FIGURE 8.1 Current source within the transistor equivalent circuit.

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EEE107
Circuit Theorems
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Current Sources
A current source is said to be the dual of a
voltage source.
Remember that a voltage source supplies a fixed
voltage and the current from it will vary
depending upon the load.
A current source will supply a fixed current to
the branch to which it is connected, while the
terminal voltage will vary depending upon the
load.
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Find VS and II for the given circuit
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Determine VS, I1 and I2
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IL
If the value of Rs is very small in comparison to
the load RL then it may be ignored, and the
voltage source in the shaded area may be
considered as an IDEAL voltage source
RS
E
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If the value of Rs is very large in comparison to
the load RL then it may be ignored, and the
current source in the shaded area may be
considered as an IDEAL current source
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Source Conversions
If the internal resistance is included with
either voltage or current sources then that
source may be converted to the other. Source
conversions are equivalent at their external
terminals.
.
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Convert the voltage source to a current source
Determine IL for both circuits
From these results you should see that the LOAD
RL does not know if the source is a voltage or
current
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Convert the current source to a voltage source
RS is the same as for the current source
Determine IL for both circuits
These results show that the LOAD RL does not know
if the source is a voltage or current.
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Current sources in parallel may be added together
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Reduce the parallel current sources to a single
current source
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Reduce to a single current source and find IL
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Invalid situation.
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Superposition Theorem
The current through, or voltage across, an
element in a linear network is equal to the
algebraic sum of the currents or voltages
produced independently by each source.
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When removing a current source it must be
replaced by an open circuit
When replacing a voltage source it must be
replaced by a short circuit
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When determining the power delivered to a
resistive element the total current through or
the total voltage across the element must be
used. NOT the simple sum of power levels
established by each source. (The currents due to
each source acting individually may not all be in
the same direction through the element, and have
to be added algebraically.)
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Using superposition find the current through the
6O resistor
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Replacing the voltage source by a short circuit
gives
The current through the resistor due to the
current source is obviously 0A due to the short
circuit
Replacing the current source by an open circuit
gives
The current through the resistor due the voltage
source gives 30V/6O 5A
Thus the total current through the resistor is
the sum from both sources i.e. 0A 5A 5A
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Using the superposition theorem determine the
current through the 4O resistor
22
Replace the 48V battery by a S/C
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Replace the 54V battery with a S/C
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Thevenin's Theorem
Any two-terminal dc network can be replaced by an
equivalent circuit consisting of a voltage source
and a series resistor
The Thevenin voltage ETh is the open circuit
voltage between the two terminals under
consideration
The Thevenin resistance RTh is the resistance
looking into the two terminals of the network
with all voltage sources acting as short
circuits and current sources acting as open
circuits.
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The network behind terminals ab may be replaced
by its Thevenin equivalent.
The voltage ETH is the open circuit voltage
between a and b
The resistance RTh is the resistance looking into
the network from a b with E1 short circuit.
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Produce Thevenin circuit for the shaded network
1 Find the Thevenin resistance RTH
looking into a b (remember to S/C E1)
?
2 Find VTH (same as V O/C at terminals a b )
Use voltage divider rule
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The equivalent Thevenin circuit is as shown
To the left of aa
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.
Find the the Thevenin equivalent circuit for the
shaded area
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i.e. the Thevenin circuit looking into the
terminals at a b
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Now applying the theorem find RTh (S/C the supply)
It should be seen that due to the S/C that R1 and
R3 are in parallel as are R2 and R4
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Applying KVL to loop shown and ETh polarities as
assumed
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Substituting the Thévenin equivalent circuit for
the network external to the resistor RL
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Norton's Theorem
Any two-terminal, linear dc network can be
replaced by an equivalent circuit consisting of a
current source and a parallel resistor
The Norton current IN is the short circuit
current between the two terminals under
consideration.
The resistance RN is the resistance looking into
the network with voltage sources acting as short
circuits and current sources acting as open
circuits.
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The network behind terminals ab may be replaced
by its equivalent Norton Circuit.
The current IN is the short circuit current
between a and b
The resistance RN is the resistance looking into
the network between terminals a and b with E1
short circuit.
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Produce Norton circuit for the shaded network
1 Find the Norton resistance RTH
looking into a b (remember to S/C E1)
Note that the Norton resistance is the same as
the Thevenin resistance
2 Find IN (same as S/C current between
terminals a b )
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Produce the equivalent Norton circuit for the
shaded network
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Produce the equivalent Norton circuit for the
shaded network
Convert to current source
RS R1
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Network redrawn with the voltage source converted
to a current source.
Add the two currents to obtain the Norton current
i.e. 8A 1.75A 6.25Awhich will flow in the
same direction as the 8A.
To find the Norton resistance look into a b
with the current sources O/C (use product over
sum) i.e. 2.4O.
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Note that conversion between Norton and Thevenin
equivalent circuits may be carried out by using
the source conversion methods introduced
earlier. Also remember that RN RTH
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The Maximum Power Transfer Theorem
A load will receive maximum power from a linear
dc network when its total resistance is exactly
equal to the Thevenin resistance of the network
as seen by the load
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Defining the conditions for maximum power to a
load using the Thévenin equivalent circuit.
Maximum power will be delivered to the load RL
when its resistance is equal to RTh
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Defining the conditions for maximum power to a
load using the Norton equivalent circuit.
Remember that RTh RN
Maximum power will be delivered to the load RL
when its resistance is equal to RN
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A tabulation of PL versus RL gives the table as
shown on the next slide.
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RL PL
0.1 4.35
0.2 8.51
0.5 19.94
1 36.00
2 59.50
3 75.00
4 85.21
5 91.84
6 96.00
7 98.44
8 99.65
9 100.00 Max.
10 99.72
11 99.00
12 97.96
13 96.69
14 95.27
15 93.75
16 92.16
17 90.53
18 88.89
19 87.24
20 85.61
Tabulated values of RL and PL
Plot of PL against RL
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Expanding the last expression with RL RTH gives
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Calculate the value of R for maximum transfer of
power and the magnitude of this power.
For max transfer of power R must RTh as shown
R
RTh
RTh (R1 // R2) R3 28 10 O
VTh Voc VR2 4V (using voltage divider
rule)