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Slides

Fundamentals of Electrical Engineering
Fundamentals of Electrical Engineering 1
Grundlagen der Elektrotechnik 1
Chapter: Basic Laws / Grundgesetze
Michael E. Auer
Source of figures:
Alexander/Sadiku: Fundamentals of Electric Circuits,
McGraw-Hill
Michael E.Auer
20.11.2009
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Course Content
Basic Concepts / Grundlagen
Basic Laws / Grundgesetze
Circuit Theorems / Zweipoltheorie
Common Methods of Network Analysis / Allgemeine Netzwerkanalyse
Operational Amplifiers / Operationsverstärker
Capacitors and Inductors / Kapazitäten und Induktivitäten
Basic RC and RL Circuits Switching / Einfache Schaltvorgänge
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Chapter Content
Ohm’s Law / Ohm'sches Gesetz
Nodes, Branches, and Loops / Knoten, Zweige und Maschen
Kirchoff's Laws / Kirchoff'sche Gesetze
Series and Parallel Resistors / Serien- und Parallelschaltung von
Wiederständen
Wye-Delta Transformations / Stern-Dreieck-Transformationen
Summary / Zusammenfassung
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Chapter Content
Wiederständen
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Resistivity
l
R =ρ⋅
A
Linear resistance
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Resistance (1)

Ohm figured out, that that the voltage across a
resistor is directly proportional to the current flowing
through the resistor, and that the relation between v
and i is constant:
v
= const. = R
i

V
= Ω (Ohm)
A
Another form for Ohm’s Law is as follows:
v = i⋅R

The resistance R of an element denotes its ability to
resist the flow of electric current; it is measured in
ohms (Ω).
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Resistance (2)

Two extreme possible values of R: 0 (zero) and
∞ (infinite) are related with two basic circuit concepts:
short circuit and open circuit.
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Conductance and Power Dissipation

Conductance is the ability of an element to conduct electric current;
it is the reciprocal of resistance R and is measured in mhos or
siemens.
1 i
G= =
R v

A
= S (Siemens)
V
The power dissipation of a resistor:
2
v
2
p = v ⋅i = i R =
R
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Linear and Non-linear Resistance
Linear resistance
dv
= const. = R
di
R = 0 : short circuit
R = ∞ : open circuit
Non-linear resistance
dv
= r ≠ const.
di
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Chapter Content
Nodes, Branches, and Loops / Knoten, Zweige
und Maschen
Wiederständen
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Definitions

A branch represents a single element such as a voltage source or a
resistor.
A node is the point of connection between two or more branches.
A loop is any closed path in a circuit.

A network with b branches, n nodes, and l independent loops will


satisfy the fundamental theorem of network topology:
l = b − n +1
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Example
Original circuit
Equivalent circuit
How many branches,
nodes and loops are
there?
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Branch Voltage and Branch Current
How many branches,
nodes and loops are
there?
V1
V6
Should we consider it as one
branch or two branches?
Application of Ohm‘s Law:
Michael E.Auer
V1
= R1 =12 Ω
I1
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etc.
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Topology
Michael E.Auer
•
Two or more elements are in series if they
exclusively share a single node and consequently
carry the same current.
•
Two or more elements are in parallel if they are
connected to the same two nodes and consequently
have the same voltage across them.
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Circuit Examples
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Chapter Content
Wiederständen
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Kirchhoff‘s Current Law (1)

1st Kirchhoff‘s Law
Kirchhoff’s current law (KCL) states that the algebraic sum of
currents entering a node (or a closed boundary) is zero.
N
Mathematically,
∑i
n =1
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n
=0
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Example
Determine the current I for the circuit shown in the figure below.
I + 4A -(-3A) -2A = 0
⇒ I = -5A
This indicates that
the actual current for
I is flowing in the
opposite direction.
We can consider the whole
enclosed area as one “node”.
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Example
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Kirchhoff‘s Voltage Law (1)

2nd Kirchhoff‘s Law
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all
voltages around a closed path (or loop) is zero.
M
Mathematically,
∑v
m =1
Michael E.Auer
n
=0
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Example
Va-V1-Vb-V2-V3 = 0
V1 = I·R1 V2 = I·R2 V3 = I·R3
⇒ Va-Vb = I·(R1 + R2 + R3)
Va − Vb
I=
R1 + R2 + R3
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Examples
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Example
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Chapter Content
Series and Parallel Resistors / Serien- und
Parallelschaltung von Wiederständen
Michael E.Auer
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Series Resistors

Series: Two or more elements are in series if
they are cascaded or connected sequentially
and consequently carry the same current.

