### Document

```ECE 3336
Introduction to Circuits & Electronics
Note Set #3
Equivalent Circuits and Tools
Spring 2015,
TUE&TH 5:30-7:00 pm
Dr. Wanda Wosik
1
Series and Parallel Resistors
Equivalent Circuits
• Equivalent circuit is used to simplify the original circuit but at the
terminals it maintains the exact same parameters: ex. voltage
and current.
• Example: Elements A||B in the circuit below are replaced by C.
• Currents iA||B=iC are the same & voltage V2 is the same
iA||B
A
B
A
Equivalent
circuit
iC
C
B
2
Equivalent Circuits
Example: The same circuit but different equivalent circuit at different points
• All elements to the right of VS2 are replaced by equivalent circuit D.
• Currents i0=iD are the same
• Voltages V2&V3 lost their meanings but VD is the same.
Equivalent
circuit
i0
iD
D
A
VD
B
VD
}
}
This part of the circuit must not “notice” any change on the right.
3
Equivalent Circuits
Summing up: Basic Requirements
• Equivalent circuits as being equivalent in terms of terminal
properties.
• The properties (voltage, current, power) within the
equivalent circuit may be different.
4
Series Connections of Elements
• Two parts of a circuit are in series if the same current flows
through both of them.
• It means there is no charge accumulation in the circuit.
current
• A hydraulic analogy: Two water pipes in series - the same
flow.
Connections may be not obvious:
• the red part and the blue part of the
pipes are in series
• but the blue part and the green part
and black are not in se | ries.
5
Series Connections of
Elements
We will substitute the
chain of resistors by one
equivalent resistor REQ
Parallel Connection of Circuit Elements
•
A hydraulic analogy
Parts of a circuit are in parallel if the same
voltage is across both of them.
Voltage
V1
•
The same exact voltage across each part of
the circuit means that the two end points must
be connected together.
voltage
+
circuit
-
Pipe
Section 1
Hight
Pipe
Section 2
circuit
The analogy is between voltage and height
V2
7
Parallel Resistors and KCL
Similarly, we will substitute
the resistors in parallel by
one equivalent resistor REQ
8
Series Resistors Equivalent Circuits
• Series resistors, R1 and
R2, can be replaced
with an equivalent
circuit (with respect to
the rest of the circuit)
with a single resistor
REQ, as long as
REQ = R1 + R2 .
+
i
vR1
-
+
R1
iR1=iR2
+
Rest
of the
Circuit
vREQ
REQ
Rest
of the
Circuit
Because:
vR1 = iR1R1
vR2 = iR2 R2
vR2
R2
No VR1
and VR2
-
vREQ = vR1 + vR2 so
vREQ =i(R1 + R2 ) = iREQ
9
More than 2 Series Resistors
• In case of N series
resistors we have
R1
REQ = R1 + R2 + ... + RN .
Any voltage drop on
individual resistor in the
equivalent circuit will be
“lost”
Rest
of the
Circuit
REQ
Rest
of the
Circuit
R2
10
The Resistors Must be in Series
R1 and R2 are not in series here.
• Resistors R1 and R2
cannot be replaced with a
single resistor REQ
R1
REQ ¹ R1 + R2 .
iX
+
vR2
Rest
of the
Circuit
REQ
Rest
of the
Circuit
R2
11
Parallel Resistors
Equivalent Circuits
Here:
Parallel resistors, R1 and R2,
can be replaced with an
equivalent circuit with a single
resistor REQ.
vREQ = vR1 = vR 2
Note that individual
currents do not exist now
vR1=vR2
iR2
iR1 = vR1 / R1
R2
iR2 = vR2 / R2
+
i=iR1+iR2
iR1
R1
vREQ
Rest
of the
Circuit
REQ
Rest
of the
Circuit
iEQ = iR1 + iR2 so
iEQ =vREQ / (R1 + R2 ) = vREQ / REQ
1
1 1
= +
REQ R1 R2
Notation R1||R2
-
12
Two and More Parallel Resistors
REQ for 2 parallel resistors:
1
1
1
=
+
REQ R1 R2
REQ = R1 || R2 =
R1 R2
R1 + R2
N parallel resistors will
have an equivalent
value:
R2
R1
Rest
of the
Circuit
REQ
Rest
of the
Circuit
1
1 1
1
= + + ... +
.
