Published On: Wed, Jul 27th, 2022

## IMPEDANCES

Introduction

When solving, calculating and analyzing electrical circuits of alternating current of all voltage levels, industrial and radio frequencies, we almost often encounter the concept of impedance and admittance. In this article I will cover in detail almost everything about electrical impedances of all types in such a way that anyone can understand and easily solve any AC electrical circuit of any voltage level and frequency.

A. What is impedance?

In the simplest terms and understandable to everyone, imedance is a type of more or less complex electrical resistance in alternating current. A general graphical symbol for impedance is shown in Figure 1. Slika 1. Standardni grafički simbol za električnu impedanciju u električnim kolima naizmenične struje Figure 1. Standard graphic symbol for electrical impedance in alternating current electrical circuits

In Figure 2.a. a graphic symbol with letter symbols for complex electrical impedance, in Figure 2.b. is shown the graphic symbol for the same with markings in the polar form. Slika 2. (a) grafički simbol električne impedancije i slovna oznaka u kompleksnom obluku, (b) takođe u isto ali sa oznakama u polarnom obliku Figure 2. (a) graphic symbol of electrical impedance and letter designation in a complex  form, (b) also in the same but with designations in polar form

In alternating current electric circuits, the impedance can only be the ohmic resistance R, only the inductive resistance (XL) of an ideal coil of inductance L whose ohmic resistance can be neglected, or only the capacitive resistance (XC) or an ideal capacitor (C) with a capacity C also of negligible ohmic resistance which we will see in the next presentation.

B. Three Basic Types of Electrical Impedances-Ideal

If impedance Z consists of ohmic resistance R in ohms, its symbol and label look like in Figure 3.a. According to IEC standards, the graphic symbol from figure 3.a. is also a graphic symbol for ohmic (active) resistance, while symbol 3.b. also another form of graphic symbol for ohmic resistance R. Impedance, as well as all electrical quantities in a complex calculation, are marked with a bold letter Z, or the letter Z with a dash above the letter or below the letter. Slika 3. Grafički simbol za čistu otpornu impedanciju Z (a), i istovremeno za omski otpor R kao i na (b), u skladu sa IEC standardima. Figure 3. Graphical symbol for pure resistive impedance Z (a), and simultaneously for ohmic resistance R as in (b), according to IEC standards.

When impedance Z consists of pure inductive resistance XL in ohms, its symbol and label look like in figure 4.a., while the symbol in figure 4.b. symbol for inductance coil L. Slika 4. (a) Grafički simbol za čistu induktivnu imepdanciju (Z) odnosno induktivnu reaktanciju (X L ) kako se u praksi naziva i (b) za namotaj (L) zanemarljivog omskog otpora. Figure 4. (a) Graphical symbol for pure inductive impedance (Z), i.e. inductive reactance (XL ), as it is also called in practice (b) for coil (L) of negligible ohmic resistance

When impedance Z consists of pure capacitive resistance XC in ohms, its symbol and label look like in figure 5.a. while the symbol in Figure 5.b. designation for capacitor capacity C. Slika 5. (a) Grafički simbol za čistu kapacitivnu imepdanciju (Z) odnosno kapacitivnu reaktanciju (X C ) kako se u praksi naziva i (b) za kondenzator (C)  zanemarljivog omskog otpora. Figure 5. (a) Graphical symbol for pure capacitive impedance (Z) or capacitive reactance (XC ) as it is also called in practice (b) for a capacitor (C) of negligible ohmic resistance

C) Four real impedances

In practice there are four real impedances, of which three are incomplete and one is complete impedance. These are:

– Inductive impedance graphically shown in Figure 6. It consists of an active ohmic resistor R and inductance coils L, ZL, respectively ZRL Slika 6. Grafički simbol i oznaka za otporno-induktivnu impedanciju koja se sastoji od omskog otpornika i namotaja induktivnosti L, ZRL Figure 6. Graphical symbol and symbol for resistive-inductive impedance consisting of an ohmic resistor and coil inductance L, ZRL

