last updated: 20210131
Song of this chapter: All songs by AC/DC ;)
Let's do a little history.
In 1887 in the United States Thomas Edison had constructed 121 DC power stations. At that time DC could not be easily converted to high voltages, needed to transport energy over longer distances. Edison promoted a system of many small, local power plants, that would power individual neighborhoods or city sections. With 110 V DC the voltage drop limited the maximum distance from the plant to the end user to 1.6 km. This limitation made power distribution in rural areas impossible.
In 1886 all Rome was electrified with AC. George Westinghouse ( entrepreneur and engineer from Pittsburgh) became aware of the new European alternating current systems in 1885 when reading an UK technical journal Engineering. He imported transformers and an AC electric generator, and began experimenting with AC networks and helped installing the first multiplevoltage AC power system. An hydroelectric generator produced 500 V AC stepped down to 100 V to light incandescent bulbs. The Westinghouse company installed more AClighting systems and by the end of 1887 it had 68 alternating current power stations.
The competition with Edison became the "War of the currents" with Thomas Edison spreading public perception that the high voltages used in AC distribution were unsafe.
Most power plants (water, wind, coal, nuclear, gas) give us mechanical (kinetic) energy of rotation (from steam turbines, gas turbines or water turbines). With generators (dynamos or alternators) this rotational energy is converted to electrical energy.
A current (moving electrons) produces a magnetic field. Bending a wire into multiple closely spaced loops to form a coil gives us an electromagnet.
On the other hand a moving wire (or a coil) in a magnetic field produces a voltage. This is called electromagnetic induction. A conductor loop (or a coil of wire) rotating in a magnetic field produces a current which changes direction with each 180° rotation, an alternating current (AC).
The inducted voltage u
depends on the changing of the magnetic flux Φ
per time and the number of windings (N
) of the coil.
The magnetic flux depends on the magnetic flux density B
of the magnetic field and the effective area of the conductor loop A
in the magnetic field.
The width w
of the effective loop area in the magnetic field is maximum at 0°
, and zero at 90°
. It can be calculated with the cosine (cos
) of the angle α
.
The changes of the magnetic flux is minimal at 0°
and 180°
and is biggest at 90°
(negative) and 270°
. The derivative of a cosine gives a sinus. That's why sinus :). The induced voltage can be calculated with:
Here is the oscilloscope screen of a measured voltage in a wall socket. The oscilloscope measured three voltages, the frequency and the period. Let's take a closer look at these physical quantities.
û
and peak to peak voltage U_{SS}
û
(U_{s}
) is the peak voltage
(V_{p}
) in Volt (V).
On the screen it is named Max. This voltage is in the European power supply system 325 V
.
The peaktopeak voltage of an AC voltage U_{SS}
(V_{pp}
) in Volt (V) is the difference between its positive peak and its negative peak. For a sinusoidal voltage (sin(x) goes from +1 to −1) the maximum values swing between +û
and −û
. The peaktopeak voltage is therefore û−(−û) = 2û
.
T
, frequency f
and angular frequency ω
The amplitude of the voltage is changing in the rhythm of the sine between the peak values. One period is one revolution of the generator (0  360°). The time needed for such a revolution depends of the rotating speed of the generator. If the generator turns with 3000 revolutions per minute or 50 revolutions per second we get 50 periods per second. One sine wave of the voltage returns 50 times in one second.
This is called the frequency f
of the voltage. The time needed for one period is called periodic time T
.
Frequency is the number of occurrences of a repeating event per unit of time. The symbol is f
and the unit is hertz Hz
or s^{1}
(cycle per second).
The period T
is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. It's unit is the second s
.
One revolution is equal to 360°
or 2π
radians. We can use the angular frequency ω
to connect the time of one revolution with the angle.
The angular frequency ω
is a scalar measure of rotation rate. It's unit is radians per second
