Simple, Practical Electronics

By Steven Bayes

The first edition of this book analyses Transistor Based Analogue Electronics, only, with concentration on Bipolar Junction Transistors ( BJT's ) in Audio. Junction Field Effect Transistors ( JFET's ) are, also, discussed.

 

The book analyses transistor circuits without any Mathematics or Physics, just, simple, analogue logic analysis.

 

Mathematics, commonly used in Electronics, is also provided, where, only, readily available formulae and theorems are provided without their derivations.

 

Parts of the book, which, contain Mathematics, are marked accordingly and the reader is advised to skip these parts. Parts of the book, which, contain simple logic analysis of Analogue Electronics, are marked accordingly and the reader is advised to read these parts.

 

• Language: English
• Publisher: Steven Bayes
• Release date: Jun 4, 2021
• ISBN: 9798201087500

Book preview

Contents :

Chapter 1 : Simple Mathematics, Physics and Electricity, Used in Electronics

RMS Mathematics

Simple Electricity

Filters

Imaginary Numbers Mathematics

Complex and Simple Filters and Bode

Integrators and Differentiators

Dividend and Divisor

Some Passive Circuit Analysis

Chapter 2 : Transistor Electronics

What is Transistor?

Inner Work of Transistors ( Transistor Micro Technology )

Chapter 3 : Transistor Application Electronics Engineering

Simple Device, Simple Explanation

External Feedback Circuit : Common Collector Circuit

Common Emitter, the Simplest Amplifier

Example with the Simplest Amplifier

The Simplest Voltage Amplifier

More on the Simple Amplifier

Common Emitter Amplifier

Combination Circuit

Common Base Amplifier ( Alternative, Standalone Schematics )

Common Base Amplifier ( Standard Schematics )

General Emitter Circuit

Chapter 4 : Examples Made in Practice

Common Emitter Push Pull Amplifier

Standalone, Transistor Based, Differential Preamplifier

Distortion by Amplification

The main objective of this book is the explanations.

Because Analogue Electronics does not contain anything, but, a simple analogue logic ( analogue common sense ) all explanations are, therefore, simple and cannot be complicated. There is no mathematics in electronics ( and in any engineering ) and, therefore, no mathematics is necessary to be known in order to know, understand, apply, design and develop Electronics.

Electronics is a separate science ( engineering ) and has nothing to do with Electrical Techniques Science ( Electricity ). Electrical Techniques Science ( Electricity ) is a separate science ( engineering ) which analyses passive components only, such as, resistors, capacitors and coils, i. e., the linear changes between voltage potentials and electrical currents. Electrical Techniques Science ( Electricity ) conforms to steady parameters of components and is based on the linearity of the main laws of Ohm and Kirchhoff.

Of course, Electronics and anything else conforms to these laws as they are general principle laws of physics with equivalents in any other branch of engineering such as hydraulics and pneumatics.

However, Electronics deals with changes of the main parameters of Electricity ( resistance, mainly, capacitance and inductance ), which changes are caused by the electrical currents and voltages. As an example, the collector emitter resistance, Rce, of a transistor changes with an electrical current ( the base emitter current ). The forward resistance of a diode also changes with the forward current and so does this of a Zener diode.

Simple Mathematics, Physics and Electricity, Used in Electronics

Mathematics has been used in this whole chapter. Therefore, this whole chapter can be skipped, although, part of this chapter uses Electricity, which, is, widely, used in Electronics.

Although not necessary and incorrect, some have decided to use mathematics in Electricity. Even worse, some have decided to use mathematics in Electronics.

As everywhere, there have been a lot of strange people in the field of Electronics.

Thankfully, the used mathematics is not very complicated, yet, still, unnecessary. Also, thankfully, there is no need to understand this mathematics. The only thing necessary for this mathematics is to know a few, readily available, formulae.

RMS Mathematics

RMS mathematics is easy and easily confusing. The confusion comes, because, normally, half of something is this thing, divided by two. In case of RMS, this may not be the case.

RMS means Root Mean Square and what this means can be derived directly from the name. Take a few values, square them, add the squares and take a square root of the sum.

An important observation is the values in the RMS calculation are squared. Therefore, negative values provide the same squared values as positive values. Thus, whether the signal is rectified or not, the RMS is the same.

Physically, RMS is called Effective Value. This is, because, RMS of a given, changing, signal ( AC ) is the effect which this changing signal has, equivalent to the effect of a constant ( DC ) signal with a value, equal to the RMS value of the changing signal.

Thus, an RMS value of a given, changing, signal is equal to the value of the equivalent constant signal.

Some RMS Rules

Um is the amplitude of a given, periodical signal.

1. Half Wave and Full Wave Dependence

For a given periodical signal with equal half periods and with equal | values | of the signal during each of the half periods ( with just opposite signs ), half wave consists of only the positive or only the negative values of this signal. The other polarity values are replaced by zeroes.

The same as with RMS of a full wave, periodical signal, to find RMS of a half wave, periodical signal, take each possible value, square each possible, add the squares to make a sum of them and take a square root of this sum. For full wave, as mentioned, each squared value is repeated twice over the period, because square of a negative is equal to square of the positive. Then then sum of all squared values is taken. Then a square root of the sum is taken. Thus, for the full wave signal, each value, Ui, is repeated twice over the period and appears twice in the sum of squares : Ui ² + Ui ² = 2 * Ui ². For half wave, only once : Ui ². Therefore, before the square root is taken, a full wave will make a sum of values, Usum, which is twice the sum made by the half wave, Usumhw. Or Usum = 2 * Usumhw, therefore Usumhw = Usum / 2. By definition, as mentioned, RMS is the square root from the sum Usum, i. e., Urms = √ Usum. And when the square root is taken from Usumhw, Urmshw = √ Usumhw = √ ( Usum / 2 ) = √ Usum / √ 2. However, Urms = √ Usum. Therefore,