– Capacitive impedance ZC graphically shown in Figure 7. It consists of an active ohmic resistor R and a capacitor of capacity C, i.e. ZRC Slika 7. Grafički simbol i oznaka za otporno-kapacitivnu impedanciju koja se sastoji od omskog otpornika i kondenzatora kapaciteta C, ZRC Figure 7. Graphical symbol and letters symbol for resistive-capacitive impedance consisting of an ohmic resistor and a capacitor of capacity C, ZRC

– The inductive-capacitive impedance ZX is shown graphically in Figure 8. It consists of a coil of inductance L and a capacitor of capacity C, i.e. ZLC. In practice, it is also called capacitive-inductive reactance. Slika 8. Grafički simbol i oznaka za kapacitivno-induktivnu impedanciju ZX koja se sastoji od namotaja induktivnosti L i kondenzatora kapaciteta C, ZLC Figure 8. Graphical symbol and designation for capacitive-inductive impedance ZX consisting of a coil of inductance L and a capacitor of capacity C, ZLC

– Finally, the third , the complete impedance consisting of an ohmic resistor R, an inductance coil L and a capacitor of capacity C, Figure 9., Z or ZRLC Slika 9. Potpuna impedancija Z koja se sastoji od omskog otpornika R, namotaja induktivnosti L i kondenaztora kapaciteta C . Figure 9. A complete impedance Z consisting of an ohmic resistor R, an inductance coil L and a capacitor

The previous four real impedances have been discussed in the sense that they consist of only one electrical component each, the ohmic resistor R, the inductive coil L and the capacitor C. When in electrical circuit analyzes and practical calculations, each of these three main electrical components depending on its structures can consist of two or all three components in the appropriate configuration, and they must be taken into account during the appropriate calculations.

On the other hand, each electrical component can be connected to each other in series or in parallel.

In Figure 10.a1. and 10.a2. is shown the impedance consisting of one ohmic resistor R and one winding L which are connected in series and form one total impedance Z or ZRL.

In Figure 10.b1. and 10.b2. the same impedance is shown where ohmic resistance R and inductive resistance XL are ie. windings connected in parallel and which form one total impedance Z, i.e. ZRL. Slika 10.a1. i 10.a2. impedancija koja se sastoji od jednog omskog otpornika R i induktivni otpor XL koji su spojeni na red i čine jednu ukupnu impedanciju Z odnosno ZRL. Slika 10.b1. i 10.b2. impedancija kod koje su omski R i induktivni otpor XL tj. namotaj spojeni paralelno i koji čine jednu ukupnu impedanciju Z odnosno ZRL. Figure 10.a1. and 10.a2. impedance consisting of one ohmic resistor R and inductive resistance XL which are connected in series and form one total impedance Z or ZRL. Figure 10.b1. and 10.b2. impedance where the ohmic R and the inductive resistance are XL i.e. windings connected in parallel and which form one total impedance Z, i.e. ZRL.

In Figure 11.a1. and 11.a2. shown is the impedance consisting of one ohmic resistor R and one capacitor C, i.e. capacitive reactance XC which are connected in series forming one total impedance Z or ZRC.

In Figure 11.b1. and 11.b2. the same impedance is shown where the ohmic resistance R and the capacitive resistance XC are i.e. of capacitors C connected in parallel and forming one total impedance Z or ZRC. Slika 11.a1. i 11.a2. impedancija koja se sastoji od jednog omskog otpornika R i jednog kondenzatora C tj. kapacitvne reaktancijeXC koji su spojeni na red čineći jednu ukupnu impedanciju Z odnosno ZRC. Slika 11.b1. i 11.b2. je prikazana ista impedancija kod koje su omski otpor R i kapacitivni otpor XC  tj. kondenzatora C spojeni paralelno i koji čine jednu ukupnu impedanciju Z odnosno ZRC.   Figure 11.a1. and 11.a2. impedance consisting of one ohmic resistor R and one capacitor C ie. capacitive reactance XC which are connected in series forming one total impedance Z or ZRC. Figure 11.b1. and 11.b2. the same impedance is shown where the ohmic resistance R and the capacitive resistance XC are i.e. of capacitors C connected in parallel and forming one total impedance Z or ZRC.

