INTRODUCTION

In this chapter application of AC & DC circuits are discussed in details

Additionally, we may encounter situations where DC is mixed with AC: where a waveform is superimposed on a steady (DC) signal. The result of such a mix is a signal varying in intensity, but never changing polarity, or changing polarity asymmetrically (spending more time positive than negative, for example). Since DC does not alternate as AC does, its “frequency” is said to be zero, and any signal containing DC along with a signal of varying intensity (AC) may be rightly called a mixed-frequency signal as well. In any of these cases where there is a mix of frequencies in the same circuit, analysis is more complex than what we've seen up to this point.

Sometimes mixed-frequency voltage and current signals are created accidentally. This may be the result of unintended connections between circuits -- called coupling -- made possible by stray capacitance and/or inductance between the conductors of those circuits. A classic example of coupling phenomenon is seen frequently in industry where DC signal wiring is placed in close proximity to AC power wiring. The nearby presence of high AC voltages and currents may cause “foreign” voltages to be impressed upon the length of the signal wiring. Stray capacitance formed by the electrical insulation separating power conductors from signal conductors may cause voltage (with respect to earth ground) from the power conductors to be impressed upon the signal conductors, while stray inductance formed by parallel runs of wire in conduit may cause current from the power conductors to electromagnetically induce voltage along the signal conductors. The result is a mix of DC and AC at the signal load. The following schematic shows how an AC “noise” source may “couple” to a DC circuit through mutual inductance (M_stray) and capacitance (C_stray) along the length of the conductors. (Figure below)

Stray inductance and capacitance couple stray AC into desired DC signal.

When stray AC voltages from a “noise” source mix with DC signals conducted along signal wiring, the results are usually undesirable. For this reason, power wiring and low-level signal wiring should always be routed through separated, dedicated metal conduit, and signals should be conducted via 2-conductor “twisted pair” cable rather than through a single wire and ground connection: (Figure below)

Shielded twisted pair minimized noise.

Circuit effects

The principle of non-sinusoidal, repeating waveforms being equivalent to a series of sine waves at different frequencies is a fundamental property of waves in general and it has great practical import in the study of AC circuits. It means that any time we have a waveform that isn't perfectly sine-wave-shaped, the circuit in question will react as though its having an array of different frequency voltages imposed on it at once.

When an AC circuit is subjected to a source voltage consisting of a mixture of frequencies, the components in that circuit respond to each constituent frequency in a different way. Any reactive component such as a capacitor or an inductor will simultaneously present a unique amount of impedance to each and every frequency present in a circuit. Thankfully, the analysis of such circuits is made relatively easy by applying the Superposition Theorem, regarding the multiple-frequency source as a set of single-frequency voltage sources connected in series, and analyzing the circuit for one source at a time, summing the results at the end to determine the aggregate total:

Circuit driven by a combination of frequencies: 60 Hz and 90 Hz.

Analyzing circuit for 60 Hz source alone:

Circuit for solving 60 Hz.

Analyzing the circuit for 90 Hz source alone:

Circuit of solving 90 Hz.

Superimposing the voltage drops across R and C, we get:

Because the two voltages across each component are at different frequencies, we cannot consolidate them into a single voltage figure as we could if we were adding together two voltages of different amplitude and/or phase angle at the same frequency. Complex number notation give us the ability to represent waveform amplitude (polar magnitude) and phase angle (polar angle), but not frequency.

What we can tell from this application of the superposition theorem is that there will be a greater 60 Hz voltage dropped across the capacitor than a 90 Hz voltage. Just the opposite is true for the resistor's voltage drop. This is worthy to note, especially in light of the fact that the two source voltages are equal. It is this kind of unequal circuit response to signals of differing frequency that will be our specific focus in the next chapter.

