EENG 2610: Circuits AnalysisClass 2: Kirchhoff’s Laws, Single-Loop Circuits, Single-Node Pair Circuits

Oluwayomi AdamoDepartment of Electrical EngineeringCollege of Engineering, University of North Texas

Some Important Concepts Lumped-Parameter Circuit

Wires in circuits are assumed perfect conductor Interconnections in circuits have zero resistance Wire doesn’t consume energy; energy in circuits is

lumped in each circuit element. Node

A point of connection of two or more circuit elements. A node is one end of a circuit element together with

all the perfect conductor that are attached to it. Loop

A loop is any closed path through the circuit in which no node is encountered more than once.

Branch A branch is a portion of a circuit containing only

a single element and the nodes at each end of the element.

Kirchhoff’s Current Law (KCL) KCL

The algebraic sum of the currents ENTERING any node is zero:

0)(1

N

jj ti

Example 2.5: Write all KCL equations

Our sign convention for KCL:- The algebraic sign of the current is ‘plus’ if the current is entering the node- The algebraic sign of the current is ‘minus’ if the current is leaving the node

Closed Surface as Super-Node Super-Node

If some set of elements are completely contained within a surface that is interconnected, the surface is called super-node.

Generalized KCL for Super-Node The algebraic sum of the currents entering any closed surface

(or super-node) is zero.

Write KCL equations for super-nodes

Kirchhoff’s Voltage Law (KVL) KVL

The algebraic sum of the voltages around any loop is zero:

Voltage is defined as the difference in energy level of a unit positive charge located at each of the two points. KVL is based on the conservation of energy: the work required to move a unit charge around any loop is zero.

Our Sign Convention for KVL As moving around a loop, the algebraic sign of voltage is

positive in KVL equation if encounter the plus sign first, and the algebraic sign of voltage is negative in KVL equation if encounter the minus sign first.

0)(1

N

jj tv

Example 2.9: Using KVL equation to find3R

V

= 18 V

= 12 V

Convention for Voltage Notation Double-subscript notation

Vab = Va – Vb

+ and – notation

Arrow notation Use an arrow between two

points, pointing from negative node to positive node.

KVL can be applied to a closed path even if part of the closed path is the arrow notation.

V = Vab

a

b

Vab

Example 2.11: Use KVL to find Vae and Vec

aeVecV

Single-Loop Circuits

Single-loop circuits Elements are connected

in series. All elements carry same

current. Voltage Divider

Source v(t) is divided between two resistors in proportion to their resistance

2

1

)(

)(

0)(

2

1

21

Rtiv

Rtiv

vvtv

R

R

RR

)(

)(

21

2

21

1

2

1

tvRR

Rv

tvRR

Rv

R

R

21

)()(

RR

tvti

Multiple Voltage Sources in Single-Loop Circuit

04321 Rvvvvv

1v

2v

3v 4v

RRv+

_

Sv RRv

+

_

0 RS vv

4321 vvvvvS

EquivalentTransform

Multiple Resistors in Single-Loop Circuit

iRv iRi

0...)(321

NRRRR vvvvtv

NR

R

R

Rtiv

Rtiv

Rtiv

N)(

...

)(

)(

2

1

2

1

)(tv SR

)(ti

EquivalentTransform

Single-Node Pair Circuits Single-Node Pair Circuits

Elements are connected in parallel Elements have the same voltage across them

Current division

)()(21

21 tiRR

RRtv

Multiple Current Sources in Single-Node Pair Circuits

EquivalentTransform

)(2 ti)(5 ti

Multiple Resistors in Single-Node Pair Circuits

)()()()(

)()(ti

R

Rti

R

tvti

tiRtv

Ok

pK

kk

OP

General current divider:

)(tio)(tv+

_

PR

EquivalentTransform

Equivalent circuit

Example 2.19: Find the current IL