Horseflies, Tractors and Mr. Kirchhoff
March 1968 Radio-Electronics

March 1968 Radio-Electronics [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.

In this 1968 Radio-Electronics magazine educational fiction article, high school student Jerry Whipple challenges Kirchhoff's voltage law in an AC circuit, convinced he's found a flaw. His experiment measures 7.1 volts across both a resistor and inductor in series, totaling 14 volts - contradicting the expected 10 V source voltage. His instructor, Mr. Bean, explains that the discrepancy arises from phase differences in AC circuits: the voltages are out of phase, not additive. Using a tractor analogy, Bean illustrates how forces (or voltages) at angles combine vectorially, not arithmetically. He introduces Pythagoras' theorem to resolve the apparent paradox - the 7.1 V readings are orthogonal components, summing to 10 V as the hypotenuse of a right triangle. The article underscores that meters measure peak voltages, not instantaneous phase-dependent values, preserving Kirchhoff's laws when properly interpreted. The exchange highlights foundational AC principles while celebrating critical thinking in electronics education.

Horseflies, Tractors and Mr. Kirchhoff

By Wayne Lemons

"I'm a little shook about Mr. Kirchhoff" this morning," Jerry Whipple told his high school electronics instructor. "Did Mr. Kirchhoff know about ac circuits?"

The instructor - a short, balding, plump man named Bean - scratched his head and looked at the student. Jerry was a glasses-wearing, baseball-pitching senior who sometimes asked embarrassingly complex questions.

"I suppose you mean the Kirchhoff who developed some laws for electrical circuits?" Mr. Bean.

"That's right," Jerry said, "the guy we studied about in dc circuits. Do his ideas hold water in ac circuits?"

"Far as I know." Mr. Bean grinned. "What'd you have in mind?"

"See this circuit here?" Jerry pointed to an inductor and a resistor: "It's a 10,000-ohm resistor and a 10-millihenry rf choke in series. The experiment calls for putting a 160-kHz, 10-volt peak-to-peak signal across the circuit."

"Fine, so what's your problem?" "Well, according to Mr. Kirchhoff the individual voltage drops across the components in a series circuit are equal to the source voltage. Isn't that so?"

"That's right."

"Then Mr. Kirchhoff better come up with something new, 'cause I think this circuit just repealed his law."

Mr. Bean chuckled. "Don't think Mr. Kirchhoff can do that; he died, I believe, in 1887. But I can't recall anyone ever disputing his conclusions about circuits. What makes you think you've found a flaw in Kirchhoff's rule?"

"I'll demonstrate. With an rf detector probe on the vtvm I measure across the inductor and resistor and I find 10 volts. See?"

"Yes."

"Now," Jerry said, "I'll measure the voltage across the resistor. The meter reads about 7.1 volts. And according to Mr. Kirchhoff I ought to have 2.9 volts across the coil. Right?"

Mr. Bean avoided an answer by asking a question, "How much do you have?"

Jerry didn't reply. He moved the vtvm leads across the coil and the meter came to rest reading again just over 7 volts. "Look at that - just about the same as across the resistor."

"So what's wrong?" Mr. Bean appeared puzzled.

"Wrong?" Jerry blurted. "What's wrong is that 7 volts and 7 volts add up to 14 volts - not 10 volts. And that proves that Mr. Kirchhoff, may he rest in peace, was all wet."

"Interesting," Mr. Bean bantered.

"I've made a quick check in your addition, and 7 and 7 are 14 if you're adding 7 and 7 of the same thing. But, if you'll excuse my farm-boy up-bringing, 7 horses and 7 flies don't make 14 horseflies."

"I understand that," said the boy. "But I'm measuring volts and volts and 7 and 7 ought to be 14."

"But obviously it isn't," said the instructor. "You said yourself that you had only 10 volts to start with. How do you explain that? Maybe we shouldn't be too hard on Kirchhoff until we look a little further into the matter."

"But how is it possible that the component voltage drops add up to more than the source voltage?" Mr. Bean went to the blackboard, motioning Jerry to follow him. He drew a battery symbol, connected one end to an s.p.d.t. switch, and drew two circles to represent meters.

"Look at this circuit," he said. "Assume this is a 9-volt battery. If I move the switch to the left, the left-hand meter will read 9 volts. Agreed?"

Jerry nodded.

If I move the switch to the right, the righthand voltmeter will read 9 volts. Okay?"

Mr. Bean grinned a little and continued: "Now if I move the switch rapidly I will have both meters reading. Because the pointer can't return to zero very fast I will have both meters reading, let's say, 6 volts. Right?"

"I see that," said Jerry, "but I don't see what you're driving at."

"The point is this," said Mr. Bean. "Just because both meters can be made to read 6 volts doesn't mean that the battery voltage is 12 - even though 6 and 6 are 12. Does it?"