Urmshw = Urms / √ 2

where,

Urmshw is the RMS value of the half wave

Urms is the RMS value of the full wave

2. RMS of a Signal, Made by Addition of Two ( or Many ) Signals

The overall signal, signal 12, can be represented as a sequence of all values of the first signal, signal 1 and a sequence of all values of the second signal, signal 2. Therefore, each value of signal 1 is squared and each value of signal 2 is squared. The sum of all squared values, Usum12 is the sum of all squared values of signal 1, Usum1, plus the sum of all squared values of signal 2, Usum2, therefore, Usum12 = Usum1 + Usum2. However, by definition, the RMS of each signal is Urms1 = √ Usum1 and Urms2 = √ Usum2. Therefore, Urms1 ² = Usum1 and Urms2 ² = Usum2. Therefore, Urms1 ² + Urms2 ² = Usum1 + Usum2 = Usum12. Therefore, Usum12 = Urms1 ² + Urms2 ². Therefore, √ Usum12 = √ ( Urms1 ² + Urms2 ² ). However, by definition, the RMS of the superimposed signal, Urms12, is : Urms12 = √ Usum12. Therefore,

Urms12 = √ ( Urms1 ² + Urms2 ² )

and

Urms12 ² = Urms1 ² + Urms2 ²

where,

Urms12 is the RMS of the overall signal, made by a superimposition of two signals, signal 1 and signal 2

Urms1 is the RMS signal 1

Urms2 is the RMS signal 2

The same applies for a superimposition of n signals.

Urmsall ² = ∑ Urmsi ² ( for i = 1 to n )

and

Urmsall = √ [ ∑ Urmsi ² ( for i = 1 to n ) ]

where,

Urmsall is the RMS of the overall signal, made by a superimposition of n signals

Urmsi is the RMS signal i

3. Any Wave with an Offset

As per rule 2, the RMS of a given, periodical signal with a constant ( DC ) offset is the RMS of the squared RMS of the DC offset plus the squared RMS of the periodical signal without any constant ( DC ) offset. However, the RMS of the DC signal is equal to the value of the DC signal. Therefore :

Urmsdcac = √ ( Udc ² + Urms ² )

where

Urms is the RMS of the signal without any offset ( zero offset, zero DC )

Udc is the DC offset

Urmsdcac is the RMS of the whole signal

An important observation is the values in the RMS calculation are squared. Therefore, negative values provide the same squared values as positive values. Thus, any combination of positives and negatives between the offset and the signal give the same value, which is sign invariant.

RMS of Some Signals

1. Sinusoidal

Figure RMS 1. shows various sinusoidal wave arrangements.

Figure RMS 1. : Some Sinusoidal Wave Arrangements

Figure RMS 1a. through Figure RMS 1c. show some sinusoidal wave arrangements without any offset. Figure RMS 1a. shows a sinusoidal wave, Figure RMS 1b. shows a full wave rectified sinusoidal wave and Figure RMS 1c. shows a half wave rectified sinusoidal. Figure RMS 1d. through Figure RMS 1f. show the same sinusoidal wave arrangements with a DC offset, Udc. All sinusoidal waves have an amplitude of Um.

a. Full Wave

Urms = Um / √ 2

b. Half Wave

Urmshw = Urms / √ 2 = ( Um / √ 2 ) / √ 2 = Um / 2

c. Full Wave with Offset

Urmsdcac = √ ( Urmsdc ² + Urms ² ) = √ ( Udc ² + Um ² / 2 )

d. Half Wave with Offset

Urmshwacdc = √ ( Urmsdc ² + Urmshw ² ) = √ ( Udc ² + Um ² / 4 )

2. Rectangular ( Square ) Wave

A rectangular ( square ) wave is a rectangular, periodical signal, with a period T, with a positive rectangle, for a period of tpos = T / 2 and an equal in value ( with only different sign ) negative rectangle for a period tneg = T / 2.

Figure RMS 2. shows similar arrangements of waves as Figure RMS 1., but, the waves are rectangular ( square ) waves with equal rectangles and, therefore, a 50% duty cycle.

Figure RMS 2. : Some Rectangular ( Square ) Wave Arrangements, Equal Rectangles, 50% Duty Cycle

a. Full Wave

Urms = Um

b. Half Wave

Urmshw = Urms / √ 2 = Um / √ 2

c. Full Wave with Offset

Urmsdcac = √ ( Urmsdc ² + Urms ² ) = √ ( Udc ² + Um ² )

d. Half Wave with Offset

Urmshwacdc = √ ( Urmsdc ² + Urmshw ² ) = √ ( Udc ² + Um ² / 2 )

3. Rectangular ( Square ) Signal with Any Duty Cycle

A rectangular ( square ) wave is a rectangular, periodical signal, with a period T, with a positive rectangle, for a period of tpos and an equal in amplitude, but, different in sign negative rectangle for a period tneg. tpos ≠ tneg. tpos + tneg = T.

Figure RMS 3. shows similar arrangements of waves as Figure RMS 1., but, the waves are rectangular ( square ) waves with non equal rectangles with equal amplitudes and a random duty cycle.

Figure RMS 3. : Some Rectangular ( Square ) Wave Arrangements, Non Equal Rectangles with Equal Amplitudes, Random Duty Cycle

a. Full Wave

Urms = Um

b. Half Wave with the Positive Rectangle ( All Negative Rectangle Values Are Zero )

Urmshwpos = Um * √ ( tpos / T )

c. Half Wave with the Negative Rectangle ( All Positive Rectangle Values Are Zero )

Urmshwneg = Um * √ ( tneg / T )

d. Full Wave with Offset

Urmsdcac = √ ( Urmsdc ² + Urms ² ) = √ ( Udc ² + Um ² )

d. Half Wave with Offset

Positive Rectangle :

Urmshwacdcpos = √ ( Urmsdc ² + Urmshwpos ² ) = √ ( Udc ² + Um ² * tpos / T )

Negative Rectangle :

Urmshwacdcpos = √ ( Urmsdc ² + Urmshwneg ² ) = √ ( Udc ² + Um ² * tneg / T )

The rectangular wave of Point 3. is exactly the same as the rectangular signal with any duty cycle of point 4., but, for the rectangular wave of Point 3., tpos = tneg = T / 2. The calculations, therefore, must and are the same.

