasor diagram can be replaced by a similar impedance diagram.

*Series: R-L circuit Impedance phasor diagram.*

**Example:**

## Given a 40 Ω resistor in series with a 79.58 millihenry inductor. Find the impedance at 60 hertz.

X_L=2\pi fL = 2\pi \times 60 \times(79.58\times10^{-3} = 30\Omega

Z_{total} = Z_R + Z_L = (40 - j0 \Omega) + (0+j30 \Omega) = 40 + j30 \Omega

|Z| = \sqrt{40^2 + 30^2} = 50 \Omega

\angle Z = arctan(\frac{30}{40}) = 36.87^o

Z = 40+j30 \Omega = 50\Omega\angle36.87^o

**REVIEW:**

*Impedance*is the total measure of opposition to electric current and is the complex (vector) sum of (“real”) resistance and (“imaginary”) reactance. It is symbolized by the letter “Z” and measured in ohms, just like resistance (R) and reactance (X).- Impedances (Z) are managed just like resistances (R) in series circuit analysis: series impedances add to form the total impedance. Just be sure to perform all calculations in complex (not scalar) form! Z_{total} = Z_1 + Z_2 + . . . Z_n
- A purely resistive impedance will always have a phase angle of exactly 0
^{o}(Z_R = R \Omega \angle 0^o). - A purely inductive impedance will always have a phase angle of exactly +90
^{o}(Z_L = X_L \Omega \angle 90^o). - Ohm’s Law for AC circuits: v = i \times Z ; \large{ i = \frac{v}{Z} } ; \large{ Z = \frac{v}{i} }
- When resistors and inductors are mixed together in circuits, the total impedance will have a phase angle somewhere between 0
^{o}and +90^{o}. The circuit current will have a phase angle somewhere between 0^{o}and -90^{o}. - Series AC circuits exhibit the same fundamental properties as series DC circuits: current is uniform throughout the circuit, voltage drops add to form the total voltage, and impedances add to form the total impedance.

## Parallel resistor-inductor circuits

Let’s take the same components for our series example circuit and connect them in parallel:

*Parallel R-L circuit.*

Because the power source has the same frequency as the series example circuit, and the resistor and inductor both have the same values of resistance and inductance, respectively, they must also have the same values of impedance.

You can follow along with the analysis in this video, or through the table analysis below

So, we can begin our analysis table with the same “given” values:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | Volts | ||

I | Amps | |||

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | Ohms |

The only difference in our analysis technique this time is that we will apply the rules of parallel circuits instead of the rules for series circuits. The approach is fundamentally the same as for DC. We know that voltage is shared uniformly by all components in a parallel circuit, so we can transfer the figure of total voltage (10 volts ∠ 0^{o}) to all components columns:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | 10+j0 10\angle0^o | 10+j0 10\angle0^o | Volts |

I | Amps | |||

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | Ohms |

Now we can apply Ohm’s Law (I = V/Z ) vertically to two columns of the table, calculating current through the resistor and current through the inductor:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | 10+j0 10\angle0^o | 10+j0 10\angle0^o | Volts |

I | 2+j0 2\angle0^o | 0-j2.6526 2.6526\angle-90^o | Amps | |

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | Ohms |

Just as with DC circuits, branch currents in a parallel AC circuit add to form the total current (Kirchhoff’s Current Law still holds true for AC as it did for DC):

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | 10+j0 10\angle0^o | 10+j0 10\angle0^o | Volts |

I | 2+j0 2\angle0^o | 0-j2.6526 2.6526\angle-90^o | 2-j2.6526 3.3221\angle-52.984^o | Amps |

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | Ohms |

Finally, total impedance can be calculated by using Ohm’s Law (Z=E/I) vertically in the “Total” column. Incidentally, parallel impedance can also be calculated by using a reciprocal formula identical to that used in calculating parallel resistances.

\large{ Z_{parallel}=\frac{1}{\frac{1}{Z_1}+\frac{1}{Z_2}+...+\frac{1}{Z_n}} }

The only problem with using this formula is that it typically involves a lot of calculator keystrokes to carry out. And if you’re determined to run through a formula like this “longhand,” be prepared for a very large amount of work! But, just as with DC circuits, we often have multiple options in calculating the quantities in our analysis tables, and this example is no different. No matter which way you calculate total impedance (Ohm’s Law or the reciprocal formula), you will arrive at the same figure:

R | L | Total | ||
---|---|---|---|---|

V | 10+j0 10\angle0^o | 10+j0 10\angle0^o | 10+j0 10\angle0^o | Volts |

I | 2+j0 2\angle0^o | 0-j2.6526 2.6526\angle-90^o | 2-j2.6526 3.3221\angle-52.984^o | Amps |

Z | 5+j0 5\angle0^o | 0+j3.7699 3.7699\angle90^o | 1.8122+j2.4035 3.0102\angle52.984^o | Ohms |

