II

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II3 DC Circuits I

• Theory Examples

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Main Topics

• Resistors in Series and Parallel.
• Resistor Networks.
• General Topology of Circuits.
• Kirchhoffs Laws Physical Meaning.
• The Use of the Kirchhoffs Laws.
• The superposition principle.
• The Use of the Loop Currents Method.

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Resistors in Series

• When resistors are connected in series, they have
  the same current passing through them. At the
  same time the total voltage on them must be a sum
  of individual voltages. So such a connection can
  be replaced by a resistor whose resistance is the
  sum of individual resistances.
• R R1 R2

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Resistors in Parallel

• When resistors are connected in parallel, there
  is the same voltage on each of them. At the same
  time the total current must be a sum of
  individual currents. So such a connection can be
  replaced by a resistor whose reciprocal
  resistance is the sum of individual reciprocal
  resistances.
• 1/R 1/R1 1/R2

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General Resistor Networks

• First we substitute resistors in the serial
  branches and then in the parallel.
• A triangle circuit we replace by a star using
  cyclic permutations of
• r? rbrc/(rarb rbrc rcra)
• This follows from cyclic permutations of
• r? r? rc(ra rb)/(rarb rbrc rcra)

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General Topology of Circuits

• Circuits are constructed of
• Branches wires with power sources and
  resistors.
• Junctions points in which at least three
  branches are connected.
• Loops all different possible closed trips
  through various branches and joints which dont
  cross.

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Solving Circuits

• To solve a circuit completely means to find
  currents in all branches. Sometimes it is
  sufficient to deal only with some of them.
• When solving circuits it is important to find
  independent loops. There are geometrical methods
  for that and usually several possibilities. In
  practice, we have to obtain enough linearly
  independent equations.

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The Kirchhoffs Laws I

• The physical background for solving circuits are
  the Kirchhoffs laws. They express fundamental
  properties the conservation of charge and
  potential energy.
• In the simplest form they are valid in the
  approximation of stationary fields and currents
  but can be generalized to some time dependent
  fields and currents as well.

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The Kirchhoffs Laws II

• The Kirchhoffs first law or junction rule states
  that at any junction point, the sum of all
  currents entering the junction must be equal the
  sum of all currents leaving the junction.
• It is a special case of conservation of charge
  which is more generally described by the equation
  of continuity of the charge.

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The Kirchhoffs Laws III

• The Kirchhoffs second law or loop rule states
  that, the sum of all the changes in potential
  around any closed path ( loop) of a circuit must
  be zero.
• It is based on the conservation of potential
  energy or more generally on the conservativity of
  the electric field.

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The Use of Kirchhoffs Laws I

• We have to build as many independent equations as
  is the number of branches
• First we name all currents and choose their
  direction. If we make a mistake they will be
  negative in the end.
• We write equations for all but one junctions. The
  last equation would be lin. dependent.
• We write equation for every independent loop.

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Example I-1

• Our circuit has 3 branches, 2 junctions and 3
  loops of which two are independent.
• Since there are sources in two branches we cant
  use simple rules for serial or parallel
  connections of resistors.

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Example I-2

• We name the currents and choose their directions.
  Here, let all leave the junction a, so at least
  one must be negative.
• It is convenient to mark polarities on resistors
  according to the supposed direction of currents.
• The equation for the junction a
• I1 I2 I3 0.

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Example I-3

• Equation for the junction b would be the same so
  we must proceed to loops.
• We e.g. start in the point a go through the
  branch 1 and return through the branch 3
• -V1 R1I1 R3I3 0
• Similarly from a via 2 and back via 3
• V2 R2I2 R3I3 0

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Example I-4

• The rule of the thumb is to put all terms on
  one side of the equation and write the sign
  according to the polarity which we approach first
  during the path.
• Then we can get -I3 I1 I2 from the first
  equation and substitute it the the other two
• V1 (R1 R3)I1 R3I2
• -V2 R3I1 (R2 R3)I2

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Example I-5

• Numerically we have
• 25I1 20I2 10
• 20I1 30I2 -6
• We can proceed several ways and finally get I1
  1.2 A, I2 -1 A, I3 -0.2 A
• We see that the current I2 and I3 run the other
  way the we had originally estimated.