4. Triangular Signal, General

a. Without Any Offset

For any triangular, periodical signal with any shape or form, half wave or full wave, and with the only condition to have equal positive and negative triangles, with equal amplitudes ( height ) and sides, regardless of their orientation, again, regardless whether half wave ( one of the triangles, either the positive or the negative = 0 ) or full wave,

Urms = Um * √ [ t / ( 3 * T ) ]

where,

Urms is the RMS value of the signal

Um is the amplitude ( the height of the triangle )

t is for how long the triangle is on ( non zero values, which do not belong to the triangle, not part of the triangle )

T is the period of the periodical signal

In case the signal is a periodical triangular full or half wave without any zero values, which are not part of a triangle ( any zero values, which are different than the zero values, which the triangle makes on the t axis ), then, t = T and, therefore,

Urms = Um * √ [ t / ( 3 * T ) ] = Um * √ [ T / ( 3 * T ) ],

Urms = Um / √ 3

b. With Offset

General,

Urmsdcac = √ ( Urmsdc ² + Urms ² ) = √ [ Udc ² + Um ² * t / ( 3 * T ) ]

In case the signal is a periodical triangular full or half wave without any zero values, which are not part of a triangle ( any zero values, which are different than the zero values, which the triangle makes on the t axis ), then, t = T and, therefore,

Urmsdcac = √ ( Urmsdc ² + Urms ² ) = √ ( Udc ² + Um ² / 3 )

The triangular signals, just analysed, are of immense importance for calculation of ripple voltages, because, the ripple voltage signal is very closely approximated to a triangular signal.

Figure RMS 4. shows some periodical, triangular signals with zero values between the triangles, which zero values do not belong to the triangles, with and without the DC offset, Udc. Figure RMS 5. shows some periodical, triangular signals without zero values between the triangles, which zero values, do not belong to the triangles, with and without the DC offset, Udc.

Figure RMS 4. : Some Periodical, Triangular Signals with Zero Values Between the Triangles ( Random Duty Cycle )

Figure RMS 5. : Some Periodical, Triangular Signals without Zero Values Between the Triangles ( 50% Duty Cycle )

Simple Electricity

Although Electronics is not a subject of this chapter, Electricity, which, is, widely, used in Electronics and cannot be avoided, is discussed.

Some Electrical Forewords

Electrical Techniques ( Electrical ) are a totally separate subject as compared to Electronic Techniques ( Electronics ).

Electrical deals with components which cannot change their parameters with the changes of the electrical parameters through or over them. A simple example is a resistor. Whatever the current through a resistor and whatever the voltage over a resistor is irrelevant. The resistor never changes the resistance based on the current through or the voltage over.

In contrast, electronic components change their parameters depending on the current through them or the voltage over them or the two thereof. Electronics deals with these components and analyses schematics considering these changes.

Electronics does not and cannot change Electrical. The laws of Ohm and Kirchhoff, for example, are laws of physics and cannot be broken.

Thus, Electronics must obey Electrical as well as must use Electrical in any analysis.

Thus, a few words best be said for the most used Electrical principles in Electronics, other than the said basic Electrical laws.

Ohm and Kirchhoff

The laws of Ohm and Kirchhoff are very simple and straight forward, yet, very important as everything else is based on them.

Ohm

Figure SE 1. illustrates the law of Ohm.

Figure SE 1 : Ohm

The law of Ohm says : as the current flows, whenever resistance is encountered, the current makes a voltage over this resistance equal to the current multiplied by this resistance.

U = R * I

where,

U is the voltage,

I is the current,

R is the resistance

This law is bidirectional : whenever voltage is applied over a resistance, a current flows through the resistance and the value of the current is equal to the value of the voltage divided by the value of the resistance. The bi directionality comes directly from the law.

I = U / R

where,

U is the voltage,

I is the current,

R is the resistance

This law is applicable to all currents, DC and AC regardless of anything else except the value of the input current ( or voltage ) and the value of the resistance.

This law is applicable for all resistances : frequency independent and frequency dependent ( called impedances ).

Kirchhoff

Kirchhoff has two simple laws : a current law and a voltage law.

Before them, an important information is currents and voltages have directions. This applies for DC and for AC.

Currents flow from positive voltage point to negative voltage point. This is their direction.

The direction of a voltage is from positive voltage point to negative ( i. e. the direction of the pressure which this voltage exerts to resistances ).

Very important is to remember, direction is a graphical representation of a sign.

Now, how can AC currents and AC voltages have signs and, thus, directions. After all they are AC because they change their directions and signs. Yes, this is true. However, a direction and, thus, a sign either of a given AC current or a given AC voltage can be accepted. Then, all of the other currents and voltages can be referenced to this accepted direction.

For example a take two voltage of a pure sinusoidal with an amplitude of 1V and a frequency of 60 Hz.  They are the same, but, one of the voltages is an inverted sinusoidal wave as compared to the other. Thus, one of the voltages can be accepted as - the other. Or, they can be accepted with the same sign, but 180 phase difference between them.

Also, graphical direction and sign are interchangeable. In case a given current has a given direction and a given sign, this current can be drawn to be of an opposite direction as long as the sign is reversed. And, in case a given current has a given direction and a given sign, The sign of this current can be reversed as long as the direction is reversed. ( Each current and voltage can only have two possible signs, plus or minus and, therefore can only have two possible directions and not more. )

Thus, current from up to down with a plus sign is the same as a current from down to up, but, with a minus sign.

Voltage from up to down with a plus sign is the same as a voltage from down to up, but, with a minus sign.

In a circuit, because of relativity, a given point can be chosen as a point of reference against which all voltages in the circuit are measured. This point of reference is called Ground and is accepted to have 0V. Thus, any voltage at any other point can be measured against this common point, called Ground. In any circuit, the voltage of a given point, measured against this common point, called Ground, is called potential. Thus, for example, in case a given point has a potential of 15VDC and another, of - 5VDC, the voltage between these points is 20VDC.