(s^{1}
).
For our sine wave we get:
with the momentary angle:
So, why 50 Hz? The alternating current needed a frequency high enough to avoid flickering (see following exercise in this chapter) and not too high for the rotational speed of the generators.
U
(U_{eff}
, U_{RMS}
)The peak voltage is not a good value to describe the effect of an AC voltage. We want an AC voltage value that has the same effect than a DC voltage (how hot gets a radiator or how bright is a lamp). This can be achieved by comparing the time averaged power P
. Instantaneous power p(t)
would be calculated with:
A time averaged power (where the averaging is performed over any integer number of cycles and with the same effect than DC power) can be calculated using a new AC voltage value called root mean square (RMS
) value U_{RMS}
(V_{RMS}
) or effective value U_{eff}
(europe) or simply a big U
.
The effective voltage of a waveform can be calculated with the corresponding crest factor:
Sine wave:
Triangle wave and sawtooth wave:
Square wave:
We want to measure the AC voltage in our socket. Because this is potentially :) dangerous, we will use a transformer to reduce the voltage and nevertheless be able to visualise the curve of the voltage. Read and copy the characteristics marked on the transformer.
We will use our multimeter to measure the AC voltage and the frequency of the voltage (frequency range). What maximum and minimum frequencies the used multimeter can measure (read the manual, look at the frequency range)? What maximum frequency the AC voltage may have to measure correct RMS values (AC voltage range)?
Measure and note the voltage(s) and frequency(ies) and compare them with the characteristic of the transformer.
To view the voltage characteristic we will use an oscilloscope. We use a mixedsignaloscilloscope from Rohde & Schwarz. The RTC 1002
has 2 channels and can display signals up to 100 MHz. Our oscilloscope is able to save the oscilloscope pictures to an USBstick. You will use this feature to document this exercise and all future exercises where the oscilloscope is needed. Document the oscilloscope picture of the AC voltage.
Measure the period time, frequency and voltages (U_{p}, U_{pp}, U_{RMS}) with the oscilloscope. Calculate the root mean square and compare with your values from the voltmeter and the oscilloscope.
Comment the measured values and the measured waveform.
Alternating voltages and currents can come in a number of different waveform's. With oscilloscopes we can see this changing amplitude over time.
Square waves are used for example in digital switching electronics, as timing references (clock signals) and in music synthesizers (square waves sound hollow, and are used as the basis for wind instrument sounds).
Triangle waves are also found in sound synthesis and are useful for testing linear electronics like amplifiers.
Sawtooth waves are often used in music synthesizers, in switchedmode power supplies and the sawtooth wave is used to produce vertical and horizontal deflection signals used to generate a raster on CRTbased television or monitor screens.
PWM waves have multiple applications, as servo control, encoding, voltage regulation and power delivery.
Lets use our ESP32 board to produce and measure alternating voltages (the chip is running with 3.3 V!). The first Arduino sketch will produce the simplest waveform, a square wave. This is done with the simple blink demo program. Connect an LED to digital pin 16 and test the following program. Calculate the period time T
and the frequency f
. Watch the signal on the oscilloscope and measure T
and f
with the oscilloscope. Save in all following exercises the oscilloscope pictures to an USBstick and use them to properly document your tasks.
// iot_jdi_AC4.ino (blink)
// weigu.lu
const byte PIN_SQUARE_OUT = 16;
unsigned int delay_time = 100;
void setup() {
pinMode (PIN_SQUARE_OUT, OUTPUT);
}
void loop() {
digitalWrite(PIN_SQUARE_OUT, LOW);
delay(delay_time);
digitalWrite(PIN_SQUARE_OUT, HIGH);
delay(delay_time);
}
Reduce the delay time until you can't see the blinking no more. Calculate the frequency. If you are interested in this phenomena read the wiki articles about the flicker fusion threshold and the persistence of vision.
Now we replace the LED with a piezoelectric speaker. Replace the delay()
