Transcript CSCI 2980: Introduction to Circuits, CAD, and Instrumentation

EENG 2610: Circuits Analysis
Class 2: Kirchhoff’s Laws, Single-Loop Circuits, SingleNode Pair Circuits
Oluwayomi Adamo
Department of Electrical Engineering
College of Engineering, University of North Texas
Some Important Concepts

Lumped-Parameter Circuit




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


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.
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: N
 i (t )  0
j 1
j
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
Example 2.5: Write all KCL equations
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:
N
 v (t )  0
j 1


j
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.
Example 2.9: Using KVL equation to find
VR3
= 18 V
= 12 V
Convention for Voltage Notation

Double-subscript notation

Vab = Va – Vb
a


+ 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.
Vab
V = Vab
b
Example 2.11: Use KVL to find Vae and Vec
Vae
Vec
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
 v(t )  vR1  vR2  0

vR1  i(t ) R1


vR2  i(t ) R2

i (t ) 
R1

v

 R1 R  R v(t )

1
2

v  R2 v(t )
 R2 R1  R2
v(t )
R1  R2
Multiple Voltage Sources in Single-Loop Circuit
v3
v2
v4
Equivalent
Transform
+
vR
R
vS
_
+
vR
_
v1
v1  v2  v3  v4  vR  0
vS  v1  v2  v3  v4
 vS  vR  0
R
Multiple Resistors in Single-Loop Circuit
i (t )
Equivalent
Transform
RS
v(t )
 v(t )  vR1  vR2  vR3  ... vRN  0
vR1  i (t ) R1

vR2  i (t ) R2

...
v  i (t ) R
N
 RN
v R  Ri i 
i
Single-Node Pair Circuits

Single-Node Pair Circuits



Elements are connected in parallel
Elements have the same voltage across them
Current division
v(t ) 
R1 R2
i(t )
R1  R2
Multiple Current Sources in Single-Node Pair
Circuits
Equivalent
Transform
i2 (t )
i5 (t )
Multiple Resistors in Single-Node Pair Circuits
Equivalent
Transform
+
v(t )
io (t )
_
General current divider:
v(t )  RP iO (t )
R
v(t )   iK (t )  p iO (t )
ik (t ) 
Rk
Rk 
RP
Example 2.19: Find the current IL
Equivalent circuit