circuit theory i basic laws - eastern mediterranean universityfaraday.ee.emu.edu.tr/eeng223/circuit...
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Circuit Theory I Basic Laws Assistant Professor Suna BOLAT
Eastern Mediterranean University
Electric and electronic department
Ref2: Anant Agarwaland Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare(http://ocw.mit.edu/), Massachusetts Institute of Technology
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Basic laws
Physics laws makes our lifes easier
1. Ohm’s Law
2. Kirchhoff’ Laws
form the foundation for electric circuit analysis
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Resistance
• All materials resist to 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:
𝑅 = 𝜌𝑙
𝐴
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• 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.
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Ohm’s Law
• There is a linear relationship between the voltage and the current
𝜈 = 𝑖 𝑅
During this course, we will assume (naively) that Ohm’s law holds!
𝜈 = voltage in volts (V), i = current in (A), R = resistance in (Ω).
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Don’t trust Ohm’s law!
• Resistivity is a strong function of temperature.
• If the temperature is increased all the electrons in the material (i.e. Copper conductor) gets faster, resistivity goes up!
• If resistivity goes up, resistance goes up as well.
• Higher temperature, higher resistance
𝜈 = 𝑖 𝑅(T)
For example, for a light bulb, which is a resistor, when current runs through it, the bulb heats up, resistance of the bulb gets higher.
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Passive sign convention (revisit)
• Note that the relationship between current and voltage are sign sensitive.
• PSC is satisfied if the current enters the positive terminal of an element: if PSC is satisfied : 𝜈 = 𝑖 𝑅
if PSC is not satisfied : 𝜈 = −𝑖 𝑅
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Equations derived from Ohm’s law:
• Ohm’s law 𝜈 = 𝑖 𝑅
• Recall 𝑝 = 𝑣 𝑖 = 𝑖2𝑅 =𝑣2
𝑅
• Resistors cannot produce power , so the power absorbed by a resistors will always be positive
𝑅 =𝑣
𝑖 𝑖 =
𝑣
𝑅
1 Ω = 1 V/A
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Short circuit
• Short circuit is zero resistance
• An element (or wire) with R = 0 is called a short circuit.
• Short circuit is just drawn as a wire (line).
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Short circuit as voltage source
• An ideal voltage source with Vs = 0 V is equivalent to a short circuit.
• Since 𝜈 = 𝑖 𝑅 and R = 0, 𝜈 = 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.
• If connected, usually wire wins and the voltage source melts (smoke comes out) if not protected.
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Open circuit
• Opposite of short circuit
• An element (or wire) with R = ∞ is called an open circuit.
• Such an element is just omitted.
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Open circuit as a current source
• An ideal current source with I = 0 A is equivalent to a open circuit.
• SinceSince 𝜈 = 𝑖 𝑅 and 𝐼 = 0 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.
• If connected, usually you blow the current source (smoke comes out) if not protected.
• The insulator (air) wins. Else, sparks fly.
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Conductance
• Conductance (G) is the ability of an element to conduct electric current. Conductance is the inverse of resistance.
𝐺 =1
𝑅=
𝑖
𝑣
• Units: siemens (S) or mho ( )
• 𝜈 = 𝑖 𝑅 & 𝑝 = 𝑣 𝑖 = 𝑖2𝑅 =𝑣2
𝑅
• 𝑖 = 𝐺 𝑣 & 𝑝 = 𝑣 𝑖 = 𝑣2𝐺 =𝑖2
𝐺
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Circuit building blocks
• Nodes,
• Branches and
• Loops
– circuits are modelled to be the same as networks.
– Networks are composed of nodes, braches and loops
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Branch
• A branch represents a single element such as voltage source, resistor or current source.
• Any two terminal element is represented by a branch (Examples: voltage source/current source/resistors)
Q: How many branches?
A: How many elements?
Wire segments are not counted as branches. 15
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Node
• A node is a point of connection between two or more branches
• 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 branches.
Q: How many nodes?
• There are 3 nodes (a, b and c)
• 2 essential nodes (b and c)
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Loop
• A loop is a closed path in a circuit.
• 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.
Q: How many loops?
• There are 6 loops
• 3 independent loops.
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Kirchhoff’s Laws
• To define equations for circuit elements: use Ohm’s law
– Defining equations (from Ohm’s law) tell us how the voltage and current within a circuit element are related.
• To define relaship between the element?
– Kirchhoff’s Current Law (KCL)
– Kirchhoff’s Voltage Law (KVL)
• Kirchhoff’s laws tell us how the voltages and currents in different branches are related.
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Kirchhoff’s current law (KCL)
• 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.
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Kirchhoff’s current law (KCL)
• Applying KCL to node a:
• Equivalent circuit can be generated as follows:
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KCL for closed boundaries
• KCL also applies to
a closed boundary
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Example:
• Apply KCL to the each essential node in the circuit.
– Essential node 1:
– Essential node 2:
– Essential node 3:
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Connecting ideal current sources
• Ideal current 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.
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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.
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KVL
Apply KVL to each loop in the following circuit:
• Loop 1: • Loop 2: • Loop 3: • Loop 4: • Loop 5: • Loop 6:
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Example
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Circuit analysis
• Goal: Find all element v’s and i’s
– write element v-i relationships (Ohm’s law)
– write KCL for all nodes
– write KVL for all loops1.2.3
• lots of unknowns
• lots of equations
• lots of fun (?)
• solve
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Element relationships
• For R
• For voltage source
• For current source
𝜈 = 𝑖 𝑅
𝜈 = 𝑉0
𝑅
𝑉0
𝑖 = I0
I0
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Apply KVL KCL
What do we do?! Apply element combination rules
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Element combination rules
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Series connection
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4321
4321
4321
4321
11111
)(
GGGGG
RRRRR
RIRRRRI
IRIRIRIRV
eq
eq
eqss
sssss
Applying KVL
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(Applying KVL)
• voltage accross each resistor, is proportional to its resistance. Larger the resistance, larger the voltage drop on that resistor:
(voltage divider circuit)
Series connection
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Parallel connection
4321
4321
4321
4321
1111
111111
)1111
(
RRRR
RRRRRR
R
V
RRRRV
IIIII
eq
eq
eq
s
s
s
• Applying KCL
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(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 divider circuit)
Parallel connection
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Equivalent resistance
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Examples
Calculate the Req for the following circuit.
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Calculate the Rab for the following circuit.
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Calculate the Geq for the following circuit.
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Sometimes connections are complicated
• There are cases where the resistors are neither in parallel nor in series
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Wye-Delta Transformations
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Wye (Y) Networks
Delta(Δ) Networks
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Wye-Delta transformation
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Converting a D network to a Y network
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Examples
• Convert the following Ynetwork to a D network.
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Examples
• Calculate Rab and and use it to calculate i.
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Examples
• Calculate Req and Power delivered by the source.
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Examples
• Calculate Rab and and use it to calculate i.
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