We can also apply the superposition theorem to the analysis of a circuit powered by a non-sinusoidal voltage, such as a square wave. If we know the Fourier series (multiple sine/cosine wave equivalent) of that wave, we can regard it as originating from a series-connected string of multiple sinusoidal voltage sources at the appropriate amplitudes, frequencies, and phase shifts. Needless to say, this can be a laborious task for some waveforms (an accurate square-wave Fourier Series is considered to be expressed out to the ninth harmonic, or five sine waves in all!), but it is possible. I mention this not to scare you, but to inform you of the potential complexity lurking behind seemingly simple waveforms. A real-life circuit will respond just the same to being powered by a square wave as being powered by an infinite series of sine waves of odd-multiple frequencies and diminishing amplitudes. This has been known to translate into unexpected circuit resonances, transformer and inductor core overheating due to eddy currents, electromagnetic noise over broad ranges of the frequency spectrum, and the like. Technicians and engineers need to be made aware of the potential effects of non-sinusoidal waveforms in reactive circuits.

Harmonics are known to manifest their effects in the form of electromagnetic radiation as well. Studies have been performed on the potential hazards of using portable computers aboard passenger aircraft, citing the fact that computers' high frequency square-wave “clock” voltage signals are capable of generating radio waves that could interfere with the operation of the aircraft's electronic navigation equipment. It's bad enough that typical microprocessor clock signal frequencies are within the range of aircraft radio frequency bands, but worse yet is the fact that the harmonic multiples of those fundamental frequencies span an even larger range, due to the fact that clock signal voltages are square-wave in shape and not sine-wave.

Electromagnetic “emissions” of this nature can be a problem in industrial applications, too, with harmonics abounding in very large quantities due to (nonlinear) electronic control of motor and electric furnace power. The fundamental power line frequency may only be 60 Hz, but those harmonic frequency multiples theoretically extend into infinitely high frequency ranges. Low frequency power line voltage and current doesn't radiate into space very well as electromagnetic energy, but high frequencies do.

Also, capacitive and inductive “coupling” caused by close-proximity conductors is usually more severe at high frequencies. Signal wiring nearby power wiring will tend to “pick up” harmonic interference from the power wiring to a far greater extent than pure sine-wave interference. This problem can manifest itself in industry when old motor controls are replaced with new, solid-state electronic motor controls providing greater energy efficiency. Suddenly there may be weird electrical noise being impressed upon signal wiring that never used to be there, because the old controls never generated harmonics, and those high-frequency harmonic voltages and currents tend to inductively and capacitively “couple” better to nearby conductors than any 60 Hz signals from the old controls used to.

• REVIEW:
• Any regular (repeating), non-sinusoidal waveform is equivalent to a particular series of sine/cosine waves of different frequencies, phases, and amplitudes, plus a DC offset voltage if necessary. The mathematical process for determining the sinusoidal waveform equivalent for any waveform is called Fourier analysis.
• Multiple-frequency voltage sources can be simulated for analysis by connecting several single-frequency voltage sources in series. Analysis of voltages and currents is accomplished by using the superposition theorem. NOTE: superimposed voltages and currents of different frequencies cannot be added together in complex number form, since complex numbers only account for amplitude and phase shift, not frequency!
• Harmonics can cause problems by impressing unwanted (“noise”) voltage signals upon nearby circuits. These unwanted signals may come by way of capacitive coupling, inductive coupling, electromagnetic radiation, or a combination thereof.

Other waveshapes

As strange as it may seem, any repeating, non-sinusoidal waveform is actually equivalent to a series of sinusoidal waveforms of different amplitudes and frequencies added together. Square waves are a very common and well-understood case, but not the only one.

Electronic power control devices such as transistors and silicon-controlled rectifiers (SCRs) often produce voltage and current waveforms that are essentially chopped-up versions of the otherwise “clean” (pure) sine-wave AC from the power supply. These devices have the ability to suddenly change their resistance with the application of a control signal voltage or current, thus “turning on” or “turning off” almost instantaneously, producing current waveforms bearing little resemblance to the source voltage waveform powering the circuit. These current waveforms then produce changes in the voltage waveform to other circuit components, due to voltage drops created by the non-sinusoidal current through circuit impedances.

Circuit components that distort the normal sine-wave shape of AC voltage or current are called nonlinear. Nonlinear components such as SCRs find popular use in power electronics due to their ability to regulate large amounts of electrical power without dissipating much heat. While this is an advantage from the perspective of energy efficiency, the waveshape distortions they introduce can cause problems.

