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Network Analysis

Network Theorems

Network theorems are circuit shortcuts with meaning. They help you turn messy circuits into smaller, clearer versions while keeping the same behavior at the part you care about.

01

Superposition Theorem

Superposition lets you study one source at a time instead of trying to understand every push together.

What Happens Physically

  • Only linear circuits can be handled this way.
  • When one source is active, the other independent sources are turned off.
  • A voltage source turned off becomes a short path.
  • A current source turned off becomes an open path.
  • The final current or voltage is the sum of all individual effects.

Formula

final response = response from source 1 + response from source 2 + ...

  • Response means the voltage or current you want to find.
  • Each source is tested alone.
  • The separate answers are added with their signs.

Step-by-Step Method

  1. 1Choose the voltage or current you want to find.
  2. 2Keep one independent source active.
  3. 3Turn off all other independent sources.
  4. 4Solve the circuit for that one source.
  5. 5Repeat for every source.
  6. 6Add all partial answers with correct direction or polarity.

Key Idea

Complex source action becomes easier when each source gets its own turn.

Animation Idea

Show multiple sources fading out one by one, then show each partial current path combining into the final answer.

02

Thevenin's Theorem

Thevenin turns a complicated two-terminal network into one voltage source with one series resistor.

What Happens Physically

  • The inside circuit may be large, but the load only feels terminal voltage and resistance.
  • The original network is replaced by a simpler equivalent source.
  • This is very useful when the load changes again and again.
  • Instead of solving the whole circuit repeatedly, you solve the equivalent once.

Formula

Vth with Rth in series

  • Vth is the open-terminal voltage seen by the load.
  • Rth is the resistance looking back into the network.
  • The load sees the same external behavior after replacement.

Step-by-Step Method

  1. 1Remove the load resistor.
  2. 2Find the open-circuit voltage across the load terminals. This is Vth.
  3. 3Turn off independent sources and find the resistance seen from the load terminals. This is Rth.
  4. 4Draw Vth in series with Rth.
  5. 5Reconnect the load and solve the simple circuit.

Key Idea

A big network can look like one voltage source and one resistor to the load.

Animation Idea

Show a large circuit collapsing into a battery and a series resistor while the load terminals stay fixed.

03

Norton's Theorem

Norton gives the same kind of simplification as Thevenin, but in current-source form.

What Happens Physically

  • The original network is replaced by one current source.
  • The equivalent resistance sits in parallel with that current source.
  • This form is helpful when branch currents are easier to think about.
  • Thevenin and Norton are two views of the same terminal behavior.

Formula

In with Rn in parallel

  • In is the short-circuit current through the output terminals.
  • Rn is the resistance looking back into the network.
  • Rn has the same value as Rth for the same network.

Step-by-Step Method

  1. 1Remove the load.
  2. 2Short the output terminals and find the current through the short. This is In.
  3. 3Turn off independent sources and find the resistance seen from the terminals. This is Rn.
  4. 4Draw In in parallel with Rn.
  5. 5Reconnect the load and solve using current division if useful.

Key Idea

A big network can also look like one current source and one parallel resistor.

Animation Idea

Show a complex network transforming into a current source with a parallel resistor feeding the load.

04

Maximum Power Transfer Theorem

This theorem tells us when a load receives the strongest possible power from a source network.

What Happens Physically

  • If the load is too small, current is high but voltage across the load drops.
  • If the load is too large, voltage is high but current becomes weak.
  • Maximum power happens at the balanced point.
  • That balance occurs when the load equals the source-side resistance.

Formula

RL = Rth, Pmax = Vth^2 / 4Rth

  • RL is the load resistance.
  • Rth is the Thevenin resistance of the source network.
  • Pmax is the highest power the load can receive in a DC resistive circuit.

Step-by-Step Method

  1. 1Find the Thevenin equivalent of the network seen by the load.
  2. 2Identify Rth.
  3. 3Choose the load value equal to Rth.
  4. 4Use the simplified circuit to find load current and power.
  5. 5For maximum power, use Pmax = Vth^2 / 4Rth.

Key Idea

The load gets maximum power when it matches the source-side resistance.