Kirchhoff’s Current Law

Figure SE 2. shows two wire intersections. On one of them, all currents go towards the intersection. On the other, all currents go away from the intersection.

Figure SE 2. : Kirchhoff’s Current Law

However, on each intersection, one or all currents can be drawn to go the other way as long as the sign is changed. Thus, I1, which goes toward the intersection can be drawn to go away from the intersection, but, then, the current becomes - I1. The same applies for any current and voltage.

Thus, Kirchhoff says : The sum of all currents which go in the same direction to or from an intersection is zero. Thus, the sum of all currents which go in the direction, towards an intersection, is zero and the sum of all currents which go in the same direction, away from an intersection is zero.

Thus, for the currents on Figure SE 2. :

I1 + I2 + I3 = 0

Generally,

∑ Ii = 0

This is elementary logic.

Now, what happens when some of the currents have a different direction in respect to the intersection, as shown on Figure SE 3a.

Figure SE 3. Different Current Directions

Well, as mentioned, any current can be drawn in the opposite direction, as long as, the sign is reversed. Figure SE 3b. illustrates this. On Figure SE 3a., I1 goes away from the intersection and the sign of I1 is plus. On Figure SE 3b. I1 goes towards the intersection, but, the sign is reversed to minus.

Logically, the two circuits on Figure SE 3. are the same.

Thus, Kirchhoff’s Current law is to be applied to the intersection on Figure SE 3b. :

( - I1 ) + I2 + I3 = 0, =>

( - I1 ) = ( - I2 ) + ( - I3 ), =>

I1 = I2 + I3

Thus, a better way to say Kirchhoff’s Current law may be : For an intersection where two currents go in and one current goes out, the current which goes out is equal to the sum of the currents which go in. Similarly, for an intersection where two currents go out and one current goes in, the current which goes in is equal to the sum of the currents which go out.

This can be generalised : For any intersection, the sum of all currents which go in is equal to the sum of all currents which go out.

Kirchhoff’s Voltage Law

Kirchhoff’s voltage law is also a subject to elementary logic and says : In a closed circuit, the sum of all voltages is zero.

This is shown on Figure SE 4.

Figure SE 4. Kirchhoff’s Voltage Law with Two Voltages

Figure SE 4. shows two voltages, U1 and U2 in a closed loop circuit. Each voltage can be represented with two potentials, one on each side. These potentials are measured against some random point which has randomly been said to have a potential of 0V.

U1 = P11 - P12

U2 = P21 - P22

Thus, in case U1 is 5V, P11 can be 4V against ground, but, then P12 must be - 1V, so, P11 - P12 is always 5V. Or, P11 can be 3V against ground, but, then P12 must be - 2V, so, P11 - P12 is always 5V. Exact values are irrelevant for now.

The same applies to U2, P12 and P22.

The important thing is : because P11 and P22 are connected with nothing but a simple wire, P11 must be equal to P22, or, else, a huge current would fly through the wire, because, as per Ohm :

I = U / R

and when R = 0, I = ∞. This infinite current is impossible. Therefore,

P11 = P22 = Pu

The same applies to P12, P21 and Pd :

P12 = P21 = Pd

As previously said,

U1 = P11 - P12,

therefore,

U1 = Pu - Pd

As previously also said,

U2 = P21 - P22,

Therefore,

U2 = Pd - Pu

Therefore,

U1 = ( - U2 )

Therefore,

U1 + U2 = ( - U2 ) + U2

U1 + U2 = - U2 + U2

U1 + U2 = 0

Figure SE 4. shows Kirchhoff’s Voltage Law with four voltages.

Figure SE 4. : Kirchhoff’s Volatge Law with Four Voltages

Figure SE 4. shows four voltages : U1, U2, U3, U4. Look at the voltages in a circle : either clockwise or counter clockwise.

Again,

U1 = P11 - P12

U2 = P21 - P22

U3 = P31 - P32

U4 = P41 - P42

And again,

P12 = P21 = Pll

P22 = P31 = Prl

P32 = P41 = Prr

P42 = P11 = Pul

Therefore,

U1 = Pul - Pll

U2 = Pll - Prl

U3 = Prl - Prr

U4 = Prr - Pul

To add them,

U1 + U2 + U3 + U4 = Pul - Pll + Pll - Prl + Prl - Prr + Prr - Pul = 0

The same can be generalised for any amount of voltages in a close circuit :

√ Ui = 0

Figure SE 5. shows two circuits which are exactly the same in order to explain the Kirchhoff’s voltage law with voltages with different directions when looking at them in a circular way : either clockwise or counter clockwise.

Figure SE 5. : Kirchhoff’s Volatge Law with Different Directions

On Figure SE 5a., U1 is with the opposite direction to all other voltages when looking at the voltage directions in a circle, either clockwise or counter clockwise. Because directions are nothing, but, visual depictions of signs, the circuit on Figure SE 5a. can be drawn in a way shown on as Figure SE 5b : the direction of Ui is changed to match the other voltage directions and, of course, the sign is reversed. The ground is shared for the two circuits.

So, the arrangement of Figure SE 5b. is the same as this on Figure SE 4., thus,

( - U1 ) + U2 + U3 + U4 = 0

( - U1 ) = - ( U2 + U3 + U4 ), =>

U1 = U2 + U3 + U4

Thus, the reversed voltage, U1 is equal to the sum of the non reversed voltages.

Thus, Kirchhoff’s voltage law can be generalised also as :

When looking at a closed loop circuit in a circle ( clockwise or counter clockwise, no difference ), the sum of all voltages in one direction is equal to the sum of all voltages in the opposite direction.

The arrangement of Figure SE 5a. is extremely heavily used in Electrical and Electronics.

Kirchhoff’s Law

As clearly, seen, the current and the voltage laws are very similar ( dual ). Therefore, there is no such a thing as Kirchhoff’s Current Law and Kirchhoff’s Voltage Law. There is only one Kirchhoff’s Law which says :

The sum of all currents at an intersection is zero as well as the sum of all voltages in a closed loop circuit is zero. Just the same.