These non-sinusoidal waveforms, regardless of their actual shape, are equivalent to a series of sinusoidal waveforms of higher (harmonic) frequencies. If not taken into consideration by the circuit designer, these harmonic waveforms created by electronic switching components may cause erratic circuit behavior. It is becoming increasingly common in the electric power industry to observe overheating of transformers and motors due to distortions in the sine-wave shape of the AC power line voltage stemming from “switching” loads such as computers and high-efficiency lights. This is no theoretical exercise: it is very real and potentially very troublesome.

In this section, I will investigate a few of the more common waveshapes and show their harmonic components by way of Fourier analysis using SPICE.

One very common way harmonics are generated in an AC power system is when AC is converted, or “rectified” into DC. This is generally done with components called diodes, which only allow the passage of current in one direction. The simplest type of AC/DC rectification is half-wave, where a single diode blocks half of the AC current (over time) from passing through the load. (Figure below) Oddly enough, the conventional diode schematic symbol is drawn such that electrons flow against the direction of the symbol's arrowhead:

Half-wave rectifier.

halfwave rectifier              
v1 1 0 sin(0 15 60 0 0)
rload 2 0 10k  
d1 1 2 mod1    
.model mod1 d  
.tran .5m 17m  
.plot tran v(1,0) v(2,0)
.four 60 v(1,0) v(2,0) 
.end   
halfwave rectifier              

Half-wave rectifier waveforms. V(1)+0.4 shifts the sinewave input V(1) up for clarity. This is not part of the simulation.

First, we'll see how SPICE analyzes the source waveform, a pure sine wave voltage: (Figure below)

fourier components of transient response v(1)   
dc component =   8.016E-04
harmonic   frequency  fourier    normalized   phase   normalized
no         (hz)      component   component    (deg)   phase (deg)
1        6.000E+01   1.482E+01     1.000000    -0.005       0.000
2        1.200E+02   2.492E-03     0.000168  -104.347    -104.342
3        1.800E+02   6.465E-04     0.000044   -86.663     -86.658
4        2.400E+02   1.132E-03     0.000076   -61.324     -61.319
5        3.000E+02   1.185E-03     0.000080   -70.091     -70.086
6        3.600E+02   1.092E-03     0.000074   -63.607     -63.602
7        4.200E+02   1.220E-03     0.000082   -56.288     -56.283
8        4.800E+02   1.354E-03     0.000091   -54.669     -54.664
9        5.400E+02   1.467E-03     0.000099   -52.660     -52.655

Fourier analysis of the sine wave input.

Notice the extremely small harmonic and DC components of this sinusoidal waveform in the table above, though, too small to show on the harmonic plot above. Ideally, there would be nothing but the fundamental frequency showing (being a perfect sine wave), but our Fourier analysis figures aren't perfect because SPICE doesn't have the luxury of sampling a waveform of infinite duration. Next, we'll compare this with the Fourier analysis of the half-wave “rectified” voltage across the load resistor: (Figure below)

fourier components of transient response v(2)   
dc component =   4.456E+00
harmonic  frequency   fourier    normalized   phase    normalized
no         (hz)      component   component    (deg)    phase (deg)
1        6.000E+01   7.000E+00     1.000000    -0.195       0.000
2        1.200E+02   3.016E+00     0.430849   -89.765     -89.570
3        1.800E+02   1.206E-01     0.017223  -168.005    -167.810
4        2.400E+02   5.149E-01     0.073556   -87.295     -87.100
5        3.000E+02   6.382E-02     0.009117  -152.790    -152.595
6        3.600E+02   1.727E-01     0.024676   -79.362     -79.167
7        4.200E+02   4.492E-02     0.006417  -132.420    -132.224
8        4.800E+02   7.493E-02     0.010703   -61.479     -61.284
9        5.400E+02   4.051E-02     0.005787  -115.085    -114.889

Fourier analysis half-wave output.