"I think maybe I'm beginning to get a twinkle," Jerry said. "Has it got anything to do with the timing in the circuit? Could that current trailing along behind the voltage, you've been talking about, have anything to do with what we see on the meters?"

"Just about everything," the instructor said. "And if you understand just how you'll have gone a long way toward mastering ac circuits."

"Do you mean to say that the voltages in my circuit are being switched around and fooling the meter?"

"Right." Mr. Bean smiled. "Nobody can say that a meter has a lot of sense - it can be fooled."

"How?"

"If you put a certain voltage on a resistor, the current will go up at the same time the voltage goes up. Agreed?"

Jerry nodded.

"What do we say the phase angle of a resistor is then?"

"Zero degrees?"

"That's right - we say that because there is zero delay between the rise time of the voltage and current. Now what about an inductor?"

"In class yesterday you said the current lags the voltage by about 90°."

"And just what does that mean to you?"

"It means, I guess, that the current does not start until after the voltage has already been on the coil a little while."

"And, as I pointed out yesterday, the reason for that is that a coil tries to oppose any change in voltage across it by developing a 'back' voltage-called a counter electromotive force - so long as the voltage is changing. The ac voltage starts to reverse after a quarter of a cycle - or 90° - and when that happens the current starts to flow in the coil. So, in a perfect coil, the current lag is 90°."

"But what happens when there's a resistor in series?"

"Just what happened to you. The phase angle shifts to somewhere between 0° and 90°. In your case the voltage drops are almost identical across the resistor and the coil, so the phase angle is approximately 45°."

"How do you know that?" Jerry asked.

"I'm afraid my farm-boy upbringing is still showing, but in one way or another you've been dealing with this phenomenon all your life," said Mr. Bean. "Suppose two tractors with the same power pull cables attached to the same tree. One tractor pulls north and the other, east. Where will the tree fall ?"

"Halfway between the tractors," Jerry said. "To the northeast."

"Right," agreed Mr. Bean, "and if we consider the tractor heading east to be at zero phase angle, then the tree will fall at a 45° angle to it. Agreed?"

"Yes, I'm beginning to see. If one of the tractors went farther or faster than the other, then the tree would fall toward it and that would change the phase angle."

"Yes. And that's what happens in an electrical circuit when the voltages aren't pulling in the same direction. If the circuit has more voltage across the resistor than across the inductor then the circuit phase angle will be less than 45°. If there is more voltage across the inductor then the phase angle will be more than 45°."

Mr. Pythagoras

"But," mused Jerry, "I'm still not sure I understand how to figure the total voltage in the circuit."

"Sure you do," said Mr. Bean.

"You probably learned it several times in elementary school. Ever hear of Mr. Pythagoras* and his theorem?"

"You mean the one about a right triangle where the square of the hypotenuse is equal to the sum of the squares of the other two sides?"

"That's exactly what I mean. And in our example here if these tractors each had 7.1 pounds of pull on the treetop then the total pull would be IMAGE HERE Does that suggest anything?

"Unfortunately it looks like you're going to get Mr. Kirchhoff out of my dilemma," laughed Jerry. "Anyway it seems that my roughly 7.1 volts across each component is going to have a hypotenuse of 10 volts. I'm still not sure, though, why the meters don't tell me."

"Remember the switch analogy I used earlier?" Mr. Bean asked.

"You mean this circuit is a kind of electronic switch?"

"You might call it that," said Mr. Bean. "You see your meter can't respond to that 160-kHz signal, so it just finds the peak voltage across the component and stays there."

"Then you mean I really never have 7.1 volts across either the coil or the resistor?"

"No, I don't mean that." Mr. Bean grinned. "It's just that if you want Kirchhoff's law - or for that matter Ohm's law - to work in ac circuits, you have sometimes to be concerned with instantaneous values, and not with values taken over a period of time.

"The meter reads the highest voltage that is ever across the circuit components but at the very instant that the voltage is 7.1 on one component, at that very instant it is only 2.9 across the other."

"In other words," Jerry said, "the meter can't fall back to 2.9 volts that fast, so it just stays up at the peak."

"That's pretty close," said Mr. Bean. "So we have to let Pythagoras take over and do the mathematics, but rest assured that Kirchhoff knew what it was all about."

"But what about Pythagoras?"

Jerry demanded. "He must have been in his grave a long time before anyone discovered electricity."

"Yes, as far as anybody knows, Pythagoras never dreamed of such a thing as electricity. But like a lot of other people who discover basic truths, he supplied the answers before we invented the problems."

"And you can't hardly get smarter than that," Jerry said and headed for the door as the class bell rang.

Mr. Bean grinned affectionately after his best student. It was nice to have someone who cared enough to question other people's conclusions ... even his own. R-E

* Gustav Robert Kirchhoff, German physicist, 1824-1887.

* Pythagoras. Greek philosopher and mathematician, 582-500 BC.