function with the delayMicroseconds()
function. Calculate the different values for delay_time
, so we can hear tones of 200 Hz, 1 kHz and 10 kHz.
Comment the two delay lines with //
and measure the maximal frequency the ESP32 can deliver. The square is no more symmetric because the ESP32 has an operating system running in the background and that consumes time.
With the next sketch we create a pulsewidth modulation (PWM). The frequency of the square stays the same, but the square will not stay symmetric (pulse width time + offtime = constant). To change the width of the pulse we connect the wiper of a 10 kΩ potentiometer to pin 4. The other two terminals go to 3.3 V and GND. We use the monitor to view the potentiometer values. Test the following sketch (oscilloscope and speaker connected). Document the oscilloscope screen for a pulse width of about 80% of the period:
// iot_jdi_AC5.ino
// weigu.lu
const byte PIN_SQUARE_OUT = 16;
const byte PIN_POTENTIOMETER = 4;
unsigned int delay_time = 4095;
unsigned int pulsewidth;
void setup() {
Serial.begin(115200);
pinMode (PIN_SQUARE_OUT, OUTPUT);
}
void loop() {
pulsewidth = analogRead(PIN_POTENTIOMETER);
Serial.println(pulsewidth);
digitalWrite(PIN_SQUARE_OUT, LOW);
delayMicroseconds(delay_time  pulsewidth);
digitalWrite(PIN_SQUARE_OUT, HIGH);
delayMicroseconds(pulsewidth);
}
Write in your words the definition of duty cycle (wiki). Find the formula to calculate the duty cycle. Draw an U = f(t)
diagram for a duty cycle of 30 %. Mark t_{i}
, t_{p}
and T
in the diagram.
Connect an LED instead of a speaker. Calculate the pulsewidth value needed for a duty cycle of 30 %. Eliminate the potentiometer and serial code to get an accurate result and document the oscilloscope screen.
The ESP32 has two internal DigitaltoAnalog Converters (DAC). These are faster than PWM. We can use our ESP32 to build a function generator that produces different waveforms. Here is an example. Analyze the program and document it. Read the following article about Arduino arrays. Why don't we calculate the array during the main loop?
Test the program:
// iot_jdi_AC6.ino
// weigu.lu
// DAC1 on IO25, DAC2 on IO26 (8 bit)
float y = 0;
byte sine[255];
byte i = 0;
void setup() {
for (i=0; i<255; i++) {
y = sin(((float)i/255)*2*PI);
sine[i] = byte(y*128)+128;
}
}
void loop() {
for (int i=0; i<255; i++) {
dacWrite(DAC1, sine[i]);
dacWrite(DAC2, 255i);
delayMicroseconds(1);
}
}
Oscilloscope the voltages on DAC1 and DAC2. What is the name of the second waveform?
Change dacWrite(DAC2,i);
to dacWrite(DAC2,255i);
. Comment the result.
Without supplementary circuitry the ESP32 delivers only voltages from 0 to 3.3 V. We get no negative voltages. To test further waveforms, we will use a two channel arbitrary waveform generator (1 µHz  10 MHz) from PeakTech, the PeakTech 4124. Connect the function generator to a piezoelectric speaker and to the oscilloscope. Test the different waveforms. Document the screens.
As seen the most common type of AC is the sine wave.
The french mathematician Joseph Fourier discovered in 1822 that the sine wave has a very special characteristic.
Any periodic waveform can be described with a sum of sine (or/and cosine) waves.
The study of the way functions may be represented or approximated by sums of e.g sine waves is called Fourier analysis. The representation of a function as the sum of simple sine waves is called a Fourier series.
The following program visualizes the synthesis of a square wave by adding sinuses. Connect the two channels of your oscilloscope to DAC1 and DAC2.
As you see we add the harmonics with 3f
, 5f
, 7f
and 9f
with a reduced amplitude of 1/3
, 1/5
, 1/7
and 1/9
. Test the program and document the screens.
// iot_jdi_AC7.ino
// weigu.lu
// DAC1 on IO25, DAC2 on IO26 (8 bit)
float y = 0;
const byte ymax = 80;
byte i = 0, sine[255],sine3[255],sine5[255],sine7[255],sine9[255];
unsigned int j = 0;
void setup() {
for (int i=0; i<255; i++) {
y = sin(((float)i/255)*2*PI);
sine[i] = byte(y*ymax);
}
for (int i=0; i<255; i++) {
y = sin(((float)i/255)*2*3*PI);
sine3[i] = byte(y*ymax/3);
}
for (int i=0; i<255; i++) {
y = sin(((float)i/255)*2*5*PI);
sine5[i] = byte(y*ymax/5);
}
for (int i=0; i<255; i++) {
y = sin(((float)i/255)*2*7*PI);
sine7[i] = byte(y*ymax/7);
}
for (int i=0; i<255; i++) {
y = sin(((float)i/255)*2*9*PI);
sine9[i] = byte(y*ymax/9);