Notice the relatively large even-multiple harmonics in this analysis. By cutting out half of our AC wave, we've introduced the equivalent of several higher-frequency sinusoidal (actually, cosine) waveforms into our circuit from the original, pure sine-wave. Also take note of the large DC component: 4.456 volts. Because our AC voltage waveform has been “rectified” (only allowed to push in one direction across the load rather than back-and-forth), it behaves a lot more like DC.

Another method of AC/DC conversion is called full-wave (Figure below), which as you may have guessed utilizes the full cycle of AC power from the source, reversing the polarity of half the AC cycle to get electrons to flow through the load the same direction all the time. I won't bore you with details of exactly how this is done, but we can examine the waveform (Figure below) and its harmonic analysis through SPICE: (Figure below)

Full-wave rectifier circuit.

fullwave bridge rectifier  
v1 1 0 sin(0 15 60 0 0)
rload 2 3 10k  
d1 1 2 mod1    
d2 0 2 mod1    
d3 3 1 mod1    
d4 3 0 mod1    
.model mod1 d  
.tran .5m 17m  
.plot tran v(1,0) v(2,3)
.four 60 v(2,3) 
.end   

Waveforms for full-wave rectifier

fourier components of transient response v(2,3) 
dc component =   8.273E+00
harmonic   frequency  fourier    normalized   phase   normalized
no         (hz)      component   component    (deg)   phase (deg)
1        6.000E+01   7.000E-02     1.000000   -93.519       0.000
2        1.200E+02   5.997E+00    85.669415   -90.230       3.289
3        1.800E+02   7.241E-02     1.034465   -93.787      -0.267
4        2.400E+02   1.013E+00    14.465161   -92.492       1.027
5        3.000E+02   7.364E-02     1.052023   -95.026      -1.507
6        3.600E+02   3.337E-01     4.767350  -100.271      -6.752
7        4.200E+02   7.496E-02     1.070827   -94.023      -0.504
8        4.800E+02   1.404E-01     2.006043  -118.839     -25.319
9        5.400E+02   7.457E-02     1.065240   -90.907       2.612

Fourier analysis of full-wave rectifier output.

What a difference! According to SPICE's Fourier transform, we have a 2nd harmonic component to this waveform that's over 85 times the amplitude of the original AC source frequency! The DC component of this wave shows up as being 8.273 volts (almost twice what is was for the half-wave rectifier circuit) while the second harmonic is almost 6 volts in amplitude. Notice all the other harmonics further on down the table. The odd harmonics are actually stronger at some of the higher frequencies than they are at the lower frequencies, which is interesting.

As you can see, what may begin as a neat, simple AC sine-wave may end up as a complex mess of harmonics after passing through just a few electronic components. While the complex mathematics behind all this Fourier transformation is not necessary for the beginning student of electric circuits to understand, it is of the utmost importance to realize the principles at work and to grasp the practical effects that harmonic signals may have on circuits. The practical effects of harmonic frequencies in circuits will be explored in the last section of this chapter, but before we do that we'll take a closer look at waveforms and their respective harmonics.

• REVIEW:
• Any waveform at all, so long as it is repetitive, can be reduced to a series of sinusoidal waveforms added together. Different waveshapes consist of different blends of sine-wave harmonics.
• Rectification of AC to DC is a very common source of harmonics within industrial power systems.

Square wave signals

It has been found that any repeating, non-sinusoidal waveform can be equated to a combination of DC voltage, sine waves, and/or cosine waves (sine waves with a 90 degree phase shift) at various amplitudes and frequencies. This is true no matter how strange or convoluted the waveform in question may be. So long as it repeats itself regularly over time, it is reducible to this series of sinusoidal waves. In particular, it has been found that square waves are mathematically equivalent to the sum of a sine wave at that same frequency, plus an infinite series of odd-multiple frequency sine waves at diminishing amplitude:

This truth about waveforms at first may seem too strange to believe. However, if a square wave is actually an infinite series of sine wave harmonics added together, it stands to reason that we should be able to prove this by adding together several sine wave harmonics to produce a close approximation of a square wave. This reasoning is not only sound, but easily demonstrated with SPICE.

