My Piece de Resistance!

May 1, 2001 12:00 PM, EDDIE CILETTI

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UNDERSTANDING AC/DC, SERIES/PARALLEL, RESISTANCE AND IMPEDANCE

We interface equipment every day, assuming and hoping that the relationship between source and destination will be a happy one. In any electrical circuit, impedance is a major part of that relationship, a rather deep subject that had me pouring through old textbooks and wishing I could still do the math. Here is my “Piece de Resistance,” an overview of basic electronic information that will prepare you for next month's further adventures.

AC/DC: THE JUICE

Electricity can be described as an Electromotive Force (EMF). You know AC — alternating current — as sound and power (the giant hum that comes from wall outlets). Batteries deliver direct current (DC). Both are expressed in volts. AC can be transformed, rectified and filtered into DC — power supplies do this most of the time.

AC implies time by way of repetition. The frequency of AC is stated in Hertz — Hz — formerly known as “Cycles Per Second,” or CPS. It is easiest to imagine one complete cycle of a sine wave starting at zero volts — going positive, then crossing zero, going negative and returning to zero. Before vacuum tubes, transistors or IC op amps can amplify AC, they must first be “turned on,” biased with DC using resistors!

STORAGE

AC cannot be contained like DC, which can be “stored” in an electrochemical form as a battery or in an electrostatic form as a capacitor. Imagine filling a bathtub with water or parking a car at the top of a very steep hill. With gravity as “the force,” throwing the switch is equivalent to pulling the drain plug or releasing the brake.

Unlike the popular water analogy, DC does not leak from the battery terminals, although, sometimes, the chemicals do. When AC and DC are put to work, however, magnetic energy is radiated into the air from the cabling, the principle behind inductors and transformers. Radiated AC, in the form of hum and buzz, finds its way into vulnerable “appliances” such as electric guitar and bass — another topic!

DC might imply a constant polarity and voltage, but time cannot be frozen. Even a battery — disposable or rechargeable — holds a charge over a defined period of time that is primarily determined by use. Within time as we know it, a rechargeable battery has a charge and recharge “cycle,” still technically DC, yet one could argue it being of subsonic frequency.

SERIES AND PARALLEL

Two circuit configurations are shown in Fig. 1a and Fig. 1b using resistors for each example, along with the respective formulae for calculating total resistance. Figure 1a shows resistors in series configuration, and Fig. 1b demonstrates parallel. In series, resistor values are simply added together. The total resistance decreases in the parallel configuration; the formula as shown matches the example but can be continued ad infinitum. Add more resistors, and the end result will eventually approach zero ohms; in essence, a piece of wire.

Note: Both “E” and “V” may be used to denote voltage.

Test No. 1: It is common knowledge that two 8-ohm speakers connected in parallel becomes 4 ohms. What if the two speakers were 8 ohms and 4 ohms? (Answer No. 1: 2.6 ohms.)

Tip: Most Windows PCs have a calculator with both Standard and Scientific modes.

Test No. 2a: Ohm's Law and the power formula are included in Fig. 1. These essential electronic tools are used to determine current flow and power consumption. Most cars have a 12-volt battery. Assuming that the sound system's amplifier can swing 10 volts peak-to-peak, what is the peak power output of one channel into an 8-ohm load? (Answer No. 2a: 12.5-watts peak power.)

Test No. 2b: (Bonus Question) What is the RMS power for the same sound system? Hint: See the February issue of this column. (Answer No. 2b: 1.56-watts RMS.)

LYNN EAR

Like the water analogy, a variable resistor can be viewed as a valve, although vacuum tubes and transistors are much more efficient when “heavy lifting” is required. In a recording console, for example, the effect and aux sends are examples of variable resistors in the form of rotary potentiometers (pots).

Most pots are three-terminal devices. Typical connections to a linear fader are detailed in Fig. 2. The input is at the top, output via the middle terminal (called the “wiper”) and the bottom terminal is the signal common, typically connected to ground. The wiper divides the resistor into two parts: The ratio of the bottom resistor to the total resistance determines the amount of input signal that is output. The very first example in the table on p.116 is the easiest to visualize.

The table defines 16-bits of dynamic range in 6dB increments as represented by the change in voltage and the equivalent resistance for a 10-kilohm fader.

A mechanically linear fader may also be electronically linear when a DC control voltage is used for VCA-type automation. The table on p. 116 shows how an audio taper pot differs. Starting with 10 volts at the top of a 10-kilohm fader, each 6dB drop represents a 50% voltage reduction from the previous value. While the first drop from 10 volts to 5 volts is half the electrical value, you know that the wiper knob will not be halfway down.

Test No. 3: Imagine a fader that is both mechanically and electrically linear. Put in 10 volts at the top, set it halfway and get 5 volts out. The fader comes in three resistance options — 1,200 ohms, 4,800 ohms and 10 kilohms. The equivalent circuit is a series resistor pair consisting of…(Answer No. 3: 600, 2,400 and 5 kilohms, respectively.)

Test No. 4: The resistive element in an audio taper pot is “logarithmic,” matching the ear's nonlinear sensitivity to level changes. It's hard to avoid the math, but I encourage those with access to a scientific calculator to engage the dB formula at the lower right corner of Fig. 2. Divide any two voltages from the table, take the log and multiply by 20. Compare your answers with those in the table. It feels good, doesn't it? I used to do this on a slide rule!

MIX MASTER

In a mixer, an IC op amp may have to feed several effect sends as well as the primary fader. If four pots will be connected to an op amp, then it is important, for the sake of efficiency, to choose an optimal resistance value that, when combined in parallel, isn't so low as to overload the op amp or so high as to be vulnerable to stray capacitance (next month's topic).

