Electric Circuits (Fall 2015)
Pingqiang Zhou
Lecture 3
Laws
9/29/2015
Reading: Chapter 2
Lecture 3
1
Electric Circuits (Fall 2015)
Pingqiang Zhou
Outline
•
Node, Branch and Loop
•
KCL and KVL
•
•
Series Resistors and Voltage Division
Parallel Resistors and Current Division
•
Wye-Delta Transformations
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Terminology: Nodes, Branches and Loops
•
•
•
Node: A point where two or more circuit elements are
connected.
Branch: A path that connects two nodes.
Loop: Any closed path in a circuit.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Network Topology
•
•
•
A loop is independent if it contains at least one branch not
shared by any other independent loops.
Two or more elements are in series if they share a single
node and thus carry the same current.
Two or more elements are in parallel if they are connected
to the same two nodes and thus have the same voltage.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Kirchhoff’s Laws
•
Ohm’s law is not sufficient for circuit
analysis
 Kirchhoff’s laws complete it.
• Kirchhoff’s Current Law (KCL):
Gustav Robert Kirchhoff
1824-1887
▪ The algebraic sum of all the currents entering any node in a circuit
equals zero.
▪ Based on conservation of charge.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
A Major Implication of KCL
• KCL tells us that all of the elements that are connected
in series carry the same current.
Current entering node = Current leaving node
i1 = i2
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Generalization of KCL
•
The sum of currents entering/leaving a closed surface is
zero. Circuit branches can be inside this surface, i.e. the
surface can enclose more than one node!
i2
i3
This could be a big
chunk of a circuit,
e.g. a “black box”
i4
i1
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Generalized KCL Examples
50 mA
5mA
2mA
i
i
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Notation: Node and Branch Voltages
• Use one node as the reference (the “common” or
“ground” node) – label it with a symbol.
• The voltage drop from node x to the reference node is
called the node voltage vx.
• The voltage across a circuit element is defined as the
difference between the node voltages at its terminals.
Example:
– v1 +
a R1 b
+
va +_ vs
_
c
Lecture 3
+
R2 vb
_
 REFERENCE NODE
9
Electric Circuits (Fall 2015)
Pingqiang Zhou
Kirchhoff’s Voltage Law (KVL)
•
The algebraic sum of all the voltages around any loop in
a circuit equals zero.
▪ Based on conservation of energy
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
KVL Example
Three closed paths:
1
+
va

b
+
vb
-

+ v2 
v3
2
+
a
c
+
vc

3
Path 1:
Path 2:
Path 3:
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
A Major Implication of KVL
•
•
KVL tells us that any set of elements which are connected
at both ends carry the same voltage.
We say these elements are connected in parallel.
+
va
_
+
vb
_
Applying KVL in the clockwise direction,
starting at the top:
vb – va = 0  vb = va
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Series Resistors
•
•
Two resistors are considered in series if
the same current pass through them
Take the circuit shown:
 Applying Ohm’s law to both resistors
v1  iR1 v2  iR2
 If we apply KVL to the loop we have
v  v1  v2  0
 Combining the two equations
v  v1  v2  i  R1  R2 
Req  R1  R2
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Voltage Division
•
•
The voltage drop across any one resistor can be known.
The current through all the resistors is the same, so using
Ohm’s law:
R1
R2
v1 
v v2 
v
R1  R2
R1  R2
•
This is the principle of voltage division.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
When can the Voltage Divider Formula be Used?
I
I
R1
R1
VSS
+

R2
+
V
– 2
R2
VSS +

R3
R3
R4
R4
𝑅2
𝑉2 =
𝑉𝑠𝑠
𝑅1 + 𝑅2 + 𝑅3 + 𝑅4
+
–V2
𝑉2 =
R5
𝑅2
𝑉 ?
𝑅1 +𝑅2 +𝑅3 +𝑅4 𝑠𝑠
Why? What is V2?
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Parallel Resistors
•
When resistors are in parallel, the
voltage drop across them is the same
v  i1R1  i2 R2
•
By KCL, the current at node a is
i  i1  i2
•
The equivalent resistance is:
R1 R2
Req 
R1  R2
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Current Division
•
Given the current entering the node, the voltage drop
across the equivalent resistance will be the same as that
for the individual resistors
iR1R2
v  iReq 
R1  R2
•
This can be used in combination with Ohm’s law to get the
current through each resistor:
iR2
i1 
R1  R2
iR1
i2 
R1  R2
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Measuring Voltage (Voltmeter)
• To measure the voltage drop across an element in a
real circuit, insert a voltmeter (digital multimeter in
voltage mode) in parallel with the element.
• Voltmeters are characterized by their “voltmeter input
resistance” (Rin). Ideally, this should be very high
(typical value 10 MW)
Ideal
Voltmeter
Rin
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Effect of Voltmeter
undisturbed circuit
circuit with voltmeter inserted
R1
VSS
+
_
R1
+
R2
–
VSS
V2
 R2 
V2  VSS 

