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close this bookIntroduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
View the document(introduction...)
View the documentPreface
View the document1. Importance of Electrical Engineering
close this folder2. Fundamental Quantities of Electrical Engineering
View the document2.1. Current
View the document2.2. Voltage
View the document2.3. Resistance and Conductance
close this folder3. Electric Circuits
View the document3.1. Basic Circuit
View the document3.2. Ohm’s Law
close this folder3.3. Branched and Unbranched Circuits
View the document3.3.1. Branched Circuits
View the document3.3.2. Unbranched Circuits
View the document3.3.3. Meshed Circuits
close this folder4. Electrical Energy
View the document4.1. Energy and Power
View the document4.2. Efficiency
View the document4.3. Conversion of Electrical Energy into Heat
View the document4.4. Conversion of Electrical Energy into Mechanical Energy
close this folder4.5. Conversion of Electrical Energy into Light
View the document4.5.1. Fundamentals of Illumination Engineering
View the document4.5.2. Light Sources
View the document4.5.3. Illuminating Engineering
View the document4.6. Conversion of Electrical Energy into Chemical Energy and Chemical Energy into Electrical Energy
close this folder5. Magnetic Field
View the document5.1. Magnetic Phenomena
View the document5.2. Force Actions in a Magnetic Field
close this folder5.3. Electromagnetic Induction
View the document5.3.1. The General Law of Induction
View the document5.3.2. Utilisation of the Phenomena of Induction
View the document5.3.3. Inductance
close this folder6. Electrical Field
View the document6.1. Electrical Phenomena in Non-conductors
close this folder6.2. Capacity
View the document6.2.1. Capacity and Capacitor
View the document6.2.2. Behaviour of a Capacitor in a Direct Current Circuit
View the document6.2.3. Types of Capacitors
close this folder7. Alternating Current
View the document7.1. Importance and Advantages of Alternating Current
View the document7.2. Characteristics of Alternating Current
View the document7.3. Resistances in an Alternating Current Circuit
View the document7.4. Power of Alternating Current
close this folder8. Three-phase Current
View the document8.1. Generation of Three-phase Current
View the document8.2. The Rotating Field
View the document8.3. Interlinking of the Three-phase Current
View the document8.4. Power of Three-phase Current
close this folder9. Protective Measures in Electrical Installations
View the document9.1. Danger to Man by Electric Shock
close this folder9.2. Measures for the Protection of Man from Electric Shock
View the document9.2.1. Protective Insulation
View the document9.2.2. Extra-low Protective Voltage
View the document9.2.3. Protective Isolation
View the document9.2.4. Protective Wire System
View the document9.2.5. Protective Earthing
View the document9.2.6. Connection to the Neutral
View the document9.2.7. Fault-current Protection
View the document9.3. Checking the Protective Measures

3.3.2. Unbranched Circuits

When connecting several load in one circuit in such a manner that the current passes through all loads one after the other, then this circuit is called series connection or series circuit.

Fig. 3.8. shows such an arrangement.

Fig. 3.8. Unbranched circuit

The two voltage sources connected in series produce a total voltage of

E = E1 + E2

The current driven by the total voltage has the same intensity at any point in the circuit and causes an adequate voltage drop at any load. Since the charge carriers give their en-tire energy to the circuit, the statement in the form of the 2nd Kirchhoff’s law or the mesh-network theorem holds:

In each unbranched circuit, the sum of all voltage drops is equal to the total voltage.

For the example shown in Fig. 3.8., thus we have

E = E1 + E2 = U = U1 + U2 + U3


As the current through all loads is the same, we have

U/I = U1/I + U2/I +U3/I

U/I, however, is the resistance of the load; the total resistance or equivalent resistance will be called Requ and obtained in the form of

Requ = R1 + R2 + R3


In a series connection of loads, the equivalent resistance is equal to the sum of the individual resistances; it is always greater than the greatest individual resistance.

For n-equal resistances (n = 2, 3, 4, ...)

Requ = n · R holds.


If conductances have to be calculated, the required equations can easily be derived from the relation G = 1/R. In practice, calculations of conductances are required very seldom for unbranched circuits.

The ratio of the partial voltages is dependent on the ratio of the partial resistances. It is obvious that the greater voltage drop occurs via the greater partial resistance and vice versa. This relation called voltage divider rule is expressed for two loads connected in aeries in the following way:

U1/U2 = R1/R2



U1/U = R1/Requ


In a voltage divider, the ratio of the partial voltages is the same as that of the partial resistances.

Example 3.5.

Two loads with the resistances R1 = 18 W and R2 = 72 W are connected in series and a voltage of 60 V is applied to them. Draw a sketch of this circuit. Calculate the equivalent resistance, the current and the partial voltages.


R1 = 18 W
R2 = 72 W
U = 60 V

To be found:

U1; U2



According to equation (3.8.):

Requ = R1 +R2
Requ = 18 W + 72 W
Requ = 90 W

the equivalent resistance is greater than the greatest individual resistance

Fig. 3.9. Circuit with two loads connected in series

According to equation (3.1.b):

I = U/Requ
I = 60 V/90 W
I = 0.0667 A = 667 mA

According to equation (3.1.a):

U1 = R1 · I
U1 = 18 W · 0.667 A
U1 = 12 V;

the smaller partial voltage drops via the smaller individual resistance

U2 = R2 · I
U2 = 72 W · 0.667 A
U2 = 48 V

U1 and U2 can also be calculated according to the voltage divider rule, equation (3.9.b), namely,

U1/U = R1/Requ


U2/U = R2/Requ

U1 = U · (R1/Requ)

U2 = U · (R2/Requ)

U1 = 60 V · 18/90

U2 = 60 V · (72/90)

U1 = 12

U2 = 48

Verification by means of equation (3.7.):

U = U1 +U2
U = 12 V + 48 V
U = 60 V;

the sum of all voltage drops is equal to the total voltage.

In the following Table 3.2., an unbranched circuit is compared with a branched circuit. In each of the two cases, two loads are included. It is obvious that the formulas of one circuit have the same structure as those of the other circuit. The only difference is that, as compared to the branched circuit, for the unbranched circuit U is put instead of I, I instead of U, R instead of G and G instead of R.

Table 3.2. Relations in Branched and Unbranched Circuits Having two Loads Each

Branched Circuit

Unbranched Circuit

U is equal across all loads

I is equal through all loads

I = I1 + I2

U = I1 + U2

Gequ = G1 + G2

Requ = R1 + R2

1/Requ = 1/R1 + 1/R2

1/Gequ = 1/G1 + 1/G2

® Requ = (R1 · R2)/(R1 + R2)

® Gequ = (G1 · G2)/(G1 + G2)

for G1 = G2 = G = 1/R is

for R1 = R2 = R = 1/G is

Gequ = 2 · G, since n = 2

Requ = 2 · R, since n = n

or Requ = R/2

or Gequ = G/2

I1/I2 = G1/G2 = R2/R1

U1/U2 = R1/R2 = G2/G1

I1/I = G1/Gequ = G1/(G1 + G2)

U1/U = R1/Requ = R1/(R1 + R2)

I1/I = Requ/R1 = R2/(R1 + R2)

U1/U = Gequ/G1 = G2/(G1 + G2)