}
}
void loop() {
for (j=0; j<1000; j++) {
for (i=0; i<255; i++) {
dacWrite(DAC1, sine[i]+128);
dacWrite(DAC2, sine3[i]+128);
delayMicroseconds(1);
}
}
for (j=0; j<1000; j++) {
for (i=0; i<255; i++) {
dacWrite(DAC1, sine5[i]+128);
dacWrite(DAC2, sine7[i]+128);
delayMicroseconds(1);
}
}
for (j=0; j<1000; j++) {
for (i=0; i<255; i++) {
dacWrite(DAC1, sine[i]+sine3[i]+128);
dacWrite(DAC2, sine[i]+sine3[i]+sine5[i]+128);
delayMicroseconds(1);
}
}
for (j=0; j<1000; j++) {
for (i=0; i<255; i++) {
dacWrite(DAC1, sine[i]+sine3[i]+sine5[i]+sine7[i]+128);
dacWrite(DAC2, sine[i]+sine3[i]+sine5[i]+sine7[i]+sine9[i]+128);
delayMicroseconds(1);
}
}
}
Change the harmonics to 2f
, 3f
, 4f
and 5f
with a reduced amplitude of 1/2
, 1/3
, 1/4
and 1/5
. Which waveform will arise? Document the screens.
Let's have a closer look on the Fourier series of a square wave:
We see that a square wave with a certain base frequency f
can be created by adding to a sine wave with the same base frequency and an amplitude of 4/π = 1.2732
to a second sine wave with 3 times the base frequency and an amplitude of 4/3π = 1.2732/3
and a third sine wave with 3 times the base frequency and an amplitude of 4/5π = 1.2732/5
and so forth.
Harmonics of a wave with a base frequency f are waves with a frequency that is a positive integer multiple of the frequency f. So the original wave is also called the 1st harmonic. All other waves with a multiple of f are named higher harmonics. For example, if the fundamental frequency is 220 Hz (a in music), the frequencies of next three higher harmonics are 440 Hz (2nd harmonic, also a in music but an octave higher), 660 Hz (3rd harmonic) and 880 Hz (4th harmonic, also a in music but 2 octaves higher). The differences of sound in music instruments playing the same note (base or fundamental frequency) are the mixture of the higher harmonics. Listen to the "additive square demo" on [wikipedia]((https://en.wikipedia.org/wiki/Square_wave).
Even by adding multiple sine waves up to very high frequencies, the square wave is not perfect! There will always be little ripples (ringing artifacts) and the rise time (time to change from a low to high) will not be infinite low.
This tells us a very important lesson.
It is impossible to achieve in physical systems an ideal square wave (instantaneously changes between the high and the low state and no under or overshooting). This would require an infinite bandwidth, but every real system has the characteristic of a lowpass filter. To transport a digital signal, the passing frequency of the used transportation channel must be much higher than the base frequency of the signal.
If we use an oscilloscope of 20 MHz to watch a perfect digital signal of 10 MHz, we will see a sine wave!! Even with an oscilloscope of 100 MHz we will not see the real signal!
Another example:
In older days :) the telephone line was limited to 3.4kHz. It was not possible to pass a digital signal directly at a reasonable frequency. So it was necessary to use a modem (modulatordemodulator). Instead of sending a digital signal four sine waves were used. The call originator used sine waves at 1070 Hz and 1270 Hz (for low and high) and the answering modem 2025 Hz and 2225 Hz. This modulation is a frequency modulation and is called Frequencyshift keying (FSK
) and allowed fullduplex at 300 bit/s! (50 years ago). If you want to see modem with acoustic coupler watch the film war games from 1983.
Square waves, often used in digital circuits, contain a wide range of harmonics. These can generate electromagnetic radiation that interferes with other nearby circuits, causing noise or errors.
Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. On the following picture we see the frequency spectrum of three different waveform's:
Our oscilloscope has the possibility to watch the frequency domain with a fast Fourier transform function FFT (read the manual). Use the frequency generator with a square wave and a frequency of 500 kHz and document the screen. Determine the amplitudes of the first, third and fifth harmonic of the square wave in decibelmilliwatts dbm
and in Volt V
.
What are dbm
? Search the net and explain. How many watt (W
) are 20dBm, 30dBm and 90dBm?
Calculate the voltages of your measured first, third and fifth harmonic.
Class  Denomination  Short  Frequency  Wavelength 

