Download Node-Voltage Analysis

Power and Energy
Consider the product of voltage and current.
VI
(J/C)(C/s) = J/s (1 J/s = 1 W: power!)
Now multiply by time.
VIt
(J/C)(C/s) s = J (energy!)
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Power and Energy
P = VI
Recall that V = IR.
So P = VI = IRI = I2 R
Also, I = V/R.
So P = VI = V(V/R) = V2/R
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Mathematical Conventions
If positive current leaves the positive voltage terminal,
the element is delivering or furnishing power (active).
If positive current enters the positive voltage terminal,
the element is absorbing or dissipating power (passive).
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Independent Sources
Independent Voltage Source
Independent Current Source
The voltage across the
source is independent of the
current drawn from the
source.
The current delivered by the
source is independent of the
voltage across the source.
Strong, fresh battery
Lightning
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Dependent Sources
Voltage-Controlled
Voltage Source
Voltage-Controlled
Current Source
Current-Controlled
Voltage Source
Current-Controlled
Current Source
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Resistors in Series - Review
6
Resistors in Parallel - Review
7
Resistors Combination - Practice
8
RECALL: Types of connections
Resistors connected in series…
When JUST two elements connect at a
SINGLE node, they are said to be in series
and series-connected elements carry the
SAME CURRENT (why? KCL)
RECALL: Types of connections
Resistors connected in parallel…
When two elements connect at a single node pair,
they are said to be in parallel and they have the
SAME VOLTAGE across their terminals ! (why? KVL)
Step by step process to find equivalent
resistance
Resistors Combination - Practice
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Voltage and Current Division
• The current and voltage in a resistive circuit
get divided between different resistors
based on their resistance values.
• How does that happen?
• KVL and KCL are universal rules and
should be satisfied in every circuit –
including (obviously) resistive circuits.
Voltage Division
From Ohm’s Law
I = VT/Req
Req = R1 + R2 + R3
v total
i
R1  R 2  R 3

v total
v1  i  R1  
 R1  R 2  R 3

R1
 v total 
 R1  R 2  R 3
Voltage applied to series circuit applies a fraction of the
voltage across each element

 R1




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Voltage
Section
2.3 Division
Voltage Divider (Series Circuits)
Voltage Division Principle: the fraction of the total voltage across a
single resistor in a series circuit is the ratio of the given resistor to the
total series resistance
So if V1 = VT R1 / (R1 + R2 + R3) , then what is V2 and V3
VOLTAGE DIVISION
•
Voltage division is a simple method/procedure that allows one to determine the voltage across a
resistor in a series combination, if the total voltage across all series-connected resistors is
known. The relationships are shown below:
R1
v1 = vT ×
R1 + R2 + R3 + R4 + R5 + R6 + R7
R2
v2 = vT ×
R1 + R2 + R3 + R4 + R5 + R6 + R7
R1
R2
R3
v1
v2
v3
vT
v4
v7
v6
v5
R7
R6
R5
R1
v2
R2
iT
v1
V1 = V T R1/ (R1 + R2)
……
….
V6 = V T R6/ (Req)
iT
vT
R4
Current Division
Total current in a parallel circuit is divided among resistances.
R eq
1
R1 R 2