Animation Idea

Show a power bar rising as RL approaches Rth, peaking at the match point, then falling after mismatch.

05

Reciprocity Theorem

Reciprocity says that in some linear circuits, source and response positions can be swapped and the response stays linked.

What Happens Physically

  • The circuit behaves symmetrically from the two selected points.
  • It works well with passive resistor networks.
  • It does not apply freely to circuits with dependent sources or unilateral devices.
  • It is useful for checking network behavior and simplifying some measurements.

Formula

response at B due to source at A = response at A due to same source at B

  • The circuit must be linear and bilateral.
  • The same source value is moved to the other location.
  • The measured response is checked at the original source location.

Step-by-Step Method

  1. 1Apply a source at one branch.
  2. 2Measure the response in another branch.
  3. 3Move the same source to the measured branch.
  4. 4Measure the response in the original branch.
  5. 5Compare both responses.

Key Idea

In a linear bilateral network, source and response locations can sometimes trade places.

Animation Idea

Show a source icon and meter icon swapping positions while the highlighted response stays equal.

06

Millman's Theorem

Millman's Theorem helps combine several parallel voltage-source branches into one equivalent voltage.

What Happens Physically

  • Parallel source branches pull the common node toward their own voltages.
  • A branch with low resistance pulls harder.
  • The final node voltage is a weighted balance of all branches.
  • This saves time in circuits with many source-resistor branches in parallel.

Formula

Veq = (V1/R1 + V2/R2 + ...)/(1/R1 + 1/R2 + ...)

  • Each voltage source contributes according to its branch resistance.
  • Smaller resistance gives that source more influence.
  • Veq is the single equivalent node voltage.

Step-by-Step Method

  1. 1Identify all parallel branches with voltage source and series resistance.
  2. 2Calculate each V/R term.
  3. 3Add all V/R terms in the numerator.
  4. 4Add all 1/R terms in the denominator.
  5. 5Divide to get the equivalent voltage.
  6. 6Use the equivalent source for further solving.

Key Idea

Parallel source branches combine into one weighted voltage source.

Animation Idea

Show several source branches pulling one common node, then merging into one equivalent source.

07

Star-Delta Transformation

Star-Delta transformation changes a three-resistor shape into another shape that is easier to solve.

What Happens Physically

  • Some resistor networks are not simple series or parallel.
  • A star shape has three resistors meeting at one center point.
  • A delta shape has three resistors forming a triangle.
  • Changing one shape into the other can reveal series-parallel simplifications.
  • The outside terminals behave the same after transformation.

Formula

Delta to Star: RA = RbRc / (Ra + Rb + Rc)

  • Each star resistor depends on the two neighboring delta resistors.
  • The denominator is the sum of all three delta resistors.
  • The outside three terminals keep the same behavior.

Step-by-Step Method

  1. 1Find the three terminals of the star or delta network.
  2. 2Decide which transformation makes the circuit simpler.
  3. 3Apply the correct conversion formulas.
  4. 4Redraw the circuit with the new resistor shape.
  5. 5Look for new series or parallel combinations.
  6. 6Continue simplifying the circuit.

Key Idea

Change the resistor shape so hidden series-parallel paths become visible.

Animation Idea

Show a triangle of resistors smoothly folding into a Y-shape, then highlight the newly simplified branches.

One Learning Flow

These theorems are not separate tricks. They are choices. First understand what makes the circuit difficult. Then choose the theorem that removes that difficulty with the least work.

Step 1

Start With Sources

If many sources are active, use Superposition to see each source effect clearly.

Step 2

Reduce The Network

Use Thevenin or Norton when a large circuit only matters at two terminals.

Step 3

Choose The Load

Use Maximum Power Transfer when the goal is strongest power delivery to the load.

Step 4

Check Symmetry

Use Reciprocity when the circuit is linear and bilateral and source-response positions can be exchanged.

Step 5

Merge Parallel Sources

Use Millman's Theorem when several source-resistor branches share the same two nodes.

Step 6

Reshape Resistors

Use Star-Delta when the network is not directly series or parallel.

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