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The Use of Kirchhoffs Laws II

• The Kirchhoffs laws are not really useful for
  practical purposes because they require to build
  and solve as many independent equations as is the
  number of branches. But it can be shown the it is
  sufficient to build and solve just as many
  equations as is the number of independent loops,
  which is always less.

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Example II-1

• Even in our simple example we had to solve a
  system of three equations which is the limit
  which can be relatively easily solved by hand.
• We can show that even for a little more
  complicated circuit the number of equations would
  be enormous and next to impossible to solve.

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Example II-2

• Now we have 6 branches, 4 junctions and many
  loops out of which 3 are independent.
• Kirhoffs laws give us 3 independent equations
  for junctions and 3 for loops.
• We have a system of 6 equations for 6 currents,
  which is in principle sufficient but it would be
  very difficult to solve it.

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The Principle of Superposition

• The superposition principle can be applied in
  such a way that all sources act independently.
• We can shortcut all sources and leave only the
  j-th on and find currents Iij in every branch. We
  repeat this for all sources. Then Ii Ii1 Ii2
  Ii3

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Example I-6

• Let us return to our first example.
• Lets leave the first source on and shorten the
  second one.
• We obtain a simple pattern of resistors which we
  easily solve
• I11 6/7 A I21 -4/7 A I31 -2/7 A

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Example I-7

• We repeat this for the second source
• I12 12/35 A I22 -3/7 A I32 3/35 A
• Totally we get
• I1 1.2 A I2 -1 A I32 -0.2 A
• Which is the same as the previous result.
• Using superposition is handy if we want to see
  what happens e.g. if we double the voltage of the
  first source.

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The Loop Currents Method

• There are several more advanced methods which use
  only the minimum number of equations necessary to
  solve the circuits.
• Probably the most elegant and easiest to
  understand and use is the method of loop
  currents.
• It is based on the idea that only currents in the
  independent loops exist and the other currents
  are their superposition.

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Example I-8

• In our first example two independent loop
  currents exist e.g. I? in the loop a(1)(3) and I?
  in the loop a(2)(3).
• All branch currents written as their
  superposition
• I1 I?
• I2 I?
• I3 -I ? - I?

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Example I-9

• Now we write loop equations.
• (R1 R3)I? R3I? V1
• R3I? (R2 R3) I? -V2
• By inserting the numerical values and solving we
  get I? 1.2 A and I? -1A which gives again
  the same branch currents I1 1.2 A, I2 -1 A,
  I3 -0.2 A

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Example I-10

• They are, of course, the same as before but we
  solved only system of two equations for two
  currents.
• To see the advantage even better lets revisit
  the second more complicated example.

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Example II-3

• Let I? be the current in the DBAD, I? in the DCBC
  and I? in the CBAC loops. Then
• I1 I? - I ?
• I2 I? - I?
• I3 I? - I?
• I4 -I?
• I5 I?
• I6 I?

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Example II-4

• The loop equation in DBAD would be
• -V1 R1(I? - I?) V3 R3(I? - I?) R5I? 0
• (R1 R3 R5)I? - R1I? - R3I ? V1 V3
• Similarly from the loops DCBD and CABC
• -R1I? (R1 R2 R4)I? - R2I ? V4 - V1 V2
• -R3I? - R2I? (R2 R3 R6)I ? V2 - V3
• It is some work but we have a system of only 3
  equations which we can solve by hand!

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Example II-5

• Numerically we get
• ?12 2 5 ? ?I?? ?51?
• ? -2 14 10 ? I?? ?-16?
• ?-5 10 25 ? ?I?? ?25?
• From here we get I?, I?, I? and then using them
  finally the branch currents I1

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Homework

• The homework from assigned on Wednesday is due
  Monday!

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Things to read

• Repeat the chapters 21 26 except 25-7 and 26-4 !