Download Jiayu Chen Ph.D. Lecture 02 Network theorems

CA2627 Building Science
Lecture 02 Network theorems
Instructor: Jiayu Chen Ph.D.
Learning Objectives
1. Compute the solution of circuits containing linear resistors and independent sources
using node analysis.
2. Compute the solution of circuits containing linear resistors and independent sources
using mesh analysis.
3. Apply the principle of superposition to linear circuits containing independent
sources.
4. Compute Thévenin and Norton equivalent circuits for networks containing linear
resistors and independent sources.
5. Use equivalent circuits ideas to compute the maximum power transfer between a
source and a load.
© Jiayu Chen, Ph.D.
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Overview
A direct application of Kirchhoff's current and voltage laws (KCL and KVL) can solve
many circuit problems. We now study other techniques that can simplify analysis of
some circuits. These techniques are based on Kirchhoff’s laws.
We can therefore find network solutions using any of these techniques:
• Kirchhoff’s current law (KCL)
• Kirchhoff’s voltage law (KVL)
• Mesh analysis
• Nodal analysis
• Superposition theorem
• Thévenin's theorem
• Norton's theorem
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Mesh Analysis
Mesh analysis of circuits uses Kirchhoff's voltage law (KVL) around a closed path. A
closed path or a loop is drawn by starting at a node and tracing a path such that we
return to the original node without passing an intermediate node more than once.
A mesh is a special case of a loop. A mesh is a loop that does not contain any other
loops within it. Mesh current analysis is applicable only to planar networks.
A planar circuit is one that can be drawn on a plane, without crossovers.
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Mesh Analysis
An example of a nonplanar circuit is shown, where the crossover is identified and
cannot be removed by redrawing the circuit. For planar networks, the meshes in the
network look like "windows." Redrawing a planar circuit can change the meshes.
There are four meshes in the circuit shown in the figure. They are identified as Mi.
Mesh 2 contains the elements R3, R4, and R5. Note that the resistor R3 is common to
both mesh I and mesh 2.
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Mesh Analysis
We define a mesh current as the current that flows through the elements constituting
the mesh. A circuit having two meshes with the mesh currents labeled as i1 and i2. Use
clockwise convention.
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Mesh Analysis
mesh equations
One of the standard methods for analyzing an electric circuit is to write and solve a set
of simultaneous equations called the mesh equations. The unknown variables in the
mesh equations are the mesh currents of the circuit. We determine the values of the
mesh currents by solving the mesh equations.
To write a set of mesh equations, we do two things:
1. Express element voltages as functions of the mesh currents.
2. Apply Kirchhoff’s voltage law (KVL) to each of the meshes of the circuit.
3. Solve the equations
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Mesh Analysis
A circuit element
(in two meshes)
Element can be
anything:
a resistor, a current
source, a dependent
voltage source etc.
A current resource
A resistor
−3 = 𝑖1 − 𝑖2
𝑖 = 𝑖1 − 𝑖2
𝑣 = 𝑅(𝑖1 − 𝑖2 )
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Mesh Analysis
Step 1
Express element voltages as functions of the mesh currents.
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Mesh Analysis
Step 2
Apply Kirchhoff’s voltage law (KVL) to each of the meshes of
the circuit.
Mesh 1:
Mesh 2:
−𝑣𝑠 + 𝑅1 𝑖1 + 𝑅3 𝑖1 − 𝑖2 = 0
−𝑅3 𝑖1 − 𝑖2 + 𝑅2 𝑖2 = 0
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Mesh Analysis
Step 3
Solve the equations
If 𝑅1 = 𝑅2 = 𝑅3 = 1Ω, we will have
2𝑖1 − 𝑖2 = 𝑣𝑠
−𝑖1 + 2𝑖2 = 0
Then we will know
𝑖1 =
2𝑣𝑠
and 𝑖2
3
=
𝑣𝑠
3
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Mesh Analysis
If we have N meshes and write N mesh equations in terms of N mesh currents, we
can obtain N independent mesh equations. This set of N equations is independent
and thus guarantees a solution for the N mesh currents. A circuit that contains only
independent voltage sources and resistors results in a specific format of equations
that can readily be obtained.