The circuit we'll be simulating is nothing more than several sine wave AC voltage sources of the proper amplitudes and frequencies connected together in series. We'll use SPICE to plot the voltage waveforms across successive additions of voltage sources, like this: (Figure below)

A square wave is approximated by the sum of harmonics.

In this particular SPICE simulation, I've summed the 1st, 3rd, 5th, 7th, and 9th harmonic voltage sources in series for a total of five AC voltage sources. The fundamental frequency is 50 Hz and each harmonic is, of course, an integer multiple of that frequency. The amplitude (voltage) figures are not random numbers; rather, they have been arrived at through the equations shown in the frequency series (the fraction 4/π multiplied by 1, 1/3, 1/5, 1/7, etc. for each of the increasing odd harmonics).

Q and bandwidth of a resonant circuit

The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit correspondes to a more narrow bandwith, which is desirable in many applications. More formally, Q is the ration of power stored to power dissipated in the circuit reactance and resistance, respectively:

       Q = Pstored/Pdissipated = I2X/I2R
       Q = X/R
       where:      X = Capacitive or Inductive reactance at resonance
                   R = Series resistance.

This formula is applicable to series resonant circuits, and also parallel resonant ciruits if the resistance is in series with the inductor. This is the case in practical applications, as we are mostly concerned with the resistance of the inductor limiting the Q. Note: Some text may show X and R interchanged in the “Q” formula for a parallel resonant circuit. This is correct for a large value of R in parallel with C and L. Our formula is correct for a small R in series with L.

A practical application of “Q” is that voltage across L or C in a series resonant circuit is Q times total applied voltage. In a parallel resonant circuit, current through L or C is Q times the total applied current.

Series resonant circuits

A series resonant circuit looks like a resistance at the resonant frequency. (Figure below) Since the definition of resonance is X_L=X_C, the reactive components cancel, leaving only the resistance to contribute to the impedance. The impedance is also at a minimum at resonance. (Figure below) Below the resonant frequency, the series resonant circuit looks capacitive since the impedance of the capacitor increases to a value greater than the decreasing inducitve reactance, leaving a net capacitive value. Above resonance, the inductive rectance increases, capacitive reactance decreases, leaving a net inductive component.

At resonance the series resonant circuit appears purely resistive. Below resonance it looks capacitive. Above resonance it appears inductive.

Current is maximum at resonance, impedance at a minumum. Current is set by the value of the resistance. Above or below resonance, impedance increases.

Impedance is at a minumum at resonance in a series resonant circuit.

Resonance in series-parallel circuits

In simple reactive circuits with little or no resistance, the effects of radically altered impedance will manifest at the resonance frequency predicted by the equation given earlier. In a parallel (tank) LC circuit, this means infinite impedance at resonance. In a series LC circuit, it means zero impedance at resonance:

However, as soon as significant levels of resistance are introduced into most LC circuits, this simple calculation for resonance becomes invalid. We'll take a look at several LC circuits with added resistance, using the same values for capacitance and inductance as before: 10 µF and 100 mH, respectively. According to our simple equation, the resonant frequency should be 159.155 Hz. Watch, though, where current reaches maximum or minimum in the following SPICE analyses:

Parallel LC circuit with resistance in series with L.

resonant circuit 
v1 1 0 ac 1 sin
c1 1 0 10u     
r1 1 2 100     
l1 2 0 100m    
.ac lin 20 100 200     
.plot ac i(v1) 
.end   

Resistance in series with L produces minimum current at 136.8 Hz instead of calculated 159.2 Hz

Minimum current at 136.8 Hz instead of 159.2 Hz!

Parallel LC with resistance in serieis with C.

Here, an extra resistor (R_bogus) (Figure below)is necessary to prevent SPICE from encountering trouble in analysis. SPICE can't handle an inductor connected directly in parallel with any voltage source or any other inductor, so the addition of a series resistor is necessary to “break up” the voltage source/inductor loop that would otherwise be formed. This resistor is chosen to be a very low value for minimum impact on the circuit's behavior.

Simple series resonance

A similar effect happens in series inductive/capacitive circuits. (Figure below) When a state of resonance is reached (capacitive and inductive reactances equal), the two impedances cancel each other out and the total impedance drops to zero!

