4. Complex DC Circuits

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5 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Complex Circuits

Presentation 2003 R. McDermott

Complex DC Circuits


A power supply with an internal resistance (lowers
terminal voltage):



Adds an additional resistor in series with the power
supply


Reduces the voltage at the terminals


Terminal voltage will depend on the external circuit



Example: A 12
-
volt battery has an internal resistance of
5.0 ohms. What is the terminal voltage of the battery when
it is connected in series to a 50 ohm resistor? A 25 ohm
resistor?

Complex Circuit #1


Example: A 12
-
volt battery has an internal resistance
of 5.0 ohms. What is the terminal voltage of the
battery when it is connected in series to a 50 ohm
resistor?






Rc
=

5.0


+

50


=

55








Ic
=

(12 v)
/
(55

)
=

.22 a


Internal drop


=

5.0


X .22 a
=

1.1 v


Terminal voltage


=

12 v


1.1 v
=

10.9 v

Complex Circuit #2


Example: A 12
-
volt battery has an internal resistance
of 5.0 ohms. What is the terminal voltage of the
battery when it is connected in series to a 25 ohm
resistor?






Rc
=

5.0


+

25


=

30








Ic
=

(12 v)
/
(30

)
=

.40 a


Internal drop


=

5.0


X .40 a
=

2.0 v


Terminal voltage


=

12 v


2.0 v
=

10 v

Complex Circuit #3


Multiple batteries in a single loop:


Voltages add if the batteries are positive to negative


Voltages subtract if the batteries are positive to
positive.

Circuit #3


The batteries are connected positive to
positive, so their voltages are subtracted
(they act against one another).






Vnet = 18v


12v = 6v



The current in the circuit is then:





Ic = Vc/Rc = 6v/6


=

1a

Complex Circuit #4



Compound series and parallel circuit:


Both sets of rules apply on various parts of the
circuit


Ratios become very important as time savers



Find the current and voltage drops for each resistor.

Circuit #4



The parallel combination has a total resistance of:

1
/
Rp
=

1
/
12
+

1
/
4
=

1
/
3

Rp
=

3 ohms



The parallel combination is in series with R
1

so:

Rc
=

6


+

3


=

9




The circuit current can now be found:

Ic
=

Vc
/
Rc
=

18v
/
9


=

2a

Circuit #4 (cont)


Resistor 1 gets the full circuit current, so its voltage drop
is:

V
1

=

I
1
R
1

=

(2a)(6

)
=

12 v



The parallel combination gets the remaining voltage:

V
P

=

Vc
-

V
1

=

18v

-

12v
=

6v



Each resistor in parallel gets the full 6 volts, so currents
are:

I
2

=

V
P
/
R
2

=

6v
/
4


=

1.5a


I
3

=

V
P
/
R
3

=

6v
/
12


=

0.5a

Complex Circuit #5



Multiple loops with multiple batteries:



Kirchoff’s junction current rule: I
IN

= I
OUT
.



Kirchoff’s loop voltage rule:

V
LOOP

= 0

Circuit #5


Kirchoff’s junction current rule: I
IN

=

I
OUT



Let’s pick junction A and assume all currents are
downward. Let the far left wire be I
1
, the middle
I
2
, and the far right be I
3
. All current is exiting the
junction, so
zero

current enters (our assumption):







I
1

+

I
2

+

I
3

=

0



We now have one equation which will allow us to
solve for one unknown

Circuit #5 (cont.)


Kirchoff’s loop voltage rule:

V
LOOP

=

0



We need two more equations, so we choose two of
the three loops in our circuit (we have one on the
left, one on the right, and the large outside loop)



Let’s take the left and right loops and arbitrarily
pick a clockwise direction to follow



If the positive side of a battery pushes current with
our loop, we assign the voltage a positive value. If
it works against our loop, we assign a negative
value

Sample #5


In the left loop, the 12v battery works with us (
+
)
and the 6v works against us (

)



In the right loop, both batteries work with us (
+
)



Current flowing through a resistor in the same
direction as we are looping produces a negative
voltage. Current flowing opposite to our loop
produces a positive voltage



I
1

flows through R
1

opposite to our loop, so I
1
R
1

is
positive. I
2

flows through R
2

in the same direction
as our loop, so I
2
R
2

is negative

#5 (cont.)


In the left loop, then:





12
+

I
1
R
1



I
2
R
2



6
=

0

and,

12
+

10I
1



5 I
2



6
=

0



This last is our second equation. We now have:


I
1

+

I
2

+

I
3

=

0

12
+

10I
1



5 I
2



6
=

0

#5


For the right loop:






6
+

I
2
R
2



I
3
R
3

+

22
=

0
and,

6
+

5I
2



20I
3

+

22
=

0




This is our third equation. We now have:


I
1

+

I
2

+

I
3

=

0

12
+

10I
1



5 I
2



6
=

0

6
+

5I
2



20I
3

+

22
=

0

This is it!


Solving:






I
1

=



0.657a

I
2

=



0.114a

I
3

=

0.771a



We were wrong about the direction of I
1

and I
2
(both up), but correct for I
3

(down)