dc circuits: basic laws · dc circuits: basic laws dr. hasan demirel . ... • conductance is the...
TRANSCRIPT
EENG223: CIRCUIT THEORY I
DC Circuits:
Basic Laws Dr. Hasan Demirel
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law • All materials resist the flow of current.. • Resistance R of an element denotes its ability to resist the flow of electric
current, which is measured in ohms (Ω). • A cylindrical material of length l and cross-sectional area A has the
following resistance:
A
lR
R= resistance of the element in ohms (Ω)
ρ= resistivity of the material in ohm-meters (Ω-m)
l = lenghth of the cylindrical material in meters (m)
A=Cross sectional area of the material in meters2 (m2)
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law • Resistance: Basic Concepts and Assumptions:
Conductors (e.g. Wires) have very low resistance (<0.1 Ω), which is usually be neglected (i.e. We will assume that wires have zero resistance).
Insulators (e.g. air) have very large resistance (>50 MΩ) that can be usually
ignored ( omitted from circuit for analysis).
Resistors have a medium range of resistance and must be accounted for the
circuit analysis.
Conceptually, a light bulb is similar to the resistor.
Properties of the bulb control how much current flows and how much power is
dissipated (absorbed then emitted as light and heat).
As with the circuit elements, we need to know the current through and voltage
across the device are ralated.
The relationship between the current and voltage can be linear or nonlinear.
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law • Ohm’s Law states that the voltage v accross a resistor is directly
proportional to the current i flowing through the resistor.
Riυ υ = voltage in volts (V),
i = current in (A),
R = resistance in (Ω).
• Sign ± is determined by passive sign convention (PSC). • Materials with linear relationships between the current and the voltage
satisfy the Ohm’s Law.
Resistor symbol
Ohm’law applies Ohm’s Law does Not apply
• Resistance R is equal to the slope m in (a).
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law • Resistors and Passive Sign Convention (PSC)
Note that the the relationship between current and voltage are sign sensitive. PSC is satisfied if the current enters the positive terminal of an element:
İf PSC is satisfied : υ =iR
İf PSC is not satisfied : υ =−iR
PSC satisfied PSC not satisfied PSC not satisfied PSC satisfied
(υ =iR) (υ =−iR) (υ =−iR) (υ =iR)
EENG223: CIRCUIT THEORY I
Equations derived from Ohm’s Law
(1 Ω = 1 V/A)
• Resistors cannot produce power , so the power absorbed by a resistors will always be positive.
• 1 Ω = 1 V/A
iRυ i
υR • Ohm’s Law:
R
υRiυip
22 • Recall:
R
υi
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit • Short Circuit as Zero resistance:
• An element (or wire) with R =0 is called a short circuit. • Short circuit is just drawn as a wire (line).
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit • Short Circuit as Voltage Source (0V):
• An ideal voltage sorce with Vs=0 V is equivalent to a short circuit. • Since υ =iR and R =0, υ =0 regardless of i. • You could draw a source with Vs=0 V, but it is not done in practice. • You cannot connect a voltage source to a short circuit. • İf connected, usually wire wins and the voltage source melts (smoke
comes out) if not protected.
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit
• An element (or wire) with R=∞ is called an open circuit. • Such an element is just omitted.
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit • Open Circuit as Current Source (0 A):
• An ideal current source with I=0 A is equivalent to a open circuit. • Since υ =iR and if I=0 A, then R = ∞.
• You could draw a source with I=0 A, but it is not done in practice. • You cannot connect a current source to an open circuit. • İf connected, usually you blow the current source (smoke comes
out) if not protected. • The insulator (air) wins. Else, sparks fly.
EENG223: CIRCUIT THEORY I
Conductance: inverse of resistance
• Conductance is the ability of an element to conduct electric current . Conductance is the inverse of resistance.
υ
i
RG
1
• Units: siemens (S) or mho ( )
Riυ
Gi
R
υRiυip
22
G
iGυυip
22
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
• In circuit analysis, we need a common language and
framework for describing circuits.
• In this course, circuits are modelled to be the same as networks.
• Networks are composed of nodes, braches and loops.
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
• A branch represents a single element such as voltage source, resistor or current source.
• How many branches? • Branch: a single two-terminal circuit element. • Wire segments are not counted as branches. • Examples: voltage source/current source/resistors.
There are 5 branches
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
• A node is a point of connection between two or more branches.
• How many nodes? • Node: a connection point between two or more branches. • May include a portion of circuit (more than a single point). • Essential Node: the point of connection between three or
more brances.
There are 3 nodes (a, b and c)
2 essential nodes (b and c)
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
• A loop is a closed path in a circuit.
• How loops? • Loop: a closed path in a circuit. • Independent Loop: A loop is independent if it contains at least
one branch which is not a part of any other independent loop.