Tevenin and Norton

The theorem of Tevenin says any electrical circuit can be replaced by an equivalent circuit with only one ideal voltage source and a resistor in series ( connected to a load, which load, is in parallel to the series connection of the ideal voltage source and the resistor ). Similarly, because of the duality of ideal voltage sources and ideal current sources, the theorem of Norton says any electrical circuit can be replaced by an equivalent circuit with only one ideal current source and a resistor in parallel ( connected to a load, which load, is in parallel to the parallel connection of the ideal current source and the resistor ).

The greatest application of these theorems is for evaluation of input and output impedances ( resistances ).

All ideal voltages sources can be replaced by a direct connection ( because the output impedance of an ideal voltage source is 0 ) and all current sources can be removed from the circuit, i. e., replaced by an open circuit ( because the output impedance of an ideal current source is 0 ).

Once these replacements have been carried out, look from the input and go through the circuit as the current would in order to find the input impedance or look from the output and go through the circuit as a possible current, which imaginary current, comes from the output in order to find the output impedance.

Sounds confusing, but, is not so. Figure SE 5.2 shows this.

Figure SE 5.2. : Input and Output Impedances with Substitution of Ideal Voltage Sources

Figure SE 5.2.o. shows a circuit with four ideal voltage sources : the power supplies Vcc and Vee, the input voltage Ui and another voltage U2.

Figure SE 5.2.a. and Figure SE 5.2.b. are used to find the input impedance : Ui is removed to be able to go through the path of the current, U2, Vcc and Vee are replaced by direct connections. R2 is, therefore, placed in parallel to R21 and RL.

Therefore, the input impedance of the schematics on figure Figure SE 5.2.O. is

Ri = R1 + ( R21 || R22 || RL )

When RL is not considered ( or when too high in comparison with R21 || R22 ),

Ri = R1 + ( R21 || R22 )

Figure SE 5.2.c., Figure SE 5.2.d. and Figure SE 5.2.e. are used to find the output impedance : RL is the load resistance, which, does not belong to the circuit, whose output resistance is to be found and, therefore, must be removed. Ui, U2, Vcc and Vee are replaced by direct connections.

When looked from the output and when the path of an imaginary current which comes from the output is found, the output impedance of the schematics on figure Figure SE 5.2.O. is,

Ro = R1 || R21 || R22

All, each and every circuit can be analysed this way regardless of how complicated a circuit can be.

Effective Values

A few words can be said on AC currents and AC voltages.

When a pure sinusoidal AC voltage, u, is applied to a light bulb ( a simple resistor ), the filament ( the simple resistor ) of the light bulb warms up, warms up the incandescent gas inside and, when warmed up, this gas converts the heat into light.

u = ua * sin ( ω * t ),

where,

u is pure sinusoidal AC voltage

ua is the amplitude of u

ω = ( 2 * π * f ) is the frequency of u

t is time

OK. What happens when DC voltage, Ů, is applied to the light bulb? Exactly the same.

At what values, the AC voltage, u and the DC voltage U make exactly the same light? When :

Ů = ua / ( √ 2 )

So, Ů is the DC voltage which creates the same effect on the light bulb filament resistor : exactly the same light.

Therefore, because the effect is the same, Ů is called the effective value of u.

For example, the nominal effective value of the mains voltage in North America is :

Ů = 120V

Therefore, the amplitude of the mains voltage is :

ua = Ů * ( √ 2 ) = 120 * ( √ 2 ) = 169.71V ≈ 170V

and the frequency of the mains voltage in North America is 60Hz, therefore, the mains voltage in North America is :

u = 170 * sin ( 2 * π * 60 * t ) = 170 * sin ( 2 * π * 60 * t ) = 170 * sin ( 377 * t )

Everywhere else in the world ( with some tiny exceptions ), the effective value of the mains voltage is 220V and the frequency is 50Hz, therefore the voltage is :

u = 220 * ( √ 2 ) * sin ( 2 * π * 50 * t ) = 311 * sin ( 2 * π * 50 * t ) = 311 * sin ( 314 * t )

Everything said for the effective value of pure sinusoidal voltage applies to a pure sinusoidal current. The effective value of the pure sinusoidal current i = ia * sin ( ω * t ) is İ = ia * / ( √ 2 ).

The power, generated by a pure sinusoidal voltage and a pure sinusoidal current with the same frequency is :

P = Ů * İ * cos ( φ )

where φ is the phase difference between the voltage and the current. When pure resistors are plugged in, φ = 0, cos ( 0 ) = 1, therefore,

P = Ů * İ

Hence one of the importances of the effective values.

Effective values are incorrectly called RMS values.

Electrical Components

Electrical components are not a subject of this paper. A quick preview is made here, however.

There are two types of electrical components reactive and non reactive. Reactive components are inductors and capacitors. Non reactive components are resistors.

Reactive components have transient processes, which means, when connected with non reactive components, i. e. resistors, the reactive components display different electrical parameters such as voltage and currents initially and, then, after a while, these parameters reach their stable values.

In a stable regime of work, reactive components work only with AC. Non reactive components can work with AC and DC.

Resistors are the most popular electrical components to be used in electronics. Important is to mention when two resistors, R1 and R2, are connected in sequence, they make one new resistance, R12 which is equal to the sum of them, i. e. R12 = R1 + R2. When they are connected in parallel, they also make one new resistance, R12, where R12 = R1 * R2 / ( R1 + R2 ).

Capacitors are also extremely popular electrical components to be used in electronics. Important is to mention when two capacitors, C1 and C2, are connected in sequence, they make one new resistance, C12 which is C12 = C1 * C2 / ( C1 + C2 ). equal to the sum of them, i. e. R12 = R1 + R2. When they are connected in parallel, they also make one new capacitance, C12, equal to the sum of them, i. e. C12 = C1 + C2.

