Circuit Analysis
Part I
ENGR 1166 Biomedical Engineering
Recap
KCL: At any node in an electrical circuit,
the algebraic sum of the currents is equal
to zero
 KVL: the directed sum of voltages along
any closed path in an electrical circuit is
zero
 Ohm’s law: the ratio between the voltage
drop at the terminals of an ideal resistor
and the current passing through it is
constant and is called “resistance”

Circuit equivalency
+
−
circuit
A
+
−
circuit
B
Two circuits are equivalent if they cannot be
distinguished from each other by voltage and current
measurements, i.e., if they behave identically
1
Circuit equivalency
+
−
circuit
A
In this case, since
+
−
circuit
B
is applied to both circuits:
Circuit A and B are equivalent ⟺
=
Circuit equivalency
+
−
circuit
A
+
−
circuit
B
Sometimes it is useful to replace a
complex circuit with an equivalent one
that is much simpler to study
Circuit equivalency
+
−
circuit
A
+
−
circuit
B
Here we will assume that the circuits to
be replaced (“reduced”) are made of
resistors only (i.e., passive circuits)
2
An example of circuit equivalency
+
circuit
A
−
 Circuit A is any combination of resistors
 The voltage at the terminals of circuit A
is
An example of circuit equivalency
+
+
circuit
A
−
−
+
−
The circuit in red is equivalent to circuit A if
=
Resistors in series
 Two resistors are in series if the same
current flows from one to another
+
−
+
+
−
−
+
−
3
Resistors in series
 Two resistors are in series if the same
current flows from one to another
+
−
+
+
−
−
+
and
and
are in series
are in series
−
Resistors in series
 Two resistors in series always share one
common node
+
−
+
+
−
−
+
and
and
are in series
are in series
−
Resistors in series
 Two resistors in series always share one
common node
+
−
+
+
−
−
+
and
and
are in series
are in series
−
4
Resistors in series
 If the same current flows through
distinct
resistors are in series
resistors then the
+
−
, and
,
are in series
+
+
−
−
−
+
Resistors in series
 If the same current flows through
distinct
resistors are in series
resistors then the
+
+
−
path 1
+
−
−
KVL at path 1:
+
−
+
+
−
=
Resistors in series
 If the same current flows through
distinct
resistors are in series
resistors then the
+
+
−
path 1
+
−
−
KVL at path 1:
+
=
−
+
+
+
−
=
+
=
5
Resistors in series
resistors in series are equivalent to a
single resistor whose resistance
is:
+
+
−
path 1
=
+
−
−
KVL at path 1:
+
−
+
+
−
=
Resistors in parallel
 Two or more resistors are in parallel if the
same voltage is across each of them
+
−
+
+
+
−
−
−
Resistors in parallel
 Two or more resistors are in parallel if the
same voltage is across each of them
+
−
To denote resistors
in parallel:
+
+
+
−
−
−
=
=
|
|
=
6
Resistors in parallel
 Resistors in parallel always share two
common nodes
node A
+
−
+
+
+
−
−
−
node B
Resistors in parallel
 Resistors in parallel always share two
common nodes
node A
+
−
+
+
+
−
−
−
KCL at node A: − +
+
+ =
Resistors in parallel
 Resistors in parallel always share two
common nodes
node A
+
−
+
+
+
−
−
−
KCL at node A: − +
+
+
=
7
Resistors in parallel
 Resistors in parallel always share two
common nodes
+=
+
−
=+
−
−
node A
+
+
KCL at node A: − +
+
−
+
+
=
Resistors in parallel
resistors in parallel are equivalent to a
single resistor whose resistance
is:
=
+
−
node A
∑
+
+
+
−
−
−
KCL at node A: − +
+
+
=
Example 1
A
B
=
Ω;
=
Ω;
=
Ω;
=
Ω;
=
Ω;
=
Ω
What is the equivalent circuit resistance
between terminals A and B?
8
Example 1
A
B
=
Ω;
=
Ω;
=
Ω;
=
Ω;
||
||
=
=
Ω
Ω
=
Ω;
=
Ω
Ω;
=
Ω
Ω;
=
Ω
Example 1
A
B
=
||
||
Ω;
=
Ω;
=
Ω;
=
Ω;
||
||
=
=
Ω
Ω
=
Example 1
A
B
=
||
||
||
Ω;
= Ω;
= Ω;
= Ω; || = Ω;
+
||
=
Ω;
=
=
Ω
9
Example 1
A
B
=
||
||
Ω;
= Ω;
= Ω;
= Ω; || = Ω;
||
Ω;
=
Ω;
=
Ω
Ω;
= Ω;
= Ω;
= Ω;
= Ω;
= Ω; || = Ω;
+ || = Ω;
=
Ω
+
||
=
+
=
Ω
Example 1
A
B
=
||
+
||
=
||
||
+
||
||
= / Ω
Example 1
A
B
=
||
Ω;
= Ω;
= Ω;
= Ω;
= Ω;
= Ω; || = Ω;
+ || = Ω;
=
||
||
+
||
=
Ω
= / Ω
10
Example 1
A
B
=
||
Ω;
= Ω;
= Ω;
= Ω;
= Ω;
=
Ω
= Ω; || = Ω;
+ || = Ω; = / Ω;
=
+
≅ .
Ω
Example 2
A
=
=
Ω;
Ω;
=
=
B
=
Ω;
Ω
Ω;
=
Ω;
=
Ω;
What is the equivalent circuit resistance
between terminals A and B?