The results are verified in this SPICE simulation using LTSpice:

**REVIEW:**

- Impedances (Z) are managed just like resistances (R) in parallel circuit analysis: parallel impedances diminish to form the total impedance, using the reciprocal formula. Just be sure to perform all calculations in complex (not scalar) form! \large{ Z_{parallel}=\frac{1}{\frac{1}{Z_1}+\frac{1}{Z_2}+...+\frac{1}{Z_n}} }
- Ohm’s Law for AC circuits: V = IZ ; I = V/Z ; Z = V/I
- When resistors and inductors are mixed together in parallel circuits (just as in series circuits), the total impedance will have a phase angle somewhere between 0
^{o}and +90^{o}. The circuit current will have a phase angle somewhere between 0^{o}and -90^{o}. - Parallel AC circuits exhibit the same fundamental properties as parallel DC circuits: voltage is uniform throughout the circuit, branch currents add to form the total current, and impedances diminish (through the reciprocal formula) to form the total impedance.

## Inductor quirks

In an ideal case, an inductor acts as a purely reactive device. That is, its opposition to AC current is strictly based on inductive reaction to changes in current, and not electron friction as is the case with resistive components. However, inductors are not quite so pure in their reactive behavior. To begin with, they’re made of wire, and we know that all wire possesses some measurable amount of resistance (unless its superconducting wire). This built-in resistance acts as though it were connected in series with the perfect inductance of the coil, like this:

*Inductor Equivalent circuit of a real inductor.*

Consequently, the impedance of any real inductor will always be a complex combination of resistance and inductive reactance.

Compounding this problem is something called the *skin effect*, which is AC’s tendency to flow through the outer areas of a conductor’s cross-section rather than through the middle. When electrons flow in a single direction (DC), they use the entire cross-sectional area of the conductor to move. Electrons switching directions of flow, on the other hand, tend to avoid travel through the very middle of a conductor, limiting the effective cross-sectional area available. The skin effect becomes more pronounced as frequency increases.

Also, the alternating magnetic field of an inductor energized with AC may radiate off into space as part of an electromagnetic wave, especially if the AC is of high frequency. This radiated energy does not return to the inductor, and so it manifests itself as resistance (power dissipation) in the circuit.

Added to the resistive losses of wire and radiation, there are other effects at work in iron-core inductors which manifest themselves as additional resistance between the leads. When an inductor is energized with AC, the alternating magnetic fields produced tend to induce circulating currents within the iron core known as *eddy currents*. These electric currents in the iron core have to overcome the electrical resistance offered by the iron, which is not as good a conductor as copper. Eddy current losses are primarily counteracted by dividing the iron core up into many thin sheets (laminations), each one separated from the other by a thin layer of electrically insulating varnish. With the cross-section of the core divided up into many electrically isolated sections, current cannot circulate within that cross-sectional area and there will be no (or very little) resistive losses from that effect.

As you might have expected, eddy current losses in metallic inductor cores manifest themselves in the form of heat. The effect is more pronounced at higher frequencies, and can be so extreme that it is sometimes exploited in manufacturing processes to heat metal objects! In fact, this process of “inductive heating” is often used in high-purity metal foundry operations, where metallic elements and alloys must be heated in a vacuum environment to avoid contamination by air, and thus where standard combustion heating technology would be useless. It is a “non-contact” technology, the heated substance not having to touch the coil(s) producing the magnetic field.

In high-frequency service, eddy currents can even develop within the cross-section of the wire itself, contributing to additional resistive effects. To counteract this tendency, special wire made of very fine, individually insulated strands called *Litz wire* (short for *Litzendraht*) can be used. The insulation separating strands from each other prevent eddy currents from circulating through the whole wire’s cross-sectional area.

Additionally, any magnetic hysteresis that needs to be overcome with every reversal of the inductor’s magnetic field constitutes an expenditure of energy that manifests itself as resistance in the circuit. Some core materials (such as ferrite) are particularly notorious for their hysteretic effect. Counteracting this effect is best done by means of proper core material selection and limits on the peak magnetic field intensity generated with each cycle.

Altogether, the stray resistive properties of a real inductor (wire resistance, radiation losses, eddy currents, and hysteresis losses) are expressed under the single term of “effective resistance:”

*Equivalent circuit of a real inductor with skin-effect, radiation, eddy current, and hysteresis losses.*

It is worthy to note that the skin effect and radiation losses apply just as well to straight lengths of wire in an AC circuit as they do a coiled wire. Usually their combined effect is too small to notice, but at radio frequencies they can be quite large. A radio transmitter antenna, for example, is designed with the express purpose of dissipating the greatest amount of energy in the form of electromagnetic radiation.