Simple series resonant circuit.

With the total series impedance equal to 0 Ω at the resonant frequency of 159.155 Hz, the result is a short circuit across the AC power source at resonance. In the circuit drawn above, this would not be good. I'll add a small resistor (Figure below) in series along with the capacitor and the inductor to keep the maximum circuit current somewhat limited, and perform another SPICE analysis over the same range of frequencies: (Figure below)

As before, circuit current amplitude increases from bottom to top, while frequency increases from left to right. (Figure above) The peak is still seen to be at the plotted frequency point of 157.9 Hz, the closest analyzed point to our predicted resonance point of 159.155 Hz. This would suggest that our resonant frequency formula holds as true for simple series LC circuits as it does for simple parallel LC circuits, which is the case:

Simple parallel (tank circuit) resonance

A condition of resonance will be experienced in a tank circuit (Figure below) when the reactances of the capacitor and inductor are equal to each other. Because inductive reactance increases with increasing frequency and capacitive reactance decreases with increasing frequency, there will only be one frequency where these two reactances will be equal.

Simple parallel resonant circuit (tank circuit).

In the above circuit, we have a 10 µF capacitor and a 100 mH inductor. Since we know the equations for determining the reactance of each at a given frequency, and we're looking for that point where the two reactances are equal to each other, we can set the two reactance formulae equal to each other and solve for frequency algebraically:

So there we have it: a formula to tell us the resonant frequency of a tank circuit, given the values of inductance (L) in Henrys and capacitance (C) in Farads. Plugging in the values of L and C in our example circuit, we arrive at a resonant frequency of 159.155 Hz.

Susceptance and Admittance

In the study of DC circuits, the student of electricity comes across a term meaning the opposite of resistance: conductance. It is a useful term when exploring the mathematical formula for parallel resistances: R_parallel = 1 / (1/R₁ + 1/R₂ + . . . 1/R_n). Unlike resistance, which diminishes as more parallel components are included in the circuit, conductance simply adds. Mathematically, conductance is the reciprocal of resistance, and each 1/R term in the “parallel resistance formula” is actually a conductance.

Whereas the term “resistance” denotes the amount of opposition to flowing electrons in a circuit, “conductance” represents the ease of which electrons may flow. Resistance is the measure of how much a circuit resists current, while conductance is the measure of how much a circuit conducts current. Conductance used to be measured in the unit of mhos, or “ohms” spelled backward. Now, the proper unit of measurement is Siemens. When symbolized in a mathematical formula, the proper letter to use for conductance is “G”.

Reactive components such as inductors and capacitors oppose the flow of electrons with respect to time, rather than with a constant, unchanging friction as resistors do. We call this time-based opposition, reactance, and like resistance we also measure it in the unit of ohms.

As conductance is the complement of resistance, there is also a complementary expression of reactance, called susceptance. Mathematically, it is equal to 1/X, the reciprocal of reactance. Like conductance, it used to be measured in the unit of mhos, but now is measured in Siemens. Its mathematical symbol is “B”, unfortunately the same symbol used to represent magnetic flux density.

The terms “reactance” and “susceptance” have a certain linguistic logic to them, just like resistance and conductance. While reactance is the measure of how much a circuit reacts against change in current over time, susceptance is the measure of how much a circuit is susceptible to conducting a changing current.

If one were tasked with determining the total effect of several parallel-connected, pure reactances, one could convert each reactance (X) to a susceptance (B), then add susceptances rather than diminish reactances: X_parallel = 1/(1/X₁ + 1/X₂ + . . . 1/X_n). Like conductances (G), susceptances (B) add in parallel and diminish in series. Also like conductance, susceptance is a scalar quantity.

When resistive and reactive components are interconnected, their combined effects can no longer be analyzed with scalar quantities of resistance (R) and reactance (X). Likewise, figures of conductance (G) and susceptance (B) are most useful in circuits where the two types of opposition are not mixed, i.e. either a purely resistive (conductive) circuit, or a purely reactive (susceptive) circuit. In order to express and quantify the effects of mixed resistive and reactive components, we had to have a new term: impedance, measured in ohms and symbolized by the letter “Z”.