For ease of head calculation, I chose four 2,400-ohm resistors. Combine the first two pairs into a single pair of 1,200-ohm resistors and then combine those into one 600-ohm load. This, too, is a magic number, vintage gear having input and output impedances of 600 ohms and being referenced to 0 dBm. The Power formula (in Fig. 1a) states that P equals E-squared divided by R. In this case, E is 0.775 volts.

Test No. 5: Apply .775 volts to the load and determine the power. (Answer No. 5: 1 milliwatt.)

Note: One mW is the reference for 0 dBm, 4 dB below “nominal” level for professional equipment.

Many op amps can comfortably drive a 600-ohm load. But within a mixer module, only the output amps need to be prepared to work this hard and only when driving vintage-style equipment. Typical pot values are between 5k and 10 kilohms. Four 10-kilohm pots represent a 2,500-ohm load to the op amp.

Test No. 6: Can you calculate the power if 0.775 volts appear across a 2,500-ohm load? (Answer No. 6: 0.24 milliwatts.)

THE SCIENTIFIC METHOD

In all of the sciences, it is common to minimize the variables to facilitate understanding and simplify calculation, if only for the moment. With that in mind, consider impedance as the full-color version of resistance. The series parallel circuit examples in this article consisted mostly of a battery and resistors — certainly not the real world, so black and white that there wasn't even a power switch! AC voltages were treated like DC, except when calculating peak and RMS power.

Oversimplified, impedance is to AC as resistance is to DC. While both quantities are expressed in “ohms,” the former requires higher math (trigonometry and calculus); the latter can pretty much be ciphered in your head. With the exception of the “dB” formula, the math presented in this article is quite basic. You calculated current using Ohm's Law — 10 volts divided by 10 ohms is 1 ampere — but if a capacitor or inductor were in the circuit, the voltage and current would change over time, requiring several calculations. Even the switch would play a key role!

Next month's math takes time into consideration, but don't get nervous about it. A demo version of “Micro-Cap 6” is available at www.spectrumsoft.com as a free download. Just draw a schematic and it does the ciphering. Meanwhile, let's look at the basic concepts comparing resistance to impedance.

Fader Top Volts Resistance (10-kilohms total) Top-to-Wiper/Wiper-to-Bottom
0 dB 10 0/10k
-6 5 5k/5k
-12 2.5 7.5k/2.5k
-18 1.25 8.75k/1.25k
-24 0.625 9,375k/625
-30 0.3125 9,687.5/312.5
-36 0.15625 9,843.75/156.25
-42 0.078125 9,921.875/78.125
-48 0.0390625 9,960.9375/39.0625
-54 0.01953125 9,980.46875/19.53125
dB millivolts
-60 9.765625 9,990.234375/9.765625
-66 4.8828125 9,995.1171875/4.8828125
-72 2.44140625 9,997.55859375/2.44140625
-78 1.220703125 9,998.779296875/1.220703125
-84 0.6103515625 9,999.3896484375/.6103515625
-90 0.30517578125 9,999.69482421875/.30517578125
-96 0.152587890625 9,999.84741209375/.15287890625
Fader Bottom millivolts All resistor pairs add up 10 kilohms
Table: Based on a typical 10-kilohm console fader, the table shows the relationship between fader position (in dB), output voltage and resistance.

SPECIFIC HEIGHTS

Resistance is a scalar quantity measured in ohms, just as the term “height” is defined by its magnitude (inches, feet, centimeters, meters) and no more. By contrast, temperature is defined by its magnitude — degrees — which alone does not tell the whole story. Wind can make a 20° day feel like 10° (wind chill), while humidity can make 70° feel like 80° (the yuck factor). Like the temperature example, impedance is a vector quantity defined by its magnitude in ohms, but instead of wind chill, there is a phase angle, a manipulation of time.

I-C-E IS N-I-C-E

Remember that DC can be stored over long periods of time in batteries and to a lesser extent in capacitors. Add a switch to the series or parallel resistor examples. At the moment, the voltage E is applied to the resistance R no matter whether the source is AC or DC, the current will be instantaneous.

Reactive components — capacitors and inductors — manipulate the time relationship between voltage and current. A fully discharged capacitor appears as a dead short at the moment it is connected to a DC voltage source, the current leading the voltage by 90°. In engineering school, simple expressions such as “E-L-I the I-C-E man” helped students remember that voltage-leads-current by 90° in an inductor L (a coil of wire).

SPARK!

Have you ever plugged or unplugged a device when its switch was in the On position? At the exact moment the plug and socket made or broke the connection, a sizeable spark most likely occurred. Connecting AC power to a reactive device such as a transformer is one of the reasons lights dim but don't stay dim when a device is turned on.

ABSOLUTE ZERO

Getting back to the temperature analogy for a moment, the materials used for making or plating wire — copper, silver, gold or aluminum — all have a defined resistance at room temperature. Absolute zero on the Kelvin scale (0 K) is the lowest temperature theoretically possible — at approximately -273.16° C (-459.69° F). Wire that cold becomes a more perfect conductor. That's why audiophiles love winter here in Minnesota!

In reality, the slight amount of resistance per foot becomes cumulative with extremes of distance or of thinness. For example, speaker cable consists of a pair of conductors separated by insulation, a capacitor by definition that becomes a contributing factor as the series resistance increases. At high frequencies, the wire also has some inductance. Hold that thought…

Next month, the circuit examples will include capacitors and inductors in real-world examples of signal corruption and failing components. In the meantime, drop by www.tangible-technology.com for a visit.


This past winter, Eddie shoveled 65 inches of snow. By the time you read this, he'll be planting a vegetable garden. Got any seeds?






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