In Figure 12.a1. and 12.a2. shown is the impedance consisting of one coil L of inductive resistance XL and one capacitor of capacity C, i.e. capacitive reactance XC which are connected in series making one total impedance Z or ZLC. This is actually the pure reactance X or the total reactance XT of the series connection of the coil and the capacitor.

In Figure 12.b1. and 12.b2. is shown the impedance where the coil L of inductive reactance XL and the condenser of capacity C of capacitive reactance XC are connected in parallel and which form one total impedance Z or ZLC or total reactance XT.

It should be emphasized here that with these coil and capacitor connections, the total impedance or reactance can be positive or negative depending on which component prevails in the total reactance value. Slika 12.a1. i 12.a2. prikazuje impedanciju koja se sastoji od jednog namotaja L induktivnog otpora XL i jednog kondenzatora kapaciteta C tj. kapacitvne reaktancijeXC koji su spojeni na red čineći jednu ukupnu impedanciju Z odnosno ZLC. Ovo je ustvari čista reaktancija X ili ukupna reaktancija  XT redne veze namotaja i kondenzatora. Slika 12.b1. i 12.b2. prikazuje impedanciji kod koje su namotaj L induktivne reaktancije XL i konddnezator kapaciteta C kapacitivne reaktancije XC spojeni paralelno čineći jednu ukupnu impedanciju Z odnosno ZLC odnosno ukupnu reaktanciju  XT . Figure 12.a1. and 12.a2. shows the impedance consisting of one coil L of inductive resistance XL and one capacitor of capacity C ie. capacitive reactance XC which are connected in series making one total impedance Z or ZLC. This is actually the pure reactance X or the total reactance XT of the series connection of the coil and the capacitor. Figure 12.b1. and 12.b2. shows the impedance at which the coil L of inductive reactance XL and the capacitor of capacity C of capacitive reactance XC are connected in parallel making one total impedance Z or ZLC or total reactance XT.

And so we arrived at the fourth and complete electrical impedance consisting of one ohmic resistor R, one coil L and one capacitor C which can be connected to each other in series or in parallel which can be clearly seen in Figure 13. and Figure 14. Slika 13. Potpuna impedancija koja se sastoji od sve tri električne komponente;  otpornika R, kalema L induktivnog otpora  i kondenzatora C kapacitivnog otpora spojenih redno (a) i paralelno (b) Figure 13. Full (complete) real impedance consisting of all three electrical components; resistor R, coil L of inductive resistance and capacitor C of capacitive resistance connected in series (a) and in parallel (b) Slika 14. Potpuna impedancija koja se sastoji od otpornika R, kalema L induktivnog otpora  i kondenzatora C kapacitivnog otpora spojenih redno (a) i paralelno (b) Slika 14. Tri električne komponente koje sačinjavaju potpunu električnu impedanciju Z, ZT , ZRLC koja se sastoji od otpornika R, kalema L tj. induktivnog optora  XL i kondenzatora C, kapacitivnog otpora XC spojenih redno (a) i paralelno (b) Figure 14. Full real impedance consisting of resistor R, coil L of inductive resistance and capacitor C of capacitive resistance connected in series (a) and in parallel (b) Figure 14. Three electrical components that make up complete electrical impedance Z, ZT , ZRLC consisting from resistor R, coil L ie. inductive optocoupler XL and capacitor C, capacitive resistance XC connected in series (a) and parallel (b)

D. Series impedances

In Figure 15.a. series connections  are shown;

– ideal ohmic resistors of different resistance values ​​RT – ZT (a1),

– ideal inductive resistances of different values ​​ZLT – XLT (a2) and

– ideal capacitive resistances – capacitive reactances ZCT – XCT (a3).

