2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
2: Resistor Circuits
Resistor Circuits: 2 – 1 / 13
Kirchoff’s Voltage Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
The five nodes are labelled
A, B, C, D, E where E is the
reference node.
Each component that links a pair of
nodes is called a branch of the
network.
• Non-ideal Voltage Source
• Summary
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Kirchoff’s Voltage Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The five nodes are labelled
A, B, C, D, E where E is the
reference node.
Each component that links a pair of
nodes is called a branch of the
network.
Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the work
done in moving a charge from one node to another does not depend on the
route you take; in particular the work done in going from one node back to
the same node by any route is zero.
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Kirchoff’s Voltage Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The five nodes are labelled
A, B, C, D, E where E is the
reference node.
Each component that links a pair of
nodes is called a branch of the
network.
Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the work
done in moving a charge from one node to another does not depend on the
route you take; in particular the work done in going from one node back to
the same node by any route is zero.
KVL: the sum of the voltage changes around any closed loop is zero.
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Kirchoff’s Voltage Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The five nodes are labelled
A, B, C, D, E where E is the
reference node.
Each component that links a pair of
nodes is called a branch of the
network.
Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the work
done in moving a charge from one node to another does not depend on the
route you take; in particular the work done in going from one node back to
the same node by any route is zero.
KVL: the sum of the voltage changes around any closed loop is zero.
Example: VDE + VBD + VAB + VEA = 0
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Kirchoff’s Voltage Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The five nodes are labelled
A, B, C, D, E where E is the
reference node.
Each component that links a pair of
nodes is called a branch of the
network.
Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the work
done in moving a charge from one node to another does not depend on the
route you take; in particular the work done in going from one node back to
the same node by any route is zero.
KVL: the sum of the voltage changes around any closed loop is zero.
Example: VDE + VBD + VAB + VEA = 0
Equivalent formulation:
VXY = VXE − VY E = VX − VY for any nodes X and Y .
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
A consequence is Kirchoff’s Current Law (KCL) which says that the current
going into any closed region of a circuit must equal the current coming out.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
A consequence is Kirchoff’s Current Law (KCL) which says that the current
going into any closed region of a circuit must equal the current coming out.
KCL: The currents flowing out of any closed region of a circuit sum to zero.
• Non-ideal Voltage Source
• Summary
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
A consequence is Kirchoff’s Current Law (KCL) which says that the current
going into any closed region of a circuit must equal the current coming out.
KCL: The currents flowing out of any closed region of a circuit sum to zero.
Green: I1 = I7
• Non-ideal Voltage Source
• Summary
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
A consequence is Kirchoff’s Current Law (KCL) which says that the current
going into any closed region of a circuit must equal the current coming out.
KCL: The currents flowing out of any closed region of a circuit sum to zero.
Green: I1 = I7
Blue: −I1 + I2 + I5 = 0
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Kirchoff’s Current Law
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Wherever charges are free to move around, they will move to ensure
charge neutrality everywhere at all times.
A consequence is Kirchoff’s Current Law (KCL) which says that the current
going into any closed region of a circuit must equal the current coming out.
KCL: The currents flowing out of any closed region of a circuit sum to zero.
Green: I1 = I7
Blue: −I1 + I2 + I5 = 0
Gray: −I2 + I4 − I6 + I7 = 0
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KCL Example
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
The currents and voltages in any linear circuit can be determined by using
KCL, KVL and Ohm’s law.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
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KCL Example
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
The currents and voltages in any linear circuit can be determined by using
KCL, KVL and Ohm’s law.
Sometimes KCL allows you to determine currents very easily without
having to solve any simultaneous equations:
How do we calculate I ?
• Non-ideal Voltage Source
• Summary
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KCL Example
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The currents and voltages in any linear circuit can be determined by using
KCL, KVL and Ohm’s law.
Sometimes KCL allows you to determine currents very easily without
having to solve any simultaneous equations:
How do we calculate I ?
KCL: −1 + I + 3 = 0
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KCL Example
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The currents and voltages in any linear circuit can be determined by using
KCL, KVL and Ohm’s law.
Sometimes KCL allows you to determine currents very easily without
having to solve any simultaneous equations:
How do we calculate I ?
KCL: −1 + I + 3 = 0
=⇒ I = −2 A
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KCL Example
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
The currents and voltages in any linear circuit can be determined by using
KCL, KVL and Ohm’s law.
Sometimes KCL allows you to determine currents very easily without
having to solve any simultaneous equations:
How do we calculate I ?
KCL: −1 + I + 3 = 0
=⇒ I = −2 A
Note that here I ends up negative which means we chose the wrong arrow
direction to label the circuit. This does not matter. You can choose the
directions arbitrarily and let the algebra take care of reality.