A very important feature of capacitors is their ability to stop DC and allow AC through them unimpeded. To carry this out, they internally deal with energy levels. Thus, when there is a signal made of 5VDC and a superimposed 1VAC, then capacitor then resistor to ground, the capacitor would have a 5VDC voltage on one side and 0VDC voltage on the other. DC cannot go through. However, there will be an AC current through the capacitor which goes from each of the sides to the other. Thus, although the capacitor has 5VDC on one side and 0VDC on the other, the AC current of 1V amplitude would go either way. How is this possible? After all, current cannot go from low potential to high potential! Yes, but, the capacitor arranges the potentials internally to be able to conduct current even from the low potential to the high. This effect is immensely important for electronics and has been explained separately where transistors are explained.

Inductors are the most unpopular electrical components to be used in electronics because they are also antennas which attract and emit noise. Important is to mention when two inductors, L1 and L2, are connected in sequence, they make one new inductance, L12 which is equal to the sum of them, i. e. L12 = L1 + L2. When they are connected in parallel, they also make one new inductance, L12, where L12 = L1 * L2 / ( L1 + L2 ).

Although inductors are mostly used in power supply and radio, they also have a special place in passive filter where one stage with one inductor and one capacitor can make a second order filter as opposed to a chain with one capacitor and one resistor or one inductor and one resistor which make a first order filter only. This is largely used in the output stage of a PWM ( pulse width modulation ) amplifier to convert the width of the pulses to an analogue signal with the same shape as the original input signal before the amplifier.

Opposite to capacitors, inductors allow the whole DC to go through, just like a simple wire, but impede and AC through because of the electromagnetic processes inside of the inductors ( because of which they also make and take noise ).

Because the capacitors and the inductors work in opposite ways, they are called dual components and the theory which deals with their duality is called Duality Theory in Electrical.

Resistors in Sequence and Parallel Derivations

Figure SE 6a. shows two resistors, R1 and R2, in sequence. Figure SE 6b. shows two resistors, R1 and R2, in parallel. The task is to find any possible combined resistance, R12, which can replace R1 and R2. Because resistors are frequency invariant, DC or AC voltage sources can be used. DC or AC current sources can be used too, but, because Voltage Based Electronics is almost always used and Current Based Electronics is almost never used, the derivations are performed with ideal voltage sources Edc = Udc.

Figure SE 6. : Resistors in Sequence and Parallel

Resistors in Sequence Derivation

Figure SE 6a., shows two resistors in sequence connected to an ideal voltage source Edc = Udc. The value of the ideal voltage source is known. The values of the resistors R1 and R2 are known.

Not known is the current Idc.

Not known is whether there is a possibility to replace these two resistors with one, combined value resistor R12 and what the value of R12 is in case possible.

In case replacement of R1 and R2 with any combined resistance, R12, is possible, than, as per Ohm :

R12 = Udc / Idc

Idc is not known and has to be found.

As per Kirchhoff,

Udc = UR1 + UR2

The same Idc goes through the two resistors, R1 and R2. Therefore,

UR1 = Idc * R1

UR2 = Idc * R2

After substitution,

Udc = Idc * R1 + Idc * R2 = Idc * ( R1 + R2 )

R1 + R2 = Udc / Idc

As per Ohm, anything equal to Udc / Idc is a resistor. Therefore, there is such a resistor equal to Udc / Idc. Thus,

R12 = R1 + R2

Therefore, a sequence of two resistors can be replaced with one resistor with a value of the sum of the values of the two resistors.

For n number of resistors, the derivation is the same :

Rn = ∑n ( Ri )

Resistors in Parallel Derivation

Figure SE 6b., shows two resistors in parallel connected to an ideal voltage source Edc = Udc. The value of the ideal voltage source is known. The values of the resistors R1 and R2 are known.

Not known are the currentc Idc, IR1 and IR2.

Not known is whether there is a possibility to replace these two resistors with one, combined value resistor R12 and what the value of R12 is in case possible.

In case replacement of R1 and R2 with any combined resistance, R12, is possible, than, as per Ohm :

R12 = Udc / Idc

Idc is not known and has to be found.

As per Kirchhoff,

Idc = IR1 + IR2

The same Udc is applied over each of the two resistors, R1 and R2. Therefore,

IR1 = Udc / R1

IR2 = Udc / R2

After substitution,

Idc = Udc / R1 + Udc / R2 = Udc * ( 1 / R1 + 1 / R2 )

1 / ( 1 / R1 + 1 / R2 ) = Udc / Idc

As per Ohm, anything equal to Udc / Idc is a resistor. Therefore, there is such a resistor equal to Udc / Idc. Thus,

R12 = 1 / ( 1 / R1 + 1 / R2 )

This formula can be worked to be :

R12 = 1 / [ ( R2 + R1 ) / ( R1 * R2 ) ], =>

R12 = R1 * R2 / ( R1 + R2 )

To check : resistor plus resistor is resistor; resistor * resistor / resistor is resistor too.

Therefore, a parallel of two resistors can be replaced with one resistor with a value of the product of the values of the two resistors divided by the sum of the values of the two resistors.

For n number of resistors, the derivation is the same :

Rn = ( 1 / [ ∑n ( 1 / Ri ) ]

Rn = R1 * R2 * .... * Rn / ( R1 + R2 + ... + Rn )

Inductors in Sequence and Parallel Derivations

Figure SE 7. : Inductors in Sequence and Parallel

Figure SE 7a. shows two inductors, L1 and L2, in sequence. Figure SE 7b. shows two inductors, L1 and L2, in parallel. The task is to find any possible combined inductance, L12, which can replace L1 and L2. Because inductors are frequency variable resistances, ( a frequency dependent resistance is called impedance, the impedance changes with a change of frequency ), only AC sources with a given frequency, f ( ω = 2 * π * f ), can be used. Ideal AC voltage sources have been used, therefore. AC current sources can be used too, but, because Voltage Based Electronics is almost always used and Current Based Electronics is almost never used, the derivations are performed with ideal voltage sources Eac = Uac.

Note : on the schematics of Figure SE 7., the inductors are encapsulated in rectangles, named XLi. This is because, at the source frequency of f, each inductor makes a resistance, XL, called impedance, which depends only on the frequency f and the value of the inductor, L, this way :

XL = j * 2 * π * f * L = j * ω * L

Important : The higher the frequency the higher the resistance ( impedance ).