It should be emphasized that an unlimited number of the same electrical components can be connected in series.

In Figure 15.b. ordinal links are shown;

– ohmic resistors and inductive resistances (windings) that make up one total incomplete impedance ZRLT, resistive-inductive type (b1),

– ohmic resistors and capacitive resistances of capacitors that make up one total incomplete impedance ZRCT, capacitive-resistive type (b2),

– inductive and capacitive resistances – reactances (coils and capacitors) which also form one incomplete impedance ZLCT, but therefore form one complete – pure reactance XT, inductive – capacitive type (b3) and

– ohmic, inductive and capacitive resistances (ohmic resistors, windings and capacitors) that make up one complete electrical impedance ZT or ZRLCT. Slika 15.a. Redne veze impedancija idealnih električnih komponenti koje čine jednu ukupnu impedanciju i  15.b. redne veze različitih električnih komponenti koje čine jednu ukupnu impedanciju Figure 15.a. Serial connections of impedances of ideal electrical components that make up one total impedance and 15.b. series connections of different electrical components that make up one total impedance

E. Parallel impedances

Figure 16 shows parallel connections;

– ideal ohmic resistors of different resistance values ​​RT – ZT (a1),

– ideal inductive resistances of different values ​​ZLT – XLT (a2) and

– ideal capacitive resistances – capacitive reactances ZCT – XCT (a3).

It should be emphasized that an unlimited number of the same electrical components can be connected in parallel.

In Figure 17.a. parallel connections of ohmic resistors and inductive resistances (coils) are shown, which make up one total incomplete impedance ZRLT, resistive-inductive type.

In Figure 17.b. parallel connections of ohmic resistors and capacitive resistances of capacitors are shown, which form one total incomplete impedance ZRCT, capacitive-resistive type.

In Figure 17.c. parallel connections of inductive and capacitive resistances – reactances (coils and capacitors) are shown, which also form one incomplete impedance ZLCT, but therefore form one complete – pure reactance XT, inductive – capacitive type.

In Figure 17.d. parallel connections of ohmic, inductive and capacitive resistances (ohmic resistors, coils and capacitors) are shown, which make up one complete electrical impedance ZT or ZRLCT. Slika 16. Paralelne  veze; idealnih omskih otpornika različitih vrednosti otpora RT  – ZT (a1), idealnih induktivnih otpora različitih vrednosti ZLT – XLT (a2) i idealnih kapacitivnih otpora – kapacitivnih reaktancija ZCT – XCT (a3). Figure 16. Parallel connections; ideal ohmic resistors of different resistance values ​​RT – ZT (a1), ideal inductive resistances of different values ​​ZLT – XLT (a2) and ideal capacitive resistances – capacitive reactances ZCT – XCT (a3). Slika 17.a. Paralelne  veze različitih vrednosti, omskih otpornika i induktivnih otpora (namotaja) koji čine jednu ukupnu nepotpunu impedanciju ZRLT , otporno-induktivnog tipa. Figure 17.a. Parallel connections of different values, ohmic resistors and inductive resistances (cols) that make one total incomplete impedance ZRLT, resistive-inductive type. Slika 17.b. prikazane su paralelne veze omskih otpornika i kapacitivnih otpora kondenzatora koji čine jednu ukupnu nepotpunu impedanciju  ZRCT , kapacitivno-otpornog tipa. Figure 17.b. Parallel connections of ohmic resistors and capacitive resistances of capacitors are shown, which form one total incomplete impedance ZRCT, capacitive-resistive type. Slika 17.c. Paralelne veze induktivnih i kapacitivnih otpora – reaktancija različitih vrednosti (namotaja i kondenzatora) koji takođe čine jednu nepotpunu impedanciju ZLCT  ali, zato čine jednu potpunu – čistu reaktanciju XT, induktivno – kapacitivnog tipa. Figure 17.c. Parallel connections of inductive and capacitive resistances – reactances of different values ​​(coils and capacitors) are shown, which also form one incomplete impedance ZLCT, but therefore form one complete – pure reactance XT, inductive – capacitive type. Slika 17.d. prikazuje paralelne veze  omskih, induktivnih i kapacitivnih otpora različitih vrednosti (omski otpornici, namotaji i kondenzatori) koji čine jednu potpunu električnu impedanciju ZT odnosno ZRLCT . Figure 17.d. The parallel connections of ohmic, inductive and capacitive resistances of different values ​​(ohmic resistors, coils and capacitors) that make up one complete electrical impedance ZT or ZRLCT