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
• Non-ideal Voltage Source
• Summary
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
Within the chain, each internal node
connects to only two branches.
• Non-ideal Voltage Source
• Summary
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
Within the chain, each internal node
connects to only two branches.
R3 and R4 are not in series and do not
necessarily have the same current.
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
Within the chain, each internal node
connects to only two branches.
R3 and R4 are not in series and do not
necessarily have the same current.
Parallel: Components that are connected to the same pair of nodes are
said to be in parallel.
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
Within the chain, each internal node
connects to only two branches.
R3 and R4 are not in series and do not
necessarily have the same current.
Parallel: Components that are connected to the same pair of nodes are
said to be in parallel.
R1 , R2 , R3 are in parallel and the same
voltage is across each resistor (even
though R3 is not close to the others).
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Series and Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Series: Components that are connected in a chain so that the same current
flows through each one are said to be in series.
R1 , R2 , R3 are in series and the same
current always flows through each.
Within the chain, each internal node
connects to only two branches.
R3 and R4 are not in series and do not
necessarily have the same current.
Parallel: Components that are connected to the same pair of nodes are
said to be in parallel.
R1 , R2 , R3 are in parallel and the same
voltage is across each resistor (even
though R3 is not close to the others).
R4 and R5 are also in parallel.
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
VX = V1 + V2 + V3
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
=
R1
R1 +R2 +R3
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
=
R1
R1 +R2 +R3
=
R1
RT
where RT = R1 + R2 + R3 is the
total resistance of the chain.
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
=
R1
R1 +R2 +R3
=
R1
RT
where RT = R1 + R2 + R3 is the
total resistance of the chain.
VX is divided into V1 : V2 : V3 in the proportions R1 : R2 : R3 .
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
=
R1
R1 +R2 +R3
=
R1
RT
where RT = R1 + R2 + R3 is the
total resistance of the chain.
VX is divided into V1 : V2 : V3 in the proportions R1 : R2 : R3 .
Approximate Voltage Divider:
A
If IY = 0, then VY = R R+R
VX .
A
B
E1.1 Analysis of Circuits (2016-8284)
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Series Resistors: Voltage Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
VX = V1 + V2 + V3
= IR1 + IR2 + IR3
= I(R1 + R2 + R3 )
V1
VX
=
IR1
I(R1 +R2 +R3 )
=
R1
R1 +R2 +R3
=
R1
RT
where RT = R1 + R2 + R3 is the
total resistance of the chain.
VX is divided into V1 : V2 : V3 in the proportions R1 : R2 : R3 .
Approximate Voltage Divider:
A
If IY = 0, then VY = R R+R
VX .
A
B
A
VX .
If IY ≪ I , then VY ≈ R R+R
A
B
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Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Parallel resistors all share the same V .
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
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Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Parallel resistors all share the same V .
I1 =
V
R1
= V G1 where G1 =
1
R1
is the conductance of R1 .
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 7 / 13
Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Parallel resistors all share the same V .
I1 =
V
R1
IX
= V G1 where G1 =
1
R1
is the conductance of R1 .
= I1 + I2 + I3
= V G1 + V G2 + V G3
= V (G1 + G2 + G3 )
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 7 / 13
Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Parallel resistors all share the same V .
I1 =
V
R1
IX
= V G1 where G1 =
1
R1
is the conductance of R1 .
= I1 + I2 + I3
= V G1 + V G2 + V G3
= V (G1 + G2 + G3 )
• Non-ideal Voltage Source
• Summary
I1
IX
=
V G1
V (G1 +G2 +G3 )
=
G1
G1 +G2 +G3
=
G1
GP
where GP = G1 + G2 + G3 is the total conductance of the resistors.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 7 / 13
Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Parallel resistors all share the same V .
I1 =
V
R1
IX
= V G1 where G1 =
1
R1
is the conductance of R1 .
= I1 + I2 + I3
= V G1 + V G2 + V G3
= V (G1 + G2 + G3 )
• Non-ideal Voltage Source
• Summary
I1
IX
=
V G1
V (G1 +G2 +G3 )
=
G1
G1 +G2 +G3
=
G1
GP
where GP = G1 + G2 + G3 is the total conductance of the resistors.
IX is divided into I1 : I2 : I3 in the proportions G1 : G2 : G3 .
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 7 / 13
Parallel Resistors: Current Divider
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Parallel resistors all share the same V .
I1 =
V
R1
IX
= V G1 where G1 =
1
R1
is the conductance of R1 .
= I1 + I2 + I3
= V G1 + V G2 + V G3
= V (G1 + G2 + G3 )
• Non-ideal Voltage Source
• Summary
I1
IX
=
V G1
V (G1 +G2 +G3 )
=
G1
G1 +G2 +G3
=
G1
GP
where GP = G1 + G2 + G3 is the total conductance of the resistors.