All inductors, represented by their impedances act as resistors at this frequency f. Therefore, the derivation for inductors is the same as the derivation for resistors. The goal is to find L12, for which :

XL12 = j * ω * L12

Thus, for inductors in sequence ( Figure SE 7a. ),

XL12 = XL1 + XL2 = j * ω * L1 + j * ω * L2 = j * ω * ( L1 + L2 ), =>

L12 = L1 + L2

Thus, for inductors in parallel ( Figure SE 7b. ),

XL12 = XL1 * XL2 / ( XL1 + XL2 ) = j * ω * L1 * j * ω * L2 / ( j * ω * L1 + ω * L2 ) = ( j * ω )² * L1 * L2 / [ ( j * ω ) * ( L1 + L2 ) ] = j * ω * [ L1 * L2 / ( L1 + L2 ) ], =>

L12 = L1 * L2 / ( L1 + L2 )

Therefore, a sequence of two inductors can be replaced with one inductor with a value of the sum of the values of the two inductors and a parallel of two inductors can be replaced with one inductor with a value of the product of the values of the two inductors divided by the sum of the values of the two inductors.

For n number of inductors in sequence, the derivation is the same as for two :

Ln = ∑n ( Li )

For n number of inductors in parallel, the derivation is the same as for two :

Ln = ( 1 / [ ∑n ( 1 / Li ) ]

Ln = L1 * L2 * .... * Ln / ( L1 + L2 + .... + Ln )

Capacitors in Sequence and Parallel Derivations

Figure SE 8. : Capacitors in Sequence and Parallel

Figure SE 8a. shows two capacitors, C1 and C2, in sequence. Figure SE 8b. shows two capacitors, C1 and C2, in parallel. The task is to find any possible combined capacitance, C12, which can replace C1 and C2. Because capacitors are frequency variable resistances, ( a frequency dependent resistance is called impedance, the impedance changes with a change of frequency ), only AC sources with a given frequency, f ( ω = 2 * π * f ), can be used. Ideal AC voltage sources have been used, therefore. AC current sources can be used too, but, because Voltage Based Electronics is almost always used and Current Based Electronics is almost never used, the derivations are performed with ideal voltage sources Eac = Uac.

Note : on the schematics of Figure SE 8., the capacitors are encapsulated in rectangles, named XCi. This is because, at the source frequency of f, each capacitor makes a resistance, XC, called impedance, which depends only on the frequency f and the value of the capacitor, C, this way :

XC = 1 / ( 2 * π * f * C ) = 1 / ( j * ω * C )

Important : The higher the frequency the lower the resistance ( impedance ).

All capacitors, represented by their impedances act as resistors at this frequency f. Therefore, the derivation for capacitors is the same as the derivation for resistors. The goal is to find C12, for which :

XC12 = 1 / ( j * ω * C12 )

Thus, for capacitors in sequence ( Figure SE 7a. ),

XC12 = XC1 + XC2 = 1 / ( j * ω * C1 ) + 1 / ( j * ω * C2 ) = ( j * w * C2 + j * w * C1 ) / [ ( j * w )² * C1 * C2 ) ] = ( j * w ) * ( C1 + C2 ) / [ ( j * w )² * C1 * C2 ] = ( C1 + C2 ) / [ ( j * w ) * C1 * C2 ] = 1 / [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ], =>

C12 = C1 * C2 / ( C1 + C2 )

Thus, for capacitors in parallel ( Figure SE 8b. ),

XC12 = XC1 * XC2 / ( XC1 + XC2 )

XC1 + XC2 has just been derived to be :

XC1 + XC2 = 1 / [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ]

XC1 * XC2 = [ 1 / ( j * ω * C1 ) ] * [ 1 / ( j * ω * C2 ) ] = 1 / [ ( j * ω )² * C1 * C2 ], =>

XC12 = XC1 * XC2 / ( XC1 + XC2 ) = { 1 / [ ( j * ω )² * C1 * C2 ] } / { 1 / [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ] } = { 1 / [ ( j * ω )² * C1 * C2 ] } * { [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ] } = { [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ] } * { 1 / [ ( j * ω )² * C1 * C2 ] } = { [ ( j * w ) * C1 * C2 / ( C1 + C2 ) ] } / { [ ( j * ω )² * C1 * C2 ] } = [ 1 / ( C1 + C2 ) ] / ( j * ω ) = 1 / [ ( j * ω ) * ( C1 + C2 ) ], =>

C1 = C1 + C2

Therefore, a sequence of two capacitors can be replaced with one capacitor with a value of the product of the values of the two capacitors divided by the sum of the values of the two capacitors and a parallel of two capacitors can be replaced with one capacitor with a value of he sum of the values of the two capacitors.

Therefore, the capacitor derivations show outcomes which are opposite to each other. This is another proof off the duality of these reactive, passive, electrical components : capacitors and inductors.

For n number of capacitors in sequence, the derivation is the same as for two :

Cn = ( 1 / [ ∑n ( 1 / Ci ) ]

Cn = C1 * C2 * .... * Cn / ( C1 + C2 + .... + Cn )

For n number of capacitors in parallel, the derivation is the same as for two :

Cn = ∑n ( Ci )

And this is of a massive, immense importance : when capacitors are connected in parallel, they add their values to one another. This is used extensively in Electronics.

Mixed Connections of Many Different Components

As mentioned, every capacitor and inductor can be represented with impedances. Thus, a complex electrical circuit which contains many resistors, capacitors and inductors can be represented as the same circuit with only resistors and impedances.

Because impedances are resistances ( although frequency dependent ) every electrical circuit can be represented with only resistances and these can be combined into one composite component by using the rules for parallel and serial connections, just derived.

Figure SE 9. : Mixed Connections of Many Different Components

Figure SE 9a. shows a bunch of resistors, capacitors and inductors in a circuit attached to Eac = Uac. Figure SE 9b. shows the same circuit, but, the capacitors, Ci, are replaced with their impedances XCi and the inductors are also replaced with their impedances XLi.

