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Some Basic Concepts

- Branch----- A branch represents a single element such as a voltage source or a resistor.
{But usually we think a branch as a path flowing the same current. So maybe includes more than one element.}

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Some Basic Concepts

- Node-------A node is the point of connection between two or more branches
- Loop-------A loop is any closed path in a circuit.
- Mesh------A mesh is a loop which does not contain any other loops within it.

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KIRCHHOFF’S LAWS

- Kirchhoff’s Current Law (KCL):

The algebraic sum of currents entering a node (or a closed boundary) is zero.

(Based on the law of conservation of charge)

The sum of the currents entering a node = the sum of the currents leaving the node.

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KCL also applies to a closed boundary.

Note: 1) KCL is available to every node at anytime.

2) KCL is related only to the currents instead of the elements.

3) Pay attention to the current direction.

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KIRCHHOFF’S LAWS

- Kirchhoff’s Voltage Law (KVL):
The algebraic sum of all voltages around a closed path (or loop) is zero.

(Based on the principle of conservation of energy)

To illustrate KVL, consider the circuit :

V1-V2-V3+V4-V5=0 or V2+V3+V5=V1+V4

Sum of voltage drops=Sum of voltage rises

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TWO SORTS OF CONSTRAINTS

- 1.Topological constraints
Determined by the way of connection among the elements.

(Such as KCL KVL)

2. Element constraints

Determined by the elements. (VAR)

Using two sorts of constraints, we can analysis any lumped circuit (solve out all the voltages and currents).

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In a circuit with b branches and n nodes, there are 2b variables should be valued. Then:

KCL for n nodes: only n-1 equations are independent.

KVL for loops: only b-n+1 equations are independent. (only KVL for meshes)

VAR for branches: b equations.

So, (n-1)+(b-n+1)+b=2b, 2b equations to value 2b variables.

That is called 2b analysis.

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SERIES RESISTORS AND VOLTAGE DIVISION variables should be valued. Then:

- As we know, Series-connected means that the same current flows in them.
- The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

See illustration with 2 resistors:

R is the equivalent resistance.

It can be applied to any number of resistors

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The equivalent power of any number of resistors connected in series is the sum of the individual powers.

Concept of Equivalent:

Be equivalent to the outside, not the inside.

Principle of voltage division:if a voltage divider has N resistors(R1,R2,…RN) in series with the source voltage v, the nth resistor(Rn)will have a voltage drop of

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G series is the sum of the individual powers.

PARALLEL RESISTORS AND CURRENT DIVISION- As we know, Parallel-connected means that the same voltage covers over them.
- The equivalent conductance of resistors connected in parallel is the sum of their individual conductances.

See illustration with 2 resistors:

G

G is the equivalent conductance.

It can be applied to any number of resistors

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The equivalent power of any number of resistors connected in parallel is the sum of the individual powers.

Principle of current division

If a current divider has N conductors (G1,G2…GN) in parallel with the source current i, the nth conductor (Gn) will have current

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MIXED CONNECTION AND ITS EQUIVALENT RESISTANCE parallel is the sum of the individual powers.

Examples (we combine resistors in series and in parallel)

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d parallel is the sum of the individual powers.

e

WYE-DELTA TRANSFORMATIONSSituations often arise in circuit analysis when the resistors are neither in parallel nor in series.

R12=?

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2 parallel is the sum of the individual powers.

1

d

e

3

So

Can you imagine another way of transformation?

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