IX is divided into I1 : I2 : I3 in the proportions G1 : G2 : G3 .
Special case for only two resistors:
I1 : I2 = G1 : G2 = R2 : R1 ⇒ I1 =
E1.1 Analysis of Circuits (2016-8284)
R2
R1 +R2 IX .
Resistor Circuits: 2 – 7 / 13
Equivalent Resistance: Series
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
We know that V = V1 + V2 + V3 = I(R1 + R2 + R3 ) = IRT
So we can replace the three resistors
by a single equivalent resistor of
value RT without affecting the
relationship between V and I .
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 8 / 13
Equivalent Resistance: Series
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
We know that V = V1 + V2 + V3 = I(R1 + R2 + R3 ) = IRT
So we can replace the three resistors
by a single equivalent resistor of
value RT without affecting the
relationship between V and I .
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 8 / 13
Equivalent Resistance: Series
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
We know that V = V1 + V2 + V3 = I(R1 + R2 + R3 ) = IRT
So we can replace the three resistors
by a single equivalent resistor of
value RT without affecting the
relationship between V and I .
Replacing series resistors by their
equivalent resistor will not affect any
of the voltages or currents in the rest
of the circuit.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 8 / 13
Equivalent Resistance: Series
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
We know that V = V1 + V2 + V3 = I(R1 + R2 + R3 ) = IRT
So we can replace the three resistors
by a single equivalent resistor of
value RT without affecting the
relationship between V and I .
Replacing series resistors by their
equivalent resistor will not affect any
of the voltages or currents in the rest
of the circuit.
However the individual voltages V1 ,
V2 and V3 are no longer accessible.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 8 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
We can use a single
equivalent resistor of
resistance RP without
affecting the
relationship between
V and I .
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
We can use a single
equivalent resistor of
resistance RP without
affecting the
relationship between
V and I .
Replacing parallel resistors by
their equivalent resistor will not
affect any of the voltages or
currents in the rest of the circuit.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
We can use a single
equivalent resistor of
resistance RP without
affecting the
relationship between
V and I .
Replacing parallel resistors by
their equivalent resistor will not
affect any of the voltages or
currents in the rest of the circuit.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
We can use a single
equivalent resistor of
resistance RP without
affecting the
relationship between
V and I .
Replacing parallel resistors by
their equivalent resistor will not
affect any of the voltages or
currents in the rest of the circuit.
R4 and R5 are also in parallel.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
Similarly we known that I = I1 + I2 + I3 = V (G1 + G2 + G3 ) = V GP .
So V = IRP where RP = G1 = G +G1 +G = 1/R +1/1R +1/R
P
1
2
3
1
2
3
We can use a single
equivalent resistor of
resistance RP without
affecting the
relationship between
V and I .
Replacing parallel resistors by
their equivalent resistor will not
affect any of the voltages or
currents in the rest of the circuit.
R4 and R5 are also in parallel.
Much simpler - although none of the original currents I1 , · · · , I5 are now
accessible. Current IS and the three node voltages are identical.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 9 / 13
Equivalent Resistance: Parallel Formulae
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
For parallel resistors GP = G1 + G2 + G3
or equivalently RP = R1 ||R2 ||R3 = 1/R +1/1R +1/R .
1
2
3
These formulae work for any number of resistors.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 10 / 13
Equivalent Resistance: Parallel Formulae
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
For parallel resistors GP = G1 + G2 + G3
or equivalently RP = R1 ||R2 ||R3 = 1/R +1/1R +1/R .
1
2
3
These formulae work for any number of resistors.
• For the special case of two parallel resistors
RP =
1
1/R +1/R
1
2
=
R1 R2
R1 +R2
(“product over sum”)
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 10 / 13
Equivalent Resistance: Parallel Formulae
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
For parallel resistors GP = G1 + G2 + G3
or equivalently RP = R1 ||R2 ||R3 = 1/R +1/1R +1/R .
1
2
3
These formulae work for any number of resistors.
• For the special case of two parallel resistors
RP =
1
1/R +1/R
1
2
=
R1 R2
R1 +R2
(“product over sum”)
• Non-ideal Voltage Source
• Summary
• If one resistor is a multiple of the other
Suppose R2 = kR1 , then
RP =
E1.1 Analysis of Circuits (2016-8284)
R1 R2
R1 +R2
=
kR12
(k+1)R1
=
k
k+1 R1
= (1 −
1
k+1 )R1
Resistor Circuits: 2 – 10 / 13
Equivalent Resistance: Parallel Formulae
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
For parallel resistors GP = G1 + G2 + G3
or equivalently RP = R1 ||R2 ||R3 = 1/R +1/1R +1/R .
1
2
3
These formulae work for any number of resistors.