R

R
 1
2
+
_
+
R2
Rin
–
V2′
 R2 || Rin 

V2  VSS 

R
||
R

R
 2
in
1
Example: VSS  10 V, R2  100K, R1  900K  V2  1V
Rin  10M , V2  ?
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Measuring Current (Ammeter)
• To measure the current flowing through an element in
a real circuit, insert an ammeter (digital multimeter in
current mode) in series with the element.
• Ammeters are characterized by their “ammeter input
resistance” (Rin). Ideally, this should be very low
(typical value 1W).
Ideal
Ammeter
Rin
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Effect of Ammeter
Measurement error due to non-zero input resistance:
undisturbed circuit
I
circuit with ammeter inserted
Imeas
ammeter
R1
R1
V1 +
_
V1 +
_
R2
V1
I
R1  R2
Rin
R2
V1
Imeas 
R1  R2  Rin
Example: V1 = 1 V, R1= R2 = 500 W, Rin = 1W
1V
I
 1mA, I meas  ?
500W  500W
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye-Delta Transformations
•
•
There are cases where resistors are neither parallel nor
series.
Consider the bridge circuit shown here. This circuit can
be simplified to a three-terminal equivalent.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye-Delta Transformations II
•
Two topologies can be interchanged:
 Wye (Y) or tee (T) networks
 Delta (Δ) or pi (Π) networks
 Transforming between these two
topologies often makes the solution of
a circuit easier.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye-Delta Transformations III
•
The superimposed wye and delta
circuits shown here will used for
reference.
•
The delta consists of the outer
resistors, labeled a, b, and c.
•
The wye network are the inside
resistors, labeled 1, 2, and 3.
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Delta to Wye
•
The conversion formula for a delta to wye transformation
are:
Rb Rc
R1 
Ra  Rb  Rc
R2 
Rc Ra
Ra  Rb  Rc
R3 
Ra Rb
Ra  Rb  Rc
𝑅𝑎𝑏 𝑌 = 𝑅𝑎𝑏 ∆ ⟹ 𝑅1 + 𝑅2 =
𝑅𝑐 (𝑅𝑎 + 𝑅𝑏 )
𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐
𝑅𝑎𝑐 𝑌 = 𝑅𝑎𝑐
𝑅𝑏 (𝑅𝑎 + 𝑅𝑐 )
∆ ⟹ 𝑅1 + 𝑅3 =
𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐
𝑅𝑏𝑐 𝑌 = 𝑅𝑏𝑐
𝑅𝑎 (𝑅𝑏 + 𝑅𝑐 )
∆ ⟹ 𝑅2 + 𝑅3 =
𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye to Delta
•
The conversion formula for a wye to delta transformation
are:
R1 R2  R2 R3  R3 R1
Ra 
R1
R1 R2  R2 R3  R3 R1
Rb 
R2
Rc 
R1 R2  R2 R3  R3 R1
R3
𝑅𝑎 𝑅𝑏 𝑅𝑐
𝑅1 𝑅2 + 𝑅1 𝑅3 + 𝑅2 𝑅3 =
𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Example
•
Find the equivalent resistance 𝑅𝑎𝑏
Wye -> Delta
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Summary
•
KCL and KVL
N
i
n 1
•
•
•
n
M
v
0
m 1
m
0
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Wye-Delta Transformations
R1 
Rb Rc
Ra  Rb  Rc
Ra 
R1 R2  R2 R3  R3 R1
R1
R2 
Rc Ra
Ra  Rb  Rc
Rb 
R1 R2  R2 R3  R3 R1
R2
R3 
Ra Rb
Ra  Rb  Rc
Rc 
R1 R2  R2 R3  R3 R1
R3
Lecture 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Next Lecture and Reminders
•
Next lecture
 Oct. 8th , Thursday
–Reading assignment – Ch. 3
•
Reminders
 HW1 due Oct. 13th, Tuesday
 Lab2 on Oct. 9th for Class 3&4
 Discussion for Class 3&4 on Oct. 9th
 NO makeup lab/discussion on Oct. 10th
Lecture 4
29