The equivalent resistance of any number of
resistors connected in a series is the sum of
the individual resistances.
N
Req = R1 + R2 + ⋅ ⋅ ⋅ + RN = ∑ Rn
n =1
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Series Resistors and Voltage Division (1)
R2
v2 =
⋅v
R1 + R2
In general:
Michael E.Auer
Req = R1 + R2
Rn
vn =
v
R1 + R2 + ⋅ ⋅ ⋅ + R N
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Series Resistors and Voltage Division (2)
Example
v2 = ?
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Parallel Resistors

Parallel: Two or more elements are in parallel if
they are connected to the same two nodes and
consequently have the same voltage across
them.

The equivalent resistance of a circuit with N
resistors in parallel is:
1
1
1
1
=
+
+ ⋅⋅⋅ +
Req R1 R2
RN
N
Geq = G1 + G2 + ⋅ ⋅ ⋅ + GN = ∑ Gn
n =1
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Parallel Resistors and Current Division (1)

The total current i is shared by the resistors in inverse proportion to
their resistances.
i1 R2 G1
= =
i2 R1 G2

The current divider can be expressed in general as:
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Req
v
in =
=i ⋅
Rn
Rn
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Extreme cases:
Short circuit branch
Open circuit branch
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Example
i1 = ?
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0,5 A
i1
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Mixed Series and Parallel Resistors
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Chapter Content
Wiederständen
Wye-Delta Transformations / Stern-DreieckTransformationen
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Problem
Bridge circuit
Req = ?
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Different Forms of the Same Network
Y
Michael E.Auer
T
(star)
Y
Δ
T
Π
Δ
Transformation
Transformation
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Π
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Wye – Delta Transformations
Star -> Delta
Delta -> Star
R1 R2 + R2 R3 + R3 R1
Ra =
R1
Rb Rc
R1 =
( Ra + Rb + Rc )
R2 =
Rc R a
( Ra + Rb + Rc )
Rb =
R1 R2 + R2 R3 + R3 R1
R2
R3 =
Ra Rb
( Ra + Rb + Rc )
Rc =
R1 R2 + R2 R3 + R3 R1
R3
Michael E.Auer
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Chapter Content
Wiederständen
Michael E.Auer
20.11.2009
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Summary (1)
•
•
•
•
•
•
•
•
Michael E.Auer
A linear resistor is a passive element in which the voltage v across it is directly proportional
to the current i through it. The v-i-relation obeys Ohm’s law v = i · R , where R is the
resistance of the resistor.
A short circuit is a resistor with zero resistance ( R = 0 ). An open circuit is a resistor with
infinite resistance ( R = ∞ )
The conductance G of a resistor is the reciprocal of its resistance G = 1 / R .
A branch is a single two terminal element in an electric circuit. A node is a point of
connection between two or more branches. A loop is a closed path in a circuit. The number
of branches b, the number of nodes n and the number of independent loops l in a network
are related as l = b – n +1 .
Kirchhoff’s Current Law (KCL) states that the currents at any node algebraically sum to zero.
In other words, the sum of the currents entering a node equals the sum of the currents
leaving the node.
Kirchhoff’s Voltage Law (KVL) states that the voltages around a closed path algebraically
sums to zero.
Two elements are in series when they are connected sequentially, end to end. When
elements are in series, the same current flows through them. The equivalent resistance of
series resistors is due to the sum of their individual resistances.
Two elements are in parallel when they are connected to the same two nodes. Elements in
parallel always have the same voltage across them. The equivalent conductance of parallel
resistors is due to the sum of their individual conductances.
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Summary (2)
•
•
•
Michael E.Auer
The total voltage across series resistors is divided among the resistors in direct proportion to
their resistances; the larger the resistance, the larger the voltage drop. This is called the
Principle of Voltage Division.
The total current entering a node with parallel resistors is shared by the resistors in inverse
proportion to their resistances ( or in direct proportion to their conductances); the larger the
conductance, the larger the current. This is called the Principle of Current Division.
Delta to Wye Conversion is often used in network analysis. Each resistor in the Y network is
the product of the resistors in the two adjacent ∆ branches, divided by the sum of the three ∆
resistors. Each resistor in the ∆ network is the sum of all possible products of Y resistors
taken two at time, divided by the opposite Y resistor.
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