To be consistent, we need a complementary measure representing the reciprocal of impedance. The name for this measure is admittance. Admittance is measured in (guess what?) the unit of Siemens, and its symbol is “Y”. Like impedance, admittance is a complex quantity rather than scalar. Again, we see a certain logic to the naming of this new term: while impedance is a measure of how much alternating current is impeded in a circuit, admittance is a measure of how much current is admitted.

Given a scientific calculator capable of handling complex number arithmetic in both polar and rectangular forms, you may never have to work with figures of susceptance (B) or admittance (Y). Be aware, though, of their existence and their meanings.

Parallel R, L, and C

We can take the same components from the series circuit and rearrange them into a parallel configuration for an easy example circuit: (Figure below)

Example R, L, and C parallel circuit.

The fact that these components are connected in parallel instead of series now has absolutely no effect on their individual impedances. So long as the power supply is the same frequency as before, the inductive and capacitive reactances will not have changed at all: (Figure below)

Example R, L, and C parallel circuit with impedances replacing component values.

With all component values expressed as impedances (Z), we can set up an analysis table and proceed as in the last example problem, except this time following the rules of parallel circuits instead of series:

Knowing that voltage is shared equally by all components in a parallel circuit, we can transfer the figure for total voltage to all component columns in the table:

Now, we can apply Ohm's Law (I=E/Z) vertically in each column to determine current through each component:

There are two strategies for calculating total current and total impedance. First, we could calculate total impedance from all the individual impedances in parallel (Z_Total = 1/(1/Z_R + 1/Z_L + 1/Z_C), and then calculate total current by dividing source voltage by total impedance (I=E/Z). However, working through the parallel impedance equation with complex numbers is no easy task, with all the reciprocations (1/Z). This is especially true if you're unfortunate enough not to have a calculator that handles complex numbers and are forced to do it all by hand (reciprocate the individual impedances in polar form, then convert them all to rectangular form for addition, then convert back to polar form for the final inversion, then invert). The second way to calculate total current and total impedance is to add up all the branch currents to arrive at total current (total current in a parallel circuit -- AC or DC -- is equal to the sum of the branch currents), then use Ohm's Law to determine total impedance from total voltage and total current (Z=E/I).

Either method, performed properly, will provide the correct answers. Let's try analyzing this circuit with SPICE and see what happens: (Figure below)

Example parallel R, L, and C SPICE circuit. Battery symbols are “dummy” voltage sources for SPICE to use as current measurement points. All are set to 0 volts.

Series R, L, and C

Let's take the following example circuit and analyze it: (Figure below)

Example series R, L, and C circuit.

The first step is to determine the reactances (in ohms) for the inductor and the capacitor.

The next step is to express all resistances and reactances in a mathematically common form: impedance. (Figure below) Remember that an inductive reactance translates into a positive imaginary impedance (or an impedance at +90^o), while a capacitive reactance translates into a negative imaginary impedance (impedance at -90^o). Resistance, of course, is still regarded as a purely “real” impedance (polar angle of 0^o):

Example series R, L, and C circuit with component values replaced by impedances.

Parallel resistor-inductor circuits

Let's take the same components for our series example circuit and connect them in parallel: (Figure below)

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

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:

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

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):

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.

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:

Series resistor-inductor circuits

In the previous section, we explored what would happen in simple resistor-only and inductor-only AC circuits. Now we will mix the two components together in series form and investigate the effects.

Take this circuit as an example to work with: (Figure below)

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

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 a real number (5 Ω ∠ 0^o, or 5 + j0 Ω), and the inductor's reactance is an imaginary number (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 resistance and reactance. In order to express this opposition succinctly, we need a more comprehensive term for opposition to current than either resistance or reactance alone. This term is called impedance, its symbol is Z, and it is also expressed in the unit of ohms, just like resistance and reactance. In the above example, the total circuit impedance is:

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

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).

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. (Figure below)

Current lags voltage in a series L-R circuit.