F. Combined impedance connections

In the case of mixed or combined connections of different types of electrical components connected in series or parallel, there are hundreds and thousands of combinations, of which I will show a few basic ones here.

Components (resistors, windings and capacitors) can be connected to each other in parallel groups of the same components but also of the same or different values ​​of resistance, inductance or capacity, and the groups are further connected in series with one or more of the same components.

Also, electrical components can be connected to each other in parallel groups with several different components, of the same or different values ​​of resistance, inductance or capacity, and groups connected in a series with one or more different components, again of the same or different values.

In Figure 18.a. Coils are shown using component graphic symbols of the same values ​​of inductance, i.e. inductive resistance, i.e. inductive reactance in a combined connection. Groups with parallel coils are connected in series with serially connected coils also of the same inductance value.

It should be emphasized here,

yes, even though the coils have the same inductance values ​​L, their impedances, i.e. inductive reactances have different values ​​due to the way they are connected to each other, which can be seen in Figure 18.b. Total inductive reactive resistance XLT ie inductive impedance ZLT is shown in Figure 18.c.

In Figure 19.a. Coils of different values ​​of inductance or inductive resistance are shown, i.e. inductive reactance in a combined connection. Groups with parallel windings connected in a series (string) with two, three or more coils  of different values ​​of inductance L.

In Figure 19.b. the equivalent calculation scheme of this group of inductive reactances after calculating their values ​​as well as the total reactance of the entire combination of inductances is presented. Slika 18.a. Namotaji istih vrednosti induktiviteta odnosno induktivnog otpora tj. induktivne reaktancije u kombinovanoj vezi, po dva i tri ista induktiviteta L spojena su međusobno paralelno a potom u seriju sa jednim, dva ili više rednih induktiviteta. Slika 18.b. Ekvivalentna proračunska šema ove grupe  induktivnih reaktancija posle proračunavanja njihovih vrednosti kao i ukupna rekatancija cele kombinacije induktivnosti. Figure 18.a. Coils of the same values ​​of inductance or inductive resistance, i.e. inductive reactance in a combined connection, two and three of the same inductance L are connected in parallel with each other and then in series with one, two or more series inductances. Figure 18.b. The equivalent calculation scheme of this group of inductive reactances after calculating their values ​​as well as the total reactance of the entire combination of inductances is presented. Slika 19.a. Namotaji različitih vrednosti induktiviteta odnosno induktivnog otpora tj. induktivne reaktancije u kombinovanoj vezi, po dva i tri ista induktiviteta L spojena su međusobno paralelno a potom u seriju sa jednim, dva ili više rednih induktiviteta. Slika 19.b. prikazana je ekvivalentna proračunska šema ove grupe  induktivnih reaktancija posle proračunavanja njihovih vrednosti kao i ukupna rekatancija cele kombinacije induktivnosti. Figure 19.a. Coils of different values ​​of inductance, i.e. inductive resistance, i.e. inductive reactance in a combined connection, two and three of the same inductance L are connected in parallel with each other and then in series with one, two or more series inductances. Figure 19.b. The equivalent calculation scheme of this group of inductive reactances after calculating their values ​​as well as the total reactance of the entire combination of inductances is presented.

In Figure 20.a. shown are capacitors of the same values ​​of capacity C, i.e. capacitive resistance, i.e. capacitive reactance in the combined connection, two and three of the same value of capacitor C connected to each other in parallel and then in series with one, two or more series capacitors.