1
1
R1  R 2

R1 R 2
v  i total R eq
R1 R 2
i
R1  R 2
 R1 R 2  1
v
1



i1 
 v
  i
R1
R1  R1  R 2  R1
What is i2?
 R2 

 i 
 R1  R 2 
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Current Division
•
Current Divider (Parallel Circuits)
Used to determine the current through one of several parallel resistors, if the total current
entering the parallel combination is known
1
R1
i1 = iT ×
1 1 1 1 1
+ + + +
R1 R2 R3 R4 R5
1
R2
i2 = iT ×
1 1 1 1 1
+ + + +
R1 R2 R3 R4 R5
iT
i1
R1
i2
i3
i4
i5
R2
R3
R4
R5
Once again…
WHEN YOU HAVE
RESISTORS
CONNECTED IN SERIES
YOU CAN USE VOLTAGE
DIVISION TO FIND INDIVIDUAL
VOLTAGE VALUES ACROSS THE
RESISTORS
WHEN YOU HAVE
RESISTORS
CONNECTED IN
PARALLEL
YOU CAN USE CURRENT
DIVISION TO FIND THE
INDIVIDUAL CURRENT VALUES
THROUGH THE RESISTORS
Let’s solve some examples…
And an example on current division…
How can we combine these techniques to solve circuits? Let’s
try to find io and vo !
First thing we can do is to assign variables to each
element – hopefully we’ll be able to solve for those !
First thing we can do is to assign variables to each
element – hopefully we’ll be able to solve for those !
Node-Voltage Analysis
In addition to analyzing circuits by combining series and parallel
resistors and applying the voltage and current – division principles,
there is the NODE-VOLTAGE Analysis.
Recall a node is a point at which two or more circuit
elements are joined together
Node-Voltage Analysis
CONVENTION: usually write
expressions for I, current,
leaving the node under
consideration and set sum to
zero
Step 1. Select or find reference node
Step 2. Label the node voltages - reference and other nodes
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Node-Voltage Analysis
CONVENTION: usually write
expressions for I, current,
leaving the node under
consideration and set sum to
zero
Step 3. Observe node voltage relationship to element voltage
(e.g. KVL says -v2 + vx + v3 = 0, then vx = v2 – v3)
Step 4. Write current equations at each of the nodes for ALL
currents leaving the node
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Node-Voltage Analysis
Step 5. Node-voltage
equations can be
written for each node
in the form of current
leaving one node and
entering another node
e.g. i4 through R4
leaving Node 2 and
entering ground gives
i4 = V2/R4
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Node-Voltage Analysis
Node Voltage Equation for Node 3
KCL iR1 + iR5 + iR3 = 0
where
iR1 = (v3 – v1)/R1
iR5 = v3/R5
iR3 =( v3 – v2)/R3
Then the Node Voltage Equation for Node 3 is
(v3 – v1)/R1 + v3/R5 + (v3 – v2)/R3 = 0
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Node-Voltage Analysis
What is the Node Voltage Equation for Node 1
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Node-Voltage Analysis
What is the Node Voltage Equation for Node 2
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Node-Voltage Analysis
RECALL: Node-voltage
equations can be written for
each node in the form of
current leaving one node and
entering another node
Another Example
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Node-Voltage Analysis
Another Example
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Section 2.4 Node Voltage Analysis
Node-Voltage Analysis
Another Example
CONVENTION: usually write expressions for I, current, leaving the node under
consideration and set sum to zero
Mesh-Current Analysis
Then there is MESH- CURRENT
Analysis.
Use KVL to write a vrises =vdrops equation for each mesh
mesh – a closed path that contains no other closed paths
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Mesh-Current Analysis
1. Label the meshes. # of independent KVL equations for
planar network are equal to # of open areas defined by the
network layout (2 OPEN areas in this circuit, hence 2 mesh
currents, i1, and i2)
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Section 2.5 Mesh Current Analysis
Mesh-Current Analysis
2. Follow KVL around each mesh
Mesh 1
-Va + i1R1 + V3 = 0
where v3 = R3(i1 - i2)
Thus
The same can be found for
Mesh 2
i1R1 + R3(i1 - i2) = Va
Section 2.5 Mesh Current Analysis
Mesh-Current Analysis
Mesh-Current Analysis
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Section 2.5 Mesh Current Analysis
Mesh-Current Analysis
Section 2.5 Mesh Current Analysis
Mesh-Current Analysis
Exercise. Find ia
Section
2.5 Mesh Current
Analysis
Mesh-Current
Analysis
Exercise. Find ib
Section 2.5 Mesh Current Analysis
Mesh-Current Analysis
Mesh with Controlled Sources
Combine MESH 1 and 2 – Supermesh
Voltage Controlled Current Source referenced as
i2 - i1 = vx/4 where vx = 2i2