ECE 2006
Electric Circuit Analysis
Fall 2011
September 08, 2011
Lecture 02
1
For Your Information
Course Webpage
http://www.d.umn.edu/~jingbai/Electric_Circuit_Analysis_(ECE_2006).html
You can find on the webpage:
Lecture: course syllabus, lecture notes, homework (& solutions)
Lab: lab syllabus (lab report format, grading matrix), lab instructions (11 labs)
Tutorials: breadboard, resistor color code, Pspice (download site and tutorial)
For unit conversion, you can learn more at
http://physics.nist.gov/cuu/Units/units.html
Lab TA: Tiger Qian, office hour: 3-4PM, Monday
2
Lab 1 Digital Multi-meter
•
Lab instructions
•
•
Available online
Print out and read before Lab
•
MWAH 391, 4:00 – 7:00 pm, next Monday or Wednesday
(Sept 13/15)
•
Work in pairs (group of 3 if odd number)
•
•
Group data, individual report
Lab safety rules
3
Lecture Outline
• Active and passive elements
• Ohm’s Law
• Kirchhoff’s current and voltage laws
4
Review: Electric Current

Created by motion of charge
I

Measured by charge change rate, unit in A (amperes)
dq
1C
i=
, 1A =
dt
1s

Convention on direction: flow direction of positive
charges, opposite to the flow of negative charges
5
Review: DC Current and AC current
Direct current (DC)
I
DC ( I ): current that remains
constant with time
Alternating current (AC)
i
AC ( i ): current that varies
sinusoidally with time
6
Voltage
Definition: electrical potential difference between two points of an
electric circuit or electromotive force (emf) required to move the
charge from one point to another point
+
a
The potential difference between points a and b:
vab= va-vb = dw/dq,
vab
b
w: energy with unit in joules (J)
q: charge with unit in coulombs (C)
v: voltage with unit in volts (V)
1V=1J/C
Polarity: “+” or “-” used to define the reference direction
Voltage drop direction: from “+” polarity to “-” polarity
The polarity can be defined by you
7
Power and Energy
Power: the time rate of expending or absorbing energy,
measured in watts (W):
p=dw/dt=(dw/dq).(dq/dt)=vi
Sign convention:
i
+
a
v
b
Current direction is the same
as the voltage drop direction ,
P=vi
i
-
a
v
+
b
When current direction is
opposite to the voltage drop
direction, P= - vi
8
Circuit Elements
Circuit element is a mathematical model of a real device
• Active element: p = vi <0, generating energy
i.e., batteries, generators
• Passive element: absorbing energy or storing energy
i.e., resistors (p=vi>0, absorbing energy);
capacitors and inductors (storing energy,
sign of power depends on the initial
condition)
9
Active Element
The most important active elements are voltage and
current sources
• Independent sources
v
+
-
• Dependent sources
+
V
i
-
Independent
voltage sources
Independent
current sources
v
+
-
Dependent
voltage source,
VCVS or CCVS
i
Dependent
current source,
VCCS or CCCS
10
Example of Dependent Sources
A
B
i
A
i
10i
+
-
C
+
5V
-
+
-
C
0.2i
CCVS
CCCS
11
Resistance
Resistance is the physical ability to resist current.
The circuit element used to model this behavior is the resistor, which is a passive
element
R
Circuit
symbol:
Unit for R: Ω, 1Ω =1V/A
l: length in m
A: crosssectional
area in m2
Resistance is dependent on the material property
and physical dimensions of the resistor
ρ: Material receptivity in Ω·m
l
R=ρ
A
12
Ohm’s Law
Statement: the voltage v across a resistor is directly
proportional to the current i flowing through the resistor, i.e.,
v∝i
v = iR
v
+
-
i
V = iR
13
Short Circuit and Open Circuit
v = iR (0 ≤ R < ∞)
(A)Short Circuit (SC)
R=0→v=0
+
i
v=0 R=0
-
(B) Open Circuit (OC)
R=∞→i=0
+
i=0
R=∞
-
14
Conductance
Conductance G is the reciprocal of resistance R, i.e.,
1 i
G= =
R v
Unit of G: siemens (S) or mhos ( )
1S=1/Ω=1A/V
15
Power
Power: the time rate of expending or absorbing energy,
measured in watts (W):
p=dw/dt=(dw/dq).(dq/dt)=vi
Instantaneous power for any circuit element: p(t)=v(t)i(t)
The power delivered to a resistor is
2
v
v
 
= v 2G
p = vi = v  =
R R
2
i
p = vi = (iR )i = i 2 R =
G
16
Branch, Nodes and Loops
One node
+
-
A branch is a single element such as a voltage (current) source or
a resistor;
A node is the point of connection between two or more branches;
A loop is any closed path in a circuit
17
Elements in Serial and Parallel
Two elements are in series if they
exclusively share a single node
Two elements are in parallel if
they are connected to the same
two nodes
18
An Example
4
1
5
2
3
6
Which circuit elements are in series? Which are in parallel?
19
Kirchoff’s Current Law (KCL)
KCL: the algebraic sum of currents entering a
node (or a closed boundary) is zero, i.e.,
N
∑i
n =1
n
=0
Algebraic sum: distinguished from arithmetical sum, the aggregate
of two or more numbers or quantities taken with regard to their signs,
as + or -, according to the rules of addition in algebra;
20
KCL (Cont’d)
Derivation of KCL is based on the conservation of charge, i.e.,
N
N
N
dqn
=
in = ∑
∑
n =1
n =1 dt
N
∑ dq
n =1
n
Example:
n =1
dt
n
0
= =0
dt
= 0 , conservation of charge
Define the currents entering the node are
“+”, those leaving are “-”
i2
i3
i1
i5
∑ dq
i4
i1 + i2 + (−i3 ) + i4 + (−i5 ) = 0
i1 + i2 + i4 = i3 + i5
The sum of currents entering the node is equal
to the sum of the currents leaving the node. 21
Application of KCL
iT
i1
i2
iT = i1 + i2 + i3
i3
i2
i1
i1 = i2
22
Kirchoff’s Voltage Law (KVL)
KVL: the algebraic sum of all voltages around a
closed path (or loop) is zero, i.e.,
N
∑v
m =1
+
v2
-
+
v3
m
=0
Define the voltage drops along clockwise
direction are “+”, those opposite are “-”
-
− v1 + v2 + v3 − i4 + v5 = 0
-
+
v1 -
+
v5
-
+
v4
v 2 + v3 + v5 = v1 + i4
Sum of voltage drop=Sum of voltage rise
23