Effective resistance in an inductor can be a serious consideration for the AC circuit designer. To help quantify the relative amount of effective resistance in an inductor, another value exists called the *Q factor*, or “quality factor” which is calculated as follows:

The symbol “Q” has nothing to do with electric charge (coulombs), which tends to be confusing. For some reason, the Powers That Be decided to use the same letter of the alphabet to denote a totally different quantity.

The higher the value for “Q,” the “purer” the inductor is. Because its so easy to add additional resistance if needed, a high-Q inductor is better than a low-Q inductor for design purposes. An ideal inductor would have a Q of infinity, with zero effective resistance.

Because inductive reactance (X) varies with frequency, so will Q. However, since the resistive effects of inductors (wire skin effect, radiation losses, eddy current, and hysteresis) also vary with frequency, Q does not vary proportionally with reactance. In order for a Q value to have precise meaning, it must be specified at a particular test frequency.

Stray resistance isn’t the only inductor quirk we need to be aware of. Due to the fact that the multiple turns of wire comprising inductors are separated from each other by an insulating gap (air, varnish, or some other kind of electrical insulation), we have the potential for capacitance to develop between turns. AC capacitance will be explored in the next chapter, but it suffices to say at this point that it behaves very differently from AC inductance, and therefore further “taints” the reactive purity of real inductors.

## More on the “skin effect”

As previously mentioned, the skin effect is where alternating current tends to avoid travel through the center of a solid conductor, limiting itself to conduction near the surface. This effectively limits the cross-sectional conductor area available to carry alternating electron flow, increasing the resistance of that conductor above what it would normally be for direct current:

*Skin effect: skin depth decreases with increasing frequency.*

The electrical resistance of the conductor with all its cross-sectional area in use is known as the “DC resistance,” the “AC resistance” of the same conductor referring to a higher figure resulting from the skin effect. As you can see, at high frequencies the AC current avoids travel through most of the conductor’s cross-sectional area. For the purpose of conducting current, the wire might as well be hollow!

In some radio applications (antennas, most notably) this effect is exploited. Since radio-frequency (“RF”) AC currents wouldn’t travel through the middle of a conductor anyway, why not just use hollow metal rods instead of solid metal wires and save both weight and cost? Most antenna structures and RF power conductors are made of hollow metal tubes for this reason.

In the following photograph you can see some large inductors used in a 50 kW radio transmitting circuit. The inductors are hollow copper tubes coated with silver, for excellent conductivity at the “skin” of the tube:

*High power inductors formed from hollow tubes.*

The degree to which frequency affects the effective resistance of a solid wire conductor is impacted by the gauge of that wire. As a rule, large-gauge wires exhibit a more pronounced skin effect (change in resistance from DC) than small-gauge wires at any given frequency. The equation for approximating skin effect at high frequencies (greater than 1 MHz) is as follows:

R_{AC}=R_{DC}(k)\sqrt{f}Where

- R_{AC} is the AC resistance at the frequency ‘f’
- R_{DC} is the resistance at DC
- k is the wire gage factor (see table below)
- f is the frequency of AC in MHz

Table below gives approximate values of “k” factor for various round wire sizes.

*“k” factor for various AWG wire sizes.*

gauge size | k factor | gauge size | k factor |
---|---|---|---|

4/0 | 124.5 | 8 | 34.8 |

2/0 | 99.0 | 10 | 27.6 |

1/0 | 88.0 | 14 | 17.6 |

2 | 69.8 | 18 | 10.9 |

4 | 55.5 | 22 | 6.86 |

6 | 47.9 | – | – |

For example, a length of number 10-gauge wire with a DC end-to-end resistance of 25 Ω would have an AC (effective) resistance of 2.182 kΩ at a frequency of 10 MHz:

R_{AC}=R_{DC}(k)\sqrt{f}

R_{AC}=(25\Omega)(27.6))\sqrt{10}

R_{AC}=2.182k\Omega

Please remember that this figure is *not* impedance, and it does *not* consider any reactive effects, inductive or capacitive. This is simply an estimated figure of pure resistance for the conductor (that opposition to the AC flow of electrons which *does* dissipate power in the form of heat), corrected for the skin effect. Reactance, and the combined effects of reactance and resistance (impedance), are entirely different matters.

## Practice Problems – Inductors in AC

This page has a number of practice problems and answers related to inductors in AC circuits:

The page includes some questions to test your foundational knowledge as well as some problems similar to the ones done on this page.

## Contributors

Contributors to this chapter are listed in chronological order of their contributions, from most recent to first.

**David Williams **(2022): Changed table pictures to tables. Incorporated content from Kuphaldt’s “Model Project” for AC series circuits. Other updates to import into website

**Jim Palmer** (June 2001): Identified and offered correction for typographical error in complex number calculation.

**Jason Starck** (June 2000): HTML document formatting, which led to a much better-looking second edition.

CC-BY 2000-2020 Tony R. Kuphaldt.