There are 6 loops.
There are 3 independent loops.
EENG223: CIRCUIT THEORY I
Kirchhoff’s Laws: Overview
• Ohm’s Law is not sufficient to analyze circuits alone.
• Kirchhoff’s Laws help Ohm’s law to form the foundation for circuit analysis: The defining equations for circuit elements (Ohm’s Law). Kirchhoff’s Current Law (KCL). Kirchhoff’s Voltage Law (KVL).
• The defining equations (from Ohm’s law) tell us how the voltage and current within a circuit element are related.
• Kirchhoff’s laws tell us how the voltages and currents in different branches are related.
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law
• Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.
• The sum of currents entering a node is equal to the sum of the currents leaving the node.
• KCL also applies to a closed boundary:
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law
• Applying KCL to node a:
312
321 0
IIII
IIII
T
T
321 IIIIT
• Equivalent circuit can be generated as follows:
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law: for Closed Boundaries
4231
4321 0
IIII
IIII
• KCL applies to a closed boundaries:
Closed boundary
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law: Example
• Apply KCL to the each essential node in the circuit.
• Essential node 1:
• Essential node 2:
• Essential node 3:
i2
i1 i3 i4
231321 0 iiiiii
mA50mA5 4343 iiii
mA50mA5 142142 iiiiii
EENG223: CIRCUIT THEORY I
Ideal Current Sources: Series
• Ideal currtent sources cannot be connected in series.
• Recall: ideal current sources guarantee the current flowing through source is at specified value.
• Recall: the current entering a circuit element must be equal to the current leaving the circuit element: Iin = Iout.
• Ideal current sources do not exist. • Tecnically allowed if : I1 = I2 , but it is a bad idea.
EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law: KVL
• Kirchhoff’s voltage law (KVL) states that the algebraic sum of voltages around a closed path (or loop) is zero.
• sum of voltage drops = sum of voltage rises
• or
EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law: KVL
• Apply KVL to each loop in the following circuit:
• Loop 1: • Loop 2: • Loop 3: • Loop 4: • Loop 5: • Loop 6:
EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law (KVL) Example:
• Calculate V2, V6 and VI.
V50301010
V30)106).(105(
V10)102).(105(
10010
33
6
33
2
6262
I
II
V
V
V
VVVVVV
EENG223: CIRCUIT THEORY I
Applying Basic Laws: Example • Calculate vo and i.
V48A8162
06412412
60642412
0
00
vii
iii
ivivi
• Apply KVL around the loop:
EENG223: CIRCUIT THEORY I
Applying Basic Laws: Example • Calculate I2, I3, I7, V3, and VI .
EENG223: CIRCUIT THEORY I
Series Resistors and Voltage Division • The equivalent resistance of any number of resistors
connected in series is the sum of individual resistance.
• Consider the following circuit:
• Applying KVL
4321
4321
4321
4321
11111
)(
GGGGG
RRRRR
RIRRRRI
IRIRIRIRV
eq
eq
eqss
sssss
(Resistors in series add)
EENG223: CIRCUIT THEORY I
Series Resistors and Voltage Division
(Applying KVL)
• voltage accross each resistor, is proportional to its resistance. Larger the resistance, larger the voltage drop on that resistor:
• voltage division principle
(voltage divider circuit)
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division • The equivalent resistance of N resistors number of resistors
can be calculated by:
• Consider the following circuit:
4321
4321
4321
4321
1111
111111
)1111
(
RRRR
RRRRRR
R
V
RRRRV
IIIII
eq
eq
eq
s
s
s
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division
4321
4321
11111
GGGGG
RRRRR
eq
eq
• Resistors in parallel have more complicated relationship • It is easier to express in conductance
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division
(Applying KCL at node a)
• Given the total current i entering to node a the current is shared by the resistors by inverse proportion to their resistance:
• Current division principle
(current divider circuit)
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance • Example: Calculate the Req for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance • Example: Calculate the Req for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance • Example: Calculate the Rab for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance • Example: Calculate the Geq for the following
circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • There are cases where the resistors are neither in parallel nor
in series.
• More tools are needed.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Three-terminal equivalent networks can be used to simplify the
analysis of the circuits. • There are two types of three-terminal networks:
Wye (Y) Networks
Delta(Δ) Networks
Two equivalent forms of (Y) Network
Two equivalent forms of (Δ) Network
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Every Wye (Y) network is functionally equivalent to a Delta (Δ)
network (vice versa).
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Example: Convert the following D network to a Y network.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Example: Convert the following Ynetwork to a D network.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Example: Calculate Req and Power delivered by the source.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations • Example: Calculate Rab and and use it to calculate i.