REQ R1 R2
RN
Notation: R1||R2||R3||…||RN
13
The Resistors NOT in Parallel
R1 and R2, can be
replaced with REQ
1
1 1
¹ + .
REQ R1 R2
iR2
R2
R1
Rest
of the
Circuit
REQ
Rest
of the
Circuit
NOT PARALLEL
14
Important Applications of
Series and Parallel
Connections
Wheatstone Bridge Circuits
Warning
• Orientation and position of the resistors in circuits may be
misleading when they just look like being connected in parallel or
in series BUT THEY ARE NOT.
16
Voltage Divider and Current Divider
Rules
These rules give us tools for important simplifications in solutions
of circuits to find fractions either of the whole
• VDR Voltage that will drop only on selected element(s)
connected in series
• CDR Current that will flow only through selected element(s)
connected in parallel
These rules are very useful but have to be carefully used:
directions and signs (YES: polarity) of current and voltages will
be critical
Voltage Divider Rule (VDR)
• The Voltage Divider Rule involves the voltages
across series resistors.
• We find the voltage on one element ex. VR1 (or VR2)
that is the fraction of the total voltage VTOTAL.
vTOTAL
iX =
.
R1 + R2
Note the voltages
polarities of in VDR
But also
For R1
For R2
vR1 vR2
ix =
=
R1 R2
vR1 = vTOTAL
R1
R1 + R2
vR2 = vTOTAL
R2
R1 + R2
Other Parts
of the Circuit
+
VR2
R2
ix
vTOTAL
+
vR1
R1
-
Other Parts
18 of
the Circuit
Voltage Divider Rule (VDR)
Negative Polarity
• The Voltage Divider Rule involves the voltages
across series resistors.
• We find the voltage on one element ex. VR1 (or VR2)
that is the fraction of the total voltage VTOTAL.
Note the voltage
polarity of in VDR;
NOW THEY ARE
CHANGED
For R1
For R2
vR1 = vTOTAL
R1
R1 + R2
vR2 = -vTOTAL
Other Parts
of the Circuit
+
VR2
R2
ix
vTOTAL
R2
R1 + R2
+
vR1
R1
-
Other Parts
19 of
the Circuit
Current Divider Rule (CDR)
This is our Second Circuit Analysis Tool to make circuit
analysis quicker and easier.
Other Parts
of the Circuit
If the current iTOTAL entering the node at two
resistors is known we can find the currents
through each of the resistors (R1&R2)
æ RR ö
v X = iTOTAL çç 1 2 ÷÷.
è R1 + R2 ø
v X = iR1R1
iR1 = iTOTAL
iTOTAL
iR1
R2
R1 + R2
(
v X = iTOTAL R1 || R2
R1 v
x
Other Parts of
the Circuit
R2
20
)
Current Divider Rule
For Each Resistor
iR1 = iTOTAL
iR 2 = iTOTAL
R2
.
R1 + R2
Other Parts
of the Circuit
iTOTAL
R1
.
R1 + R2
Note the polarities of all currents and
the voltage.
+
iR1
R1
vX iR2
R2
Other Parts of
the Circuit
21
The Current Divider Rule
Other Parts
of the Circuit
Direct write-up for the
Current Divider Rule (CDR).
iTOTAL
This is: voltage divided by resistance
vx/R1
+
R2
) R1 /R1
iR = iTOTAL(
1
R1 + R2
iR1
R1
R2
vX
-
Other Parts of
the Circuit
22
Negative Signs in the Current
Divider Rule
Change of the sign of the current
iQ in resistor R1 to have relative
polarity opposite to iTOTAL.
iQ = -iTOTAL
Other Parts
of the Circuit
R2
.
R1 + R2
iTOTAL
iQ
R1
Other Parts of
the Circuit
R2
23
Polarities
Voltage Divider and Current Divider Rules
• Correct polarities are critically important for correct
solutions of the circuits.
• VDR and CDR confirm the importance of reference
polarities.
24
25
Example Problem
37[kW]
2.2[kW]
4.7[kW]
+
8.2[kW]
5[mA]
27[kW]
6.2[kW]
iX
3.3[kW]
vW
-
Find iX and vW.
26
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