For example, consider a circuit with three meshes. Assign the clockwise direction to
all of the mesh currents. Using KVL, we obtain the three mesh equations
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Mesh Analysis
Mesh 1: −𝑣𝑠 + 𝑅1 𝑖1 + 𝑅4 𝑖1 − 𝑖2 = 0
Mesh 2: 𝑅2 𝑖2 + 𝑅5 𝑖2 − 𝑖3 + 𝑅4 𝑖2 − 𝑖1 = 0
Mesh 3: 𝑅5 𝑖3 − 𝑖2 + 𝑅3 𝑖3 + 𝑣𝑔 = 0
𝑄
In general,
−
𝑃
𝑅𝑘 𝑖𝑞 +
𝑞=1
𝑁
𝑅𝑗 𝑖𝑛 = −
𝑗=1
𝑣𝑠𝑛
𝑛=1
𝑹𝒊 = 𝒗𝒔
𝑖1
𝑖2
𝒊= …
…
𝑖𝑁
𝑣𝑠1
𝑣𝑠2
𝒗𝒔 = …
…
𝑣𝑠𝑁
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Node Analysis
Identify nodes
Analyzing a connected circuit containing n nodes will require n-1 KCL equations.
One way to obtain these equations is to apply KCL at each node of the circuit except
for one. The node at which KCL is not applied is called the reference node.
Any node of the circuit can be selected to be the reference node. We will often choose
the node at the bottom of the circuit to be the reference node. (When the circuit
contains a grounded power supply, the ground node of the power supply is usually
selected as the reference node.)
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Node Analysis
To write a set of node equations, we do two things:
1.Express element currents as functions of the node voltages.
2. Apply Kirchhoff's current law (KCL) at each of the nodes of the circuit, except for
the reference node.
𝑉𝑠 = 𝑣1 − 𝑣2
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Node Analysis
Example:
The node equations represent in the circuit is
𝑣𝑎 𝑣𝑎 − 𝑣𝑏
𝑖𝑠 =
+
𝑅2
𝑅1
Similarly, KCL equation at node b is
𝑣𝑎 − 𝑣𝑏 𝑣𝑏
=
𝑅1
𝑅3
If we assume 𝑅1 = 1Ω, 𝑅2 = 𝑅3 = 0.5Ω, 𝑎𝑛𝑑 𝑖𝑠 = 4 𝐴
Then we solve the equations as
𝑣𝑎 − 𝑣𝑏 𝑣𝑎
4=
+
1
0.5
𝑣𝑎 − 𝑣𝑏
𝑣𝑏
=
1
0.5
3
1
And 𝑣𝑎 = 2 V and 𝑣𝑏 = 2 V
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Node Analysis
Example:
The node equations of node b represent in the circuit is
𝑣𝑎 𝑣𝑎 − 𝑣𝑏
4+
+
=0
10
5
Using 𝑣𝑏 = 5 V gives 𝑣𝑎 = −10 V
Apply KCL at node b then
𝑣𝑏
𝑣𝑎 − 𝑣𝑏
−4−
=0
𝑅
5
Then we can find
𝑅 = 5Ω
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Node Analysis
Example:
Apply KCL at node a we can have
−
𝑣𝑎 − 𝑣𝑐
𝑣𝑎 − 𝑣𝑐
𝑣𝑎 − 𝑣𝑏
+ 𝑖1 −
+ 𝑖2 −
=0
𝑅1
𝑅2
𝑅5
Then we can have
1
1
1
1
1
1
+
+
𝑣𝑎 −
𝑣𝑏 −
+
𝑣 = 𝑖1 + 𝑖2
𝑅1 𝑅2 𝑅5
𝑅5
𝑅1 𝑅2 𝑐
Apply to the node b also arrange the equation
1
𝑅3
+
1
𝑅4
+
1
𝑅5
𝑣𝑏 −
1
𝑅5
𝑣𝑎 −
1
𝑅3
𝑣𝑐 = 𝑖3 − 𝑖2
Finally we will have the node equation at node c:
1
1
1
1
1
1
1
−
+
𝑣𝑎 −
𝑣𝑏 +
+
+
+
𝑣 = −𝑖1
𝑅1 𝑅2
𝑅3
𝑅1 𝑅2 𝑅3 𝑅6 𝑐
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Node Analysis
Comparison of node voltage analysis with mesh current analysis
The analysis of a complex circuit can usually be accomplished by either the node
voltage or the mesh current method. The advantage of using these methods is the
systematic procedures provided for obtaining the simultaneous equations.