In Figure 20.b. the equivalent calculation scheme of this group of capacitive reactances after calculating their values ​​as well as the total reactance of the entire combination of capacitances is presented. Slika 20.a. Kondenzatori istih vrednosti kapaciteta C odnosno kapacitivnog otpora tj. kapacitivne reaktancije u kombinovanoj vezi, po dva i tri istih vrednosti kondnenzatora C međusobno spojenih paralelno a potom u seriju sa jednim, dva ili više rednih kondnezatora. Slika 20.b. Ekvivalentna proračunska šema ove grupe  kapacitivnih reaktancija posle proračunavanja njihovih vrednosti kao i ukupna rekatancija cele kombinacije kapacitivnosti. Figure 20.a. The capacitors of the same values ​​of capacity C, i.e. capacitive resistance, i.e. capacitive reactance in the combined connection, two and three of the same value of capacitor C connected to each other in parallel and then in series with one, two or more series capacitors. Figure 20.b. The equivalent calculation scheme of this group of capacitive reactances after calculating their values ​​as well as the total reactance of the entire combination of capacitances Figure 20.c.

In Figure 21.a. capacitors of different values ​​of capacity C, i.e. capacitive resistance, ie. capacitive reactance in the combined connection, two and three different values ​​of capacitor C connected in parallel and then in series with one, two or more series capacitors .

In Figure 21.b. the equivalent calculation scheme of this group of capacitive reactances after calculating their values ​​as well as the total reactance of the entire combination of capacitances is presented. Slika 21.a. Kondenzatori različitih vrednosti kapaciteta C odnosno kapacitivnog otpora tj. kapacitivne reaktancije u kombinovanoj vezi, po dva i tri različitih  vrednosti kondnenzatora C međusobno spojenih paralelno a potom u seriju sa jednim, dva ili više rednih kondnezatora. Slika 21.b. Eekvivalentna proračunska šema ove grupe  kapacitivnih reaktancija posle proračunavanja njihovih vrednosti kao i ukupna rekatancija cele kombinacije kapacitivnosti. Figure 21.a. The capacitors of different values ​​of capacity C, i.e. capacitive resistance, ie. capacitive reactance in the combined connection, two and three different values ​​of capacitor C connected in parallel and then in series with one, two or more series capacitors . Figure 21.b. The equivalent calculation scheme of this group of capacitive reactances after calculating their values ​​as well as the total reactance of the entire combination of capacitances is presented.

Figure 22.a. shows a combination of ohmic resistors R of different resistance values ​​and inductive coils of different inductance values ​​L. Shown are two parallel groups connected in series, with two series groups of resistors and coils of different resistance values ​​R and inductance L. Of course, there can be more than two components.

In Figure 22.b. the equivalent calculation scheme of this group of impedances after calculating their values ​​is shown, as well as the total impedance of the entire combination of these two electrical elements. The total impedance of the whole combination can be seen in Figure 22.c. Everything else is completely clear from the picture.

Figure 23.a. shows a combination of ohmic resistors R of different resistance values ​​and capacitors of different capacitance values ​​C. Shown are two parallel groups connected in series, with two series groups of resistors and capacitors of different resistance values ​​R and capacitance C. Of course, there can be more than two in groups components.

In Figure 23.b. the equivalent calculation scheme of this group of impedances after calculating their values ​​is shown, as well as the total impedance of the entire combination of these two electrical elements. The total impedance of the whole combination can be seen in Figure 22.c. Everything else is completely clear from the picture.

Figure 24.a. shows a combination of coils of different values ​​of inductance L and capacitors of different values ​​of capacitance C. Shown are two parallel groups connected in series, with two series groups of coils of different values ​​of inductance L and capacitor C. Of course, there can be more than two components in the groups.