Example 2
A
=
=
Ω;
Ω;
=
=
Ω;
Ω
B
=
||
||
Ω;
=
=
=
Ω;
=
Ω;
Ω
Ω
11
Example 2
||
=
=
Ω;
Ω;
A
=
=
B
=
Ω;
Ω
||
=
Ω;
||
||
=
=
=
Ω;
Ω;
Ω
Ω
Example 2
||
=
=
Ω;
Ω;
A
=
=
Ω;
Ω;
||
B
= Ω;
= Ω;
= Ω;
|| = Ω; || = Ω
=
Ω
Example 2
||
||
=
=
Ω;
Ω;
||
A
=
=
Ω;
Ω;
||
||
B
= Ω;
= Ω;
= Ω;
|| = Ω; || = Ω
=
Ω
12
Example 2
||
||
=
=
||
=
Ω;
=
Ω;
=
= Ω
||
A
||
B
Ω;
Ω;
= Ω;
= Ω;
= Ω;
|| = Ω; || = Ω;
+
||
+
||
=
Ω
Example 2
A
=
=
||
=
Ω;
=
Ω;
=
= Ω
||
B
Ω;
Ω;
= Ω;
= Ω;
= Ω;
|| = Ω; || = Ω;
+
||
+
||
=
Ω
Example 2
A
=
=
||
B
Ω;
= Ω;
= Ω;
= Ω;
= Ω;
Ω;
= Ω; || = Ω; || = Ω;
= Ω; = Ω;
= ||
= .
Ω
13
Voltage divider rule
 It allows to calculate the voltage across
any individual resistor connected in a
series of resistors
+
−
+
+
−
−
Voltage divider rule
 It allows to calculate the voltage across
any individual resistor connected in a
series of resistors
+
−
Series:
=
+
+
+
−
−
Voltage divider rule
 It allows to calculate the voltage across
any individual resistor connected in a
series of resistors
+
+
−
−
Series:
+
−
=
=
=
+
+
14
Voltage divider rule
=
=
+
+
−
=
=
=
=
+
+
+
+
−
−
Voltage divider rule
 The voltage drop across each resistor in a
series is a fraction of the original voltage
across the entire series
+
−
+
+
−
=
=
=
=
+
+
−
Voltage divider rule (VDR)
resistors are in series ( , , …,
)
and the voltage across the series is , the
voltage drop across each resistor is:
 If
=
+
+⋯+
= , , ,…,
15
Current divider rule
 It allows to calculate the current through
any resistor connected in parallel resistor
circuits
+
+
−
−
Current divider rule
 It allows to calculate the current through
any resistor connected in parallel resistor
circuits
=
Parallel:
+
+
−
−
Current divider rule
 It allows to calculate the current through
any resistor connected in parallel resistor
circuits
=
Parallel:
+
+
−
−
=
=
+
16
Current divider rule
=
=
+
+
+
−
−
=
=
=
=
=
=
=
=
+
+
Current divider rule
=
=
+
+
+
−
−
+
+
Current divider rule
 The current through each resistor in a
parallel circuit is a fraction of the original
current and depends on the resistance
+
+
−
−
=
=
=
=
+
+
17
Current divider rule (CDR)
resistors are in parallel ( ,…,
) and
the current entering (leaving) the common
nodes is , then the current through each
resistor is:
 If
=
+
+⋯+
= , , ,…,
Example 3
+
=
=
/ Ω;
Ω; =
=
A
Ω;
=
−
=
Ω;
What is the value of
Ω;
, , and
=
Ω;
=
Ω;
?
Example 3
+
=
=
/ Ω;
Ω; =
=
A
Ω;
=
=
||
−
Ω;
||
=
=
Ω;
Ω
18
Example 3
+
=
=
/ Ω;
Ω; =
= Ω;
A;
=
CDR:
=
Ω
−
=
Ω;
Ω;
=
Ω;
=
Ω;
=
Ω;
=
Ω;
=
Ω;
=
=
Example 3
+
=
=
/ Ω;
Ω; =
= Ω;
A;
=
=
Ω
−
Ω;
Example 3
+
=
=
/ Ω;
Ω; =
= Ω;
A;
=
=
=
Ω
+
−
Ω;
=
Ω
19
Example 3
+
=
=
/ Ω;
Ω; =
= Ω;
A;
=
VDR:
=
Ω;
−
Ω;
=
=
Ω
Ω;
=
Ω;
=
Ω;
Ω;
=
=
Ω
Ω;
=
Ω;
=
Ω;
Ω;
=
=
Ω
Ω;
=
Ω;
=
Example 3
=
=
/ Ω;
Ω; =
= Ω;
A;
=
Example 3
=
=
/ Ω;
Ω; =
CDR:
= Ω;
A;
=
=
= .
= . A
20
Example 3
=
=
/ Ω;
Ω; =
CDR:
= Ω;
A;
=
=
Ω;
Ω;
=
=
Ω
= .
=
Ω;
=
Ω;
= . A
Example 3
=
=
/ Ω;
Ω; =
= Ω;
A;
=
CDR:
=
Ω;
Ω;
=
=
Ω
Ω;
=
Ω;
Ω;
=
Ω;
≅ . A
=
Example 3
=
=
/ Ω;
Ω; =
VDR:
= Ω;
A;
=
=
=
Ω;
Ω;
=
=
=
Ω
= .
V
21