In some cases one method is clearly preferred over another:
Use node analysis or Kirchhoff’s current law (KCL) when:
• the circuit contains only current sources
• the circuit has fewer nodes than meshes
• you need to know several voltages but fewer currents
• the circuit is non-planar
Use mesh analysis when:
• the circuit contains only voltage sources,
• has fewer meshes than nodes
• you need to know several currents but fewer voltages
• not for nonplanar circuits
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Node Analysis
Comparison of node voltage analysis with mesh current analysis
This circuit can be analyzed using mesh equations or using node equations.
To decide which will be easier, we first count the nodes and meshes.
This circuit has five nodes. Selecting a reference node and then applying KCL at the
other four nodes will produce a set of four node equations.
The circuit has three meshes. Applying KVL to these three meshes will produce a set of
three mesh equations.
Hence, analyzing this circuit using mesh equations instead of node equations will
produce a smaller set of equations. Further, notice that two of the three mesh currents
can be determined directly from the current source currents. This makes the mesh
equations easier to solve.
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Superposition theorem
The Superposition theorem states that in any network containing more than one
source, the current in, and the voltage across, any branch can be found by considering
each source separately and adding their effects; omitted sources of voltages are replaced
by resistances equal to their internal resistances.
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Superposition theorem
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Superposition theorem
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Superposition theorem
The principle of superposition requires that we deactivate (disable) all but one
independent source and find the response due to that source. We then repeat the process
by disabling all but a second source and determining the response. We find the response
to each source acting alone, and then the total response is the sum of all these
responses.
Deactive a voltage source:
an independent voltage source appears as a short circuit with zero voltage across it.
Deactivate a current source:
if an independent current source is set to zero, no current flows through it and it appears
as an open circuit.
Important: dependent source are never deactivated and must remain active (unaltered)
during the superposition process.
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Superposition theorem
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Superposition theorem
Example:
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Source Transformation
Two sources are equivalent if the voltages or currents of an external circuit do not
change when connect to either of them.
The process of transforming a voltage source into an equivalent current source, or a
current source into an equivalent voltage source, is called a source transformation.
Ideal Source
Ideal voltage source has 0 Ω output impedance so it can maintain the output voltage
regardless of load. Ideal current source has infinite output impedance so it can drive
the same current regardless of load and regardless of the output voltage.
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Source Transformation
Nonideal Voltage Sources
Nonideal Current Sources
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Source Transformation
Source Transformation
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Source Transformation
Example: determine the values of i_p and R_p and then v_a and v_b
Use the source transformation,
12
𝑖𝑝 =
= 2 𝑚𝐴 𝑎𝑛𝑑 𝑅𝑝 = 6 𝑘Ω
6000
The we can calculate
2000
12 = 3 𝑉
2000 + 6000
The voltage across the parallel resister is
𝑣𝑎 =
𝑣𝑏 =
200𝑅𝑝
𝑖 =3𝑉
2000 + 𝑅𝑝 𝑝
© Jiayu Chen, Ph.D.