• For the special case of two parallel resistors
RP =
1
1/R +1/R
1
2
=
R1 R2
R1 +R2
(“product over sum”)
• Non-ideal Voltage Source
• Summary
• If one resistor is a multiple of the other
Suppose R2 = kR1 , then
RP =
R1 R2
R1 +R2
=
kR12
(k+1)R1
Example: 1 kΩ || 99 kΩ =
E1.1 Analysis of Circuits (2016-8284)
=
k
k+1 R1
99
100
= (1 −
kΩ = 1 −
1
k+1 )R1
1
100
kΩ
Resistor Circuits: 2 – 10 / 13
Equivalent Resistance: Parallel Formulae
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
For parallel resistors GP = G1 + G2 + G3
or equivalently RP = R1 ||R2 ||R3 = 1/R +1/1R +1/R .
1
2
3
These formulae work for any number of resistors.
• For the special case of two parallel resistors
RP =
1
1/R +1/R
1
2
=
R1 R2
R1 +R2
(“product over sum”)
• Non-ideal Voltage Source
• Summary
• If one resistor is a multiple of the other
Suppose R2 = kR1 , then
RP =
R1 R2
R1 +R2
=
kR12
(k+1)R1
Example: 1 kΩ || 99 kΩ =
=
k
k+1 R1
99
100
= (1 −
kΩ = 1 −
1
k+1 )R1
1
100
kΩ
Important: The equivalent resistance of parallel resistors is always less
than any of them.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 10 / 13
Simplifying Resistor Networks
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Many resistor circuits can be
simplified by alternately combining
series and parallel resistors.
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 11 / 13
Simplifying Resistor Networks
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Many resistor circuits can be
simplified by alternately combining
series and parallel resistors.
Series: 2 k + 1 k = 3 k
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 11 / 13
Simplifying Resistor Networks
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Many resistor circuits can be
simplified by alternately combining
series and parallel resistors.
Series: 2 k + 1 k = 3 k
• Non-ideal Voltage Source
• Summary
Parallel: 3 k || 7 k = 2.1 k
Parallel: 2 k || 3 k = 1.2 k
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 11 / 13
Simplifying Resistor Networks
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Many resistor circuits can be
simplified by alternately combining
series and parallel resistors.
Series: 2 k + 1 k = 3 k
• Non-ideal Voltage Source
• Summary
Parallel: 3 k || 7 k = 2.1 k
Parallel: 2 k || 3 k = 1.2 k
Series: 2.1 k + 1.2 k = 3.3 k
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 11 / 13
Simplifying Resistor Networks
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
Many resistor circuits can be
simplified by alternately combining
series and parallel resistors.
Series: 2 k + 1 k = 3 k
• Non-ideal Voltage Source
• Summary
Parallel: 3 k || 7 k = 2.1 k
Parallel: 2 k || 3 k = 1.2 k
Series: 2.1 k + 1.2 k = 3.3 k
Sadly this method does not always
work: there are no series or parallel
resistors here.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 11 / 13
Non-ideal Voltage Source
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
An ideal battery has a characteristic that is
vertical: battery voltage does not vary with
current.
Normally a battery is supplying energy so V
and I have opposite signs, so I ≤ 0.
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 12 / 13
Non-ideal Voltage Source
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
An ideal battery has a characteristic that is
vertical: battery voltage does not vary with
current.
Normally a battery is supplying energy so V
and I have opposite signs, so I ≤ 0.
An real battery has a characteristic that has
a slight positive slope: battery voltage
decreases as the (negative) current
increases.
Model this by including a small resistor in
series. V = VB + IRB .
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 12 / 13
Non-ideal Voltage Source
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
• Non-ideal Voltage Source
• Summary
An ideal battery has a characteristic that is
vertical: battery voltage does not vary with
current.
Normally a battery is supplying energy so V
and I have opposite signs, so I ≤ 0.
An real battery has a characteristic that has
a slight positive slope: battery voltage
decreases as the (negative) current
increases.
Model this by including a small resistor in
series. V = VB + IRB .
The equivalent resistance for a battery
increases at low temperatures.
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 12 / 13
Summary
2: Resistor Circuits
• Kirchoff’s Voltage Law
• Kirchoff’s Current Law
• KCL Example
• Series and Parallel
• Dividers
• Equivalent Resistance:
Series
• Equivalent Resistance:
Parallel
• Equivalent Resistance:
Parallel Formulae
• Simplifying Resistor
Networks
•
•
•
•
•
Kichoff’s Voltage and Current Laws
Series and Parallel components
Voltage and Current Dividers
Simplifying Resistor Networks
Battery Internal Resistance
For further details see Hayt et al. Chapter 3.
• Non-ideal Voltage Source
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resistor Circuits: 2 – 13 / 13