         
Gamma rays  γ  300 EHz 30 EHz 
1 pm 10 pm 

Ionizing radiation 
Hard Xrays  HX  300 EHz 
1 pm 
Soft Xrays  SX  300 PHz  1 nm  
Extreme ultraviolet  EUV  30 PHz 3 PHz 
10 nm 100 nm  
         
Visible radiation => 
Near ultraviolet  NUV  3 PHz 300 THz 
100 nm 1 µm 
Near infrared  NIR  30 THz 
10 µm 

Mid infrared  MIR  3 THz 
100 µm 

Far infrared  FIR  300 GHz 
1 mm 

         
Extremely high frequency  EHF  300 GHz 30 GHz 
1 mm 1 cm 

Microwaves 
Super high frequency  SHF  3 GHz 
1 dm 
Ultra high frequency  UHF  300 MHz 
1 m 

Very High frequency  VHF  30 MHz 
10 m 

and         
High frequency  HF  30 MHz 3 MHz 
10 m 100 m 

Medium frequency  MF  300 kHz 
1 km 

Low frequency  LF  30 kHz 
10 km 

Radio waves  Very low frequency  VLF  3 kHz 
100 km 
        
Ultra low frequency  ULF  3 kHz 300 Hz 
100 km 1 Mm 

Super low frequency  SLF  30 Hz 
10 Mm 

Extremly low frequency  ELF  3 Hz 
100 Mm 
In the war of currents Edison used the danger of currents to argue against AC voltage. A pamphlet claimed DC had not caused a single death, and included newspaper stories of accidental electrocutions caused by alternating current. Westinghouse on the other side pointed out in letters to various newspapers the number of fires caused by DC equipment.
If you touch the electric wire around a meadow, you touch a voltage of 2000 V  5000 V. Why does this high voltage does not kill you on the spot?
It's not the voltage that kills (voltage can hurt without direct damage), but the height of the current and the duration the current passes the heart. The current of the meadow wire is limited to 100 mA  300 mA and the duration to 20 ms  100 ms.
The IEC publication 604791 defines four zones of currentmagnitude /timeduration. In each of these four zones the physiological effects are described.
Source: Graphic by Cmglee under Creative Commons AttributionShareAlike 3.0 Unported license(CC BYSA 3.0).
Time/current zones for AC 15 Hz to 100 Hz for hand to feet pathway
Zones  Boundaries  Physiological effects 

AC1  Up to 0.5 mA  Perception possible but usually no ‘startled’ reaction 
AC2 
0.5 mA up to green curve 
Perception and involuntary muscular contractions likely but usually no harmful electrical physiological effects 
AC3 
Green curve up to red curve 
Strong involuntary muscular contractions. Difficulty in breathing. Reversible disturbances of heart function. Immobilization may occur. Effects increasing with current magnitude. Usually no organic damage to be expected 
AC4 _{1)} 
Above red curve 
Pathophysiological effects may occur such as cardiac arrest, breathing arrest, and burns or other cellular damage. Probability of ventricular fibrillation increasing with current magnitude and time 
AC4.1  Probability of ventricular fibrillation increasing up to about 5 %  
AC4.2  Probability of ventricular fibrillation up to about 50 %  
AC4.3  Probability of ventricular fibrillation above 50 % 
_{1)}
For duration's of current flow below 200 ms, ventricular fibrillation is only initiated within the vulnerable period if the relevant thresholds are surpassed. As regards ventricular fibrillation, this figure relates to the effects of current which flows in the path left hand to feet. For other current paths, the heart current factor has to be considered.
As we see above, the differences between AC and DC are not huge.
Simplified we could state that for a short time (less than 200 ms) a human can stand 50 mA (AC or DC).
Ohm's law states that the current drawn depends on the resistance of the body. Every person has his own resistance as a lot of different factors are involved (e.g. like for wires as resistors, the resistance of an arm or leg depends on it' length and diameter). Men tend to have lower resistance than women (thicker arms and legs).
A rough value for the internal resistance of the human body is 300 Ω  1000 Ω. The path that electricity takes through the body is also important (e.g. the resistance from hand to foot is bigger than the distance from thumb to ringfinger on the same hand).
But the main resistance comes from the skin, witch is a poor conductor if it's not wet or burnt/blistered. The resistance from dry skin is between 1000 Ω  100000 Ω.
The total resistance is:
For simplification we will memorize that the total resistance of the human body is 1000 Ω.
It is the current that kills, but anyway we can state:
Voltages above 50 V AC and above 120 V DC are dangerous!
For children and animals voltages above 25 V AC and above 60 V DC are dangerous!
Voltages under 25 V AC and under 60 V DC schould be used, they are not dangerous!
A current through a wire causes heat, and so a third of all fires is caused by short circuits or overheating of electric equipment. Other dangers of electricity are the high forces generated through magnetism at high currents, or flash burns of the eyes flash arcs caused in short circuits. A current through a fluid can also produce toxic or explosive gases.