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
Basic laws
Physics laws makes our lifes easier
1. Ohm’s Law
2. Kirchhoff’ Laws
form the foundation for electric circuit analysis
2
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:
𝑙
𝑅=𝜌
𝐴
3
• 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.
4
Ohm’s Law
• There is a linear relationship between the voltage and the current
𝜈=𝑖𝑅
𝜈 = voltage in volts (V),
i = current in (A),
R = resistance in (Ω).
During this course, we will assume (naively) that Ohm’s law holds!
5
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.
6
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
7
Equations derived from Ohm’s law:
• Ohm’s law
𝜈=𝑖𝑅
𝑣
𝑅=
𝑖
𝑣
𝑖=
𝑅
1 Ω = 1 V/A
• Recall 𝑝
= 𝑣 𝑖 = 𝑖2𝑅 =
𝑣2
𝑅
• Resistors cannot produce power , so the power absorbed by a
resistors will always be positive
8
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.
10
Open circuit
• Opposite of short circuit
• An element (or wire) with R = ∞ is called an open circuit.
• Such an element is just omitted.
11
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.
12
Conductance
• Conductance (G) is the ability of an element to conduct electric
current. Conductance is the inverse of resistance.
1 𝑖
𝐺= =
𝑅 𝑣
• 𝜈=𝑖𝑅
• 𝑖=𝐺𝑣
&
&

• Units: siemens (S) or mho (
)
𝑝 = 𝑣 𝑖 = 𝑖2𝑅 =
2
𝑝=𝑣𝑖=𝑣 𝐺=
𝑣2
𝑅
𝑖2
𝐺
13
Circuit building blocks
• Nodes,
• Branches and
• Loops
– circuits are modelled to be the same as networks.
– Networks are composed of nodes, braches and loops
14
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
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)
16
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.
17
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.
18
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.
19
Kirchhoff’s current law (KCL)
• Applying KCL to node a:
• Equivalent circuit can be generated as follows:
20
KCL for closed boundaries
• KCL also applies to
a closed boundary
21
Example:
• Apply KCL to the each essential node in the circuit.
– Essential node 1:
– Essential node 2:
– Essential node 3:
22
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.
23
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.
24
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
26
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
𝑅
• For R
𝜈=𝑖𝑅
𝑉0
𝜈 = 𝑉0
• For voltage source
I0
• For current source
𝑖 = I0
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Apply KVL KCL
What do we do?!
Apply element combination rules
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Element combination rules
30
31
Series connection
Applying KVL
Vs  R1 I s  R2 I s  R3 I s  R4 I s
 I s ( R1  R2  R3  R4 )  I s Req
Req  R1  R2  R3  R4
1
1
1
1
1




Geq G1 G2 G3 G4
32
Series connection
(Applying KVL)
(voltage divider circuit)
• voltage accross each resistor, is proportional to its resistance. Larger the
resistance, larger the voltage drop on that resistor:
Parallel connection
• Applying KCL
I s  I1  I 2  I 3  I 4
1
1
1
1
 Vs ( 

 )
R1 R2 R3 R4
Vs

Req
1
1
1
1
1
 


Req R1 R2 R3 R4
 Req 
1
1
1
1
1



R R R R
Parallel connection
(Applying KCL at node a)
(current divider circuit)
• Given the total current i entering to node a the current is shared by the
resistors by inverse proportion to their resistance:
Equivalent resistance
36
Examples
Calculate the Req for the following circuit.
37
Calculate the Rab for the following circuit.
38
Calculate the Geq for the following circuit.
39
Sometimes connections are complicated
• There are cases where the resistors are neither in parallel nor in
series
40
Wye-Delta Transformations
 Wye (Y) Networks
 Delta(Δ) Networks
41
Wye-Delta transformation
42
Converting a D network to a Y network
43
Examples
•
Convert the following Ynetwork to a D network.
44
Examples
•
Calculate Rab and and use it to calculate i.
45
Examples
•
Calculate Req and Power delivered by the source.
46
Examples
•
Calculate Rab and and use it to calculate i.
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