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Source Transformation
R=10Ω
is = 1.2A
R=10Ω
is = -1.2A
R=8Ω
vs=24V
R=8Ω
vs= - 24V
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Thévenin's theorem
Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter
how complex, to an equivalent circuit with just a single voltage source and series
resistance connected to a load.
If we're dealing with passive components (such as resistors, and later, inductors and
capacitors), the circuit is linear. However, there are some components (especially
certain gas-discharge and semiconductor components) which are nonlinear: that is,
their opposition to current changes with voltage and/or current. As such, we would
call circuits containing these types of components, nonlinear circuits.
*Note:
Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one
particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit
is necessary with each trial value of load resistance, to determine voltage across it and current through it.
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Thévenin's theorem
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Thévenin's theorem
Example: Determine the Thevenin’s equivalent circuit for the following circuit
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Thévenin's theorem
Example: Determine the Thevenin’s equivalent circuit for the following circuit
𝑅 = 2; 𝑖 = 4.8
Type equation here.
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Norton's Theorem
Norton's Theorem states that it is possible to simplify any linear circuit, no matter how
complex, to an equivalent circuit with just a single current source and parallel
resistance connected to a load.
Contrasting our original example circuit against the Norton equivalent: it looks
something like this:
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Norton's Theorem
The Norton equivalent is simply the source transformation of the Thévenin equivalent.
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Norton's Theorem
Example: Determine the Norton’s equivalent circuit for the following circuit
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Delta and Star Transformation
R AB = R A + R B
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Delta and Star Transformation
Star - Delta Transformation
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Delta and Star Transformation
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Maximum power transfer
What load will extract the maximum power from a supply source?
The maximum power is transferred from the source to the load when the load resistance
is equal to the Thevenin’s equivalent resistance (or the Norton equivalent resistance).
This is called resistance matching.
the maximum power delivered to a load by a source is attained when
𝑹𝒕 = 𝑹𝑳
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Maximum power transfer
Power of the Load:
𝑝 = 𝑖 2 𝑅𝐿
Current is:
𝑣𝑠
𝑅𝐿 + 𝑅𝑡
Then we will find the power is
2
𝑣𝑠
𝑝=
𝑅𝐿
𝑅𝐿 + 𝑅𝑡
To find the maximum p
𝑑𝑝
𝑅𝑡 + 𝑅𝐿 2 − 2 𝑅𝑡 + 𝑅𝐿 𝑅𝐿
2
= 𝑣𝑠
=0
𝑑𝑅𝐿
𝑅𝐿 + 𝑅𝑡 4
𝑖=
𝒑𝒎𝒂𝒙
𝒗𝒔
=
𝑹𝒕 + 𝑹𝒕
© Jiayu Chen, Ph.D.
𝟐
𝒗𝟐𝒔
𝑹𝒕 =
𝟒𝑹𝒕
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Maximum power transfer
Example: Find the maximum power delivered to the load resistor and load could result in
maximum power
𝑅𝑡 =
𝑝𝑚𝑎𝑥
30 × 150
= 25Ω
30 + 150
2
𝑣𝑜𝑐
1502
=
=
= 225 𝑊
4𝑅𝐿 4 × 25
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Summary
• A source transformation allows us to replace a voltage source and series
resistor by a current source and parallel resistor. Doing so does not change
the element current or voltage of any other element of the circuit.
• Superposition theorems says that the response of a linear circuit to several
inputs working together is equal to the sum of the responses to each of the
inputs working separately.
• Thevenin’s theorem allows us to replace part of a circuit by a voltage
source and series resistor. Doing so does not change the element current or
voltage of any element in the rest of the circuit.
• Norton’s theorem allows us to replace part of a circuit by a current source
and parallel resistor. Doing so does not change the element current or
voltage of any element in the rest of the circuit.
• The maximum power transfer theorem describes the condition under
which one circuit transfers as much power as possible to another circuit.
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Thank You!
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