In Figure 24.b. the equivalent calculation scheme of this group of impedances is shown, that is, in this case, the reactance after calculating their values, as well as the total reactance of the entire combination of these two electrical elements. It should be noted that certain reactances can be of a positive or negative character, depending on which of the two predominates. The same applies to the total reactance of this combination. The total impedance of the whole combination can be seen in Figure 24.c. Everything else is completely clear from the picture. Slika 22.a. Kombinacija omskih otpornika R različitih vrednosti otpora i induktivnih namotaja različitih vrednosti induktivnosti L. Vidimo dve paralelne grupe koje su spojene u seriju, sa dve serijske grupe otpornika i namotaja različitih vrednosti otpora R i induktivnosi L. Slika 22.b. prikazuje ekvivalentnu proračunsku šema ove grupe  impedancija posle proračunavanja njihovih vrednosti kao i ukupna impedancija cele kombinacije ova dva električna elementa slika 22.c. Figure 22.a. A combination of ohmic resistors R of different resistance values ​​and inductive coils of different inductance values ​​L. We see two parallel groups connected in series, with two series groups of resistors and coils of different resistance values ​​R and inductance L Figure 22.b. shows the equivalent calculation scheme of this group of impedances after calculating their values, as well as the total impedance of the entire combination of these two electrical elements, Figure 22.c. Slika 23.a. prikazuje kombinaciju omskih otpornika R različitih vrednosti otpora I induktivnih namotaja različitih vrednosti induktivnosti L. Prikazane su dve paralelne grupe koje su spojene u seriju, sa dve serijske grupe otpornika I namotaja različitih vrednosti otpora R i induktivnosi L. Naravno, u grupama može biti više od dve komponente.Na slici 23.b. prikazana je ekvivalentna proračunska šema ove grupe  impedancija posle proračunavanja njihovih vrednosti kao i ukupna impedancija cele kombinacije ova dva električna elementa. Ukupnu impedanciju cele kombinacije vidimo na slici 23.c. Sve ostalo je potpuno jasno sa slike. Figure 23.a. shows a combination of ohmic resistors R of different resistance values ​​and capacitors of different capacitance values ​​C. Shown are two parallel groups connected in series, with two series groups of resistors and capacitors of different resistance values ​​R and capacitance C. Of course, there can be more than two in groups components. In Figure 23.b. the equivalent calculation scheme of this group of impedances after calculating their values ​​as well as the total impedance in Figure 23.c. Slika 24.a. Kombinacija namotaja različitih vrednosti induktivnosti L i kondenzatora različitih vrednosti kapacitivnosti C. Prikazane su dve paralelne grupe koje su spojene u seriju, sa dve serijske grupe namotaja različitih vrednosti induktivnosi L i kondnenzatora C. Slika 24.b. prikazauje ekvivalentnu proračunsku šemu ove grupe  impedancija odnosno, u ovom slučaju reaktancija posle proračunavanja njihovih vrednosti kao i ukupnu reaktanciju cele slika 24.c Figure 24.a. Combination of coils with different values ​​of inductance L and capacitors with different values ​​of capacitance C. Two parallel groups connected in series are shown, with two series groups of coils with different values ​​of inductance L and capacitor C. Of course, there can be more than two components in the groups. Figures 24.b. shows the equivalent calculation scheme of this group of impedances, that is, in this case, the reactance after calculating their values, as well as the total reactance of the entire figure 24.c

