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

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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

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

(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

Parallel connection

4321

4321

4321

4321

1111

111111

)1111

(

RRRR

RRRRRR

R

V

RRRRV

IIIII

eq

eq

eq

s

s

s

• Applying KCL

(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

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

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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