Now, all currents, voltages impedances, combined impedances, etcetera can be derived using analogous approaches to the ones just derived.

Based on the parallel and sequel rules, there are a few mathematical methods which help the analysis of electrical circuits. These are OK for circuits with hundreds of components, even, thousands.

In most cases a simple approach is good enough.

This simple approach is to start to combine components from the back to the front.

For example, XC2 is in parallel to the sequence of R3 and L3. R3 and L3 are in sequence and can be replaced with one XR3L3. XC2 and X3RL3 are in parallel and can be replaced by one XC2R3L3. XC2R3L3 is in parallel to the sequence of R2 and C1. R2 and C1 are in sequence and can be replaced with one XR2C1. XC2R3L3 and XR2C1 are in parallel and can be replaced with Xp. Xp is in sequence with R1 and L1. R1 and L1 are in sequence and can be replaced with XR1L1. Xp and XR1L1 are in sequence and can be replaced by Xps. Thus, the circuit would have only one load Xps. The current drawn from Eac can then be easily calculated.

To calculate any current or any voltage in the schematics combine all possible components except the ones to be analysed. Then, just use Ohm and Kirchhoff to find the sought parameter.

For example, to find the current through R2 and C1, combine R2 and C1, R1 and XL1, R3 and XL2, the combined of R3 and XL2 with XC2 and R1 and XL1. This way, the circuit would have only three components : one in sequence to Eac and, positioned after this, the other two in parallel to each other. Then, use Kirchhoff and Ohm to find the current.

To find the voltage over C1, carry the same out but do not combine r2 and XC1. Then, use Kirchhoff and Ohm to find the current.

Ideal Current and Voltage Sources

The symbols of ideal current and voltage sources are shown on Figure SE 10. Figure SE 10a. shows and ideal DC current source. Figure SE 10b. shows and ideal DC voltage source. Figure SE 10c. shows and ideal AC current source. Figure SE 10d. shows and ideal AC voltage source.

Figure SE 10 : Ideal Current and Voltage Sources

In some cases, only the DC symbols are used to represent AC and DC. Which one exactly is either written next to the symbol or explained. This is incorrect, but, is largely used.

The symbols depict important information.

Figure SE 10a. shows an ideal DC current source and, as mentioned, can be used for AC, although this is incorrect. An important hint is there is no line through the circle. This is because ideal current sources have infinite output impedances ( resistances ) ( what an output impedance is will be explained soon ). Thus, there is no connection through, as the output impedance is infinite, i. e., an open circuit. Ideal current source can be assumed as nothing, just air which magically drives current through a load.

Figure SE 10b. shows an ideal DC current source and, as mentioned, can be used for AC too, although this is incorrect. An important hint is there is a line through the circle. This is because ideal voltage sources have zero output impedances ( resistances ) ( what an output impedance is will be explained soon ). Thus, there is a connection through, as the output impedance is zero, i. e., a simple wire. Ideal voltage source can be assumed as a simple wire which magically drives provides voltage to a load.

Note : the arrow inside a DC voltage source points to the opposite direction of the polarity of the voltage. This is because the arrow shows the direction of the current which will fly through any attached load and not the direction of the generated voltage. And this is because ideal voltage source is called source of electrical movement voltage, i. e., source of a voltage which moves current ( electrons ) through towards the load.

Why are these sources called ideal? There are two ways to provide exactly the same answer to this question :

1. Because the generated current or voltage does NOT depend on the load. Here are simple examples. Assume an ideal current source, which provides, say 10mA current when a load of 10Ω is attached. When this load is changed to, say, 10MΩ, the current is exactly the same, 10mA and no load can change this. Assume an ideal voltage source, which provides, say 10V voltage when a load of 10MΩ is attached ( or, even, when no load is attached ). When this load is changed to, say, 10Ω, the voltage is exactly the same, 10V and no load can change this.

2. Another way to say exactly the same thing is : because ideal current sources have infinite output impedances and ideal voltage sources have zero output impedances.

Current and Voltage Dividers

Figure SE 11a. shows a current divider made by resistors R1 and R2 and Figure SE 11b. shows a voltage divider made by resistors R1 and R2.

Figure SE 11. : Current and Voltage Divider

Current Divider

To analyse the current divider on Figure SE 11a., one must consider the current source Idc is an ideal current source and the generated current, Idc, does not change with the load. This means whatever the resistor values R1 and R2 ( which make the load ) are, the current is always the same : Idc.

The resistors R1 and R2 are in parallel. This means they can, for a while be replaced by one resistor R21 with a value :

R21 = R1 || R2 = R1 * R2 / ( R1 + R2 )

As per Ohm, when the current Idc goes through these resistors, or, better said through the combined resistor R21, this current makes a voltage of Udc and

Udc = Idc * R21

This voltage is present to the upper pins of the resistors R1 and R2 ( positive ) as well as to the lower pins of the resistors R1 and R2 ( negative ).

Because this voltage, Udc, is present to the upper and lower pin of R1, this voltage will make a current through R1 with a value :

IR1 = Udc / R1

Because this voltage is present to the upper and lower pin of R2, this voltage will make a current through R2 with a value :

IR2 = Udc / R2

After substitution,

IR1 = Idc * R21 / R1 = Idc * [ R1 * R2 / ( R1 + R2 ) ] / R1 = Idc * R2 / ( R1 + R2 )

IR2 = Idc * R21 / R2 = Idc * [ R1 * R2 / ( R1 + R2 ) ] / R2 = Idc * R1 / ( R1 + R2 )

Also, as per Kirchhoff,

Idc = IR1 + IR2

Therefore, resistors R1 and R2 divide the current Idc into two parts. One of them goes through R1 and has a value which depends on Idc and the values of the two resistors. The other one goes through R2 and has a value which also depends on Idc and the values of the two resistors. When the resistors are of different values, these two currents are different. When the resistors are of equal values, R1 = R2 = R, these currents are of the same value : IR1 = IR2 = Idc / 2.

Voltage Divider

To analyse the voltage divider on Figure SE 11b., one must consider the voltage source Edc = Udc is an ideal voltage source and the generated