Figure 25.a. shows a combination of resistors, coils and capacitors of different values ​​of resistance R, inductance L and capacitance C. Three parallel groups are shown; RL, RC and LC which are connected in series with coil, capacitor and resistor, again of different values ​​of inductance L and capacitance C and ohmic resistance R. Of course, there can be more than two components in groups. In Figure 25.b. the equivalent calculation scheme of individual reactances of this combination is shown in Figure 25.c. calculation scheme of impedances, of which there are five in this combination, and at the end, picture 25.d. the total impedance of the whole combination with all three electrical elements, which means that ZT is the total impedance. Everything else is completely clear from the picture. Slika 25.a. Kombinacija otpornika, namotaja i kondenzatora različitih vrednosti otpora R, induktivnosti L i kapacitivnosti C. Vidimo tri paralelne grupe; RL, RC i LC koje su spojene u seriju sa namotajem, kondenzatorom i otpornikom, opet različitih vrednosti vrednosti induktivnosi L i kapacitivnosti C i omskog otpora R. Sliki 25.b. prikazuje ekvivalentnu proračunsku šemu pojedinačnih reaktancija ove kombinacije a slika 25.c. proračunsku šemu impedancija kojih u ovoj kombinaciji ima pet i na kraju, slika 25.d. pikazuje ukupnu impedanciju cele kombinacije sa sva tri električna elementa što znači da je ZT potpuna impedancija. Sve ostalo je potpuno jasno sa slike. Figure 25.a. A combination of resistors, coils and capacitors of different values ​​of resistance R, inductance L and capacitance C. We see three parallel groups; RL, RC and LC which are connected in series with a coil, capacitor and resistor, again with different values ​​of inductance L and capacitance C and ohmic resistance R. Figures 25.b. shows the equivalent calculation scheme of individual reactances of this combination and Figure 25.c. calculation scheme of impedances, of which there are five in this combination, and finally, picture 25.d. shows the total impedance of the whole combination with all three electrical elements, which means that ZT is the total impedance. Everything else is completely clear from the picture.

And at the end of this consideration of electrical impedances we have Figure 26.

Figure 26.a. shows a combination of resistors, coils and capacitors with different values ​​of resistance R, inductance L and capacitance C. Two parallel RLC and two series RLC groups are shown. It is clear here that it is about four complete electrical impedances because they are made up of three main electrical components of resistor R, voltage of inductance L and capacitor of capacity C. In Figure 26.b. the equivalent calculation scheme of individual reactances of this combination is shown in Figure 26.c. the calculation scheme of the impedances of which there are four in this combination, and at the end, picture 26.d. the total impedance of the whole combination with three electrical elements each, which means that ZT is the total impedance. Everything else is completely clear from the picture. r-source:0;} div.Section1 {page:Section1;} –> Slika 26.a. Kombinacija otpornika, namotaja i kondenzatora različitih vrednosti otpora R, induktivnosti L i kapacitivnosti C. Kombinacija se sastoji od po dve paralelne RLC i dve serijske RLC grupe. Slikai 26.b. prikazaje ekvivalentnu proračunsku šemu pojedinačnih reaktancija ove kombinacije a slika 26.c. proračunsku šemu impedancija kojih u ovoj kombinaciji ima četiti i na kraju, slika 26.d. prikazje ukupnu impedanciju cele kombinacije sa po tri električna elementa što znači da je ZT potpuna impedancija. Figure 26.a. A combination of resistors, coils and capacitors with different values ​​of resistance R, inductance L and capacitance C. The combination consists of two parallel RLC and two series RLC groups. Pictures 26.b. shows the equivalent calculation scheme of individual reactances of this combination and Figure 26.c. the calculation scheme of the impedances in this combination can be read and at the end, Figure 26.d. shows the total impedance of the whole combination with three electrical elements each, which means that ZT is the total impedance.

And, let’s conclude!

Plain, clear, simple, understandable for everyone from student to professional.

### About the Author - Radoje - Rade Jankovic Electrician, Electrical Technician, Electrical Engineer, PhD, Ecologist, Environmentalist, Designer, Educator, Investigator... Today PROFESSIONAL TECHNICAL WRITER AND DOCUMENTATOR Massive practical experience in almost all fields of electrical engineering over 40 years. My International education group in Facebook: EEEW - ELECTRICAL & ELECTRONIC EDUCATION WORLD where published more than 8000 small or bigger articles, lessons, technical advice, projects, technical calculations, test questions with and without answers, illustrated test questions with and without answers; all my original works and few thousands of may original photos from practice, drawings, circuit diagrams, environmental lessons and examples from everyday practice etc., etc.

1. Theo Samber says: