DE4401&APTE 5601
Topic 4
NETWORK ANALYSIS
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Introduction
• Review:
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Ohm’s Law
Resistors connected in series, in parallel or combination
Kirchhoff’s Voltage Law
Kirchhoff’s Current Law
Voltage and current dividers
• Today:
– Network analysis:
• Branch current method
• Loop current method (AK only)
• Node voltage method
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Revision
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Branch, Nodes and Loop
• KVL can be used to determine an unknown voltage in a
complex circuit, where all other voltages around a
particular "loop" are known.
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Kirchhoff's Current Law (KCL)
• The algebraic sum of all currents entering and
exiting a node must equal zero.
or
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Kirchhoff's Voltage Law
• "The algebraic sum of all voltages in a loop must
equal zero“
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Voltage divider
• The ratio of individual resistance to total resistance is the
same as the ratio of individual voltage drop to total
supply voltage in a voltage divider circuit. This is known
as the voltage divider formula
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Current divider
It is sometimes necessary to find the individual branch currents in
a parallel circuit if the resistances and total current are known, but
the voltage across the resistance bank is not known. When only
two branches are involved, the current in one branch will be some
fraction of the total current. This fraction is the quotient of the
second resistance divided by the sum of the resistances.
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Network Analysis
• Generally speaking, network analysis is any structured
technique used to mathematically analyze a circuit (a
“network” of interconnected components).
• Usually, a single equation will not be useful and we will
need a system of equations.
– The rule is: we need as many equations as we have unknown
currents.
• The techniques developed for DC circuits will be used for
AC circuits as well.
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Network Analysis Methods
• Branch current method
• Loop current method
• Node voltage method
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Branch current method
• The first and most straightforward network analysis
technique is called the Branch Current Method.
• In this method, we assume directions of currents in a
network, then write equations describing their
relationships to each other through Kirchhoff's and
Ohm's Laws.
• Once we have one equation for every unknown current,
we can solve the simultaneous equations and determine
all currents, and therefore all voltage drops in the
network.
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Branch current method steps
Steps to follow for the “Branch Current” method of analysis:
(1) Choose a node and assume directions of currents.
(2) Write a KCL equation relating currents at the node.
(3) Label resistor voltage drop polarities based on assumed
currents.
(4) Write KVL equations for each loop of the circuit, substituting
the product IR for E in each resistor term of the equations.
(5) Solve for unknown branch currents (simultaneous equations).
– If any solution is negative, then the assumed direction of current for that
solution actually flows the opposite way!
(6) Solve for voltage drops across all resistors (E=IR).
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Example 1
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(1) Choose a node and assume directions of currents.
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(2) Write a KCL equation relating currents at the node.
If any for current solution is negative, then the assumed
direction of current for that solution is wrong!
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(3) Label resistor voltage drop polarities based
on assumed currents.
•
The following is not the case in our example here, but just so you know: It is
OK if the polarity of a resistor's voltage drop doesn't match with the polarity of
the nearest battery, so long as the resistor voltage polarity is correctly based
on the assumed direction of current through it. In some cases we may
discover that current will be forced backwards through a battery, causing this
very effect.
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(4) Write KVL equations for each loop of the circuit, substituting
the product IR for E in each resistor term of the equations.
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(6) Solve for voltage drops across all resistors (V=IR).
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(5) Solve for unknown branch currents
(simultaneous equations).
• If any solution is negative, then the assumed direction of
current for that solution is wrong!
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Loop (Mesh) Current Method
• The Loop Current Method, also known as the Mesh
Current Method, is quite similar to the Branch Current
method in that it uses simultaneous equations,
Kirchhoff's Voltage Law, and Ohm's Law to determine
unknown currents in a network.
•
It differs from the Branch Current method in that it does
not use Kirchhoff's Current Law.
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Example
• We will analyze the same circuit:
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Step one: identify “loops”
• Identify “loops” within the circuit, encompassing all
components.
– In our example circuit:
• First loop is formed by E1, R1, and R2
• Second loop is formed by E2, R2, and R3
– The choice of each current's direction is arbitrary, but the
equations are easier to solve if the currents are going the same
direction through intersecting components (here, R2 ).
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Step two: label all voltage drop polarities
• The next step is to label all voltage drop polarities across
resistors according to the assumed directions of the loop
currents.
– The battery polarities, of course, are dictated by their symbol
orientations in the diagram, and may or may not “agree” with the
resistor polarities (assumed current directions)
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Step three: write equation for each loop
• Write KVL equations for each loop, substituting the
product IR for E in each resistor term of the equation.
– Where two loop currents intersect through a component, express
the current as the algebraic sum of those two loop currents (i.e.
I1 + I2 if the currents go in the same direction through that
component or I 1 - I2 if not.)
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Step four: solve the equations
• In this example we have two equations with two
unknowns
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Step five: calculating branch currents
• The result we have obtained is for the loop currents, not
branch currents. So, in this step, we must go back to our
diagram to see how they fit together to give currents
through all components (branch currents).
– Remember: negative current value means that the current flows
in the direction opposite from the assumed. (see figure below)
This change of current direction
from what was first assumed
will alter the polarity of the
voltage drops across R2 and R3
due to current I2.
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Step five (cont.)
• We can say that the current
through R1 is 5 amps. Also,
we can safely say that the
current through R3 is 1 amp,
with a voltage drop of 1 volt
(E=IR), positive on the left
and negative on the right.
• To determine the actual
current through R2, we must
see how mesh currents I1
and I2 interact
• in this case they're in opposition
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We algebraically add them
(minding the sign +/-) :
Since I1 is going “down” at
5 amps, and I2 is going
“up” at 1 amp, the real
current through R2 must
be the difference, 4 amps,
going “down”
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Step six: calculate voltages
• Using Ohm’s law, calculate voltages on all resistors.
• V = IR
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Loop current method: advantages
• The primary advantage of Loop Current analysis is that it
generally allows for the solution of a large network with
fewer unknown values and fewer simultaneous
equations than Branch Current method.
– Our example problem took three equations to solve the Branch
Current method and only two equations using the Loop Current
method.
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Revise Steps for Loop Current Method
(1) Draw currents in loops of circuit, to account for all components.
(2) Label resistor voltage drop polarities based on assumed directions
of loop currents.
(3) Write KVL equations for each loop, substituting the product IR for E
in each resistor term of the equation. Where two mesh currents
intersect through a component, express the current as the algebraic
sum of those two mesh currents (i.e. I1 + I2 if the currents go in the
same direction through that component or I 1 - I2 if not.)
(4) Solve for unknown loop currents (simultaneous equations). If any
solution is negative, the assumed current direction is wrong!
(5) Algebraically add loop currents to find current in components which
share multiple loop currents.
(6) Solve for voltage drops across all resistors (V=IR).
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Task 1
Analyse the following circuit using:
– Loop current method
– Branch current method
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Task 2
• Analyse the following circuit using:
– Loop current method
– Branch current method
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Node Voltage Method
• Node Voltage Method is one of the major techniques in
circuit analysis. It reduces the number of equations you
have to deal with.
• Node Voltage Method for solving a circuit uses node
voltage drops to specify the currents at a node.
Then, node equations of currents are written to satisfy
Kirchhoff's current law. By solving the node equations,
we can calculate the unknown node voltages.
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Terminology
• A node is a common connection for two or more
components in a circuit.
• A principal node has three or more connections (some
call it ‘junction’)
• One of the principal nodes is chosen as a reference
node and this node is connected to the ground (defined
as 0 volts).
– Because a reference node has 0 volts, you can simplify analysis
by choosing a node where a large number of devices are
connected as your reference node.
• A node voltage is the voltage of a given node with
respect to the reference node (i.e.to the ground).
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Node, principal node and reference node
• A node voltage is the voltage of a given node with
respect to the reference node
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Notation used in this class
• To each node in a circuit, a letter or number is assigned.
Example: A, B, G, and N are nodes, and G and N are
principal nodes.
• Select node G connected to ground as the reference
node.
– Then VAG is the voltage between nodes A and G, VBG is the
voltage between nodes B and G, and VNG is the voltage between
nodes N and G.
– Since the node voltage is always determined with respect to a
specified reference node, the notations VA for VAG, VB for VBG,
and VN for VNG are used.
•
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Steps for Node Voltage Method
1. Mark nodes (use letters i. e. A, B, C, G). Select a reference (ground) node.
2. Assume the direction of currents. Mark the voltage polarity across each
resistor, consistent with assumed direction of current.
3. Formulate a Kirchhoff’s Current Law (KCL) equation for each non-reference
principal node.
4. Express all branch currents in terms of voltage drops on components (e.g.
resistor in the branch) by using relationships such as Ohm’s law ( I = V/R)
and the rule for the voltage in parallel branches (“ all parallel branches have
the same voltage”). Resulting equations are now expressed in terms of
unknown principal voltages.
5. With the equations from step 4, go back to the KLC equations from step 3.
Simplify the equations to put them in standard form.
6. Solve the system of equations to find principal node voltages.
7. Next, find all voltage drops and currents in the branches.
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Example: we analyse the same circuit
• Step one: Mark nodes and select a reference (ground)
node.
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Step two: current directions, polarities
• Assume the direction of currents. Mark the voltage
polarity across each resistor, consistent with assumed
direction of current.
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Step three: KCL
• Formulate a Kirchhoff’s Current Law (KCL) equation for
each non-reference principal node.
– In this case, only one principal node, N.
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Step four: Voltage equations
• Express all branch currents in terms of voltage drops on
components (e.g. resistor in the branch) by using
relationships such as Ohm’s law ( I = V/R) and the rule for
the voltage in parallel branches (“ all parallel branches
have the same voltage”). Resulting equations are now
expressed in terms of unknown principal voltages.
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Step five and step six:
• Step five: With the equations from step 5, go back to the
KLC equations from step 4. Simplify the equations to put
them in standard form.
– In this example, we have a single equation only.
• Step six: Solve the system of equations to find principal node
voltages. (again: here it’s only one equation, not a system)
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Step seven
• Next, find all voltage drops and currents in the branches.
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Task 3
•
Analyse the circuit from Task 1 using Node Voltage
Method
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Task 4
• Analyse the circuit from Task 2 using Node Voltage
Method
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Literature for Node Voltage Method
• Chapter 5 in book: ‘Circuit Analysis For Dummies’
Santiago, John; available online from Unitec Library
• Page 105 Schaum’s Basic Electricity ; available from
Moodle in pdf format for download or a book from Unitec
Library
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Maths revision
• If algebra is a problem for you, please look up “system of
equations” in Khan academy.
– In the class, we will use Elimination method.
– Please practice Substitution and Matrix methods as well.
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