Multi-source synthesis & bridge to Thévenin/Norton
Combine nodal analysis, superposition, source transformation and port resistance on one network. Derive its complete terminal law, reduce it to 5 V + 1 kΩ or 5 mA || 1 kΩ, verify a finite load, then identify precisely what the general Thévenin/Norton theorem adds.
- Analyze one multi-source linear one-port directly and obtain its open-circuit terminal voltage.
- Recover the same open-circuit voltage by signed superposition of independent-source contributions.
- Use local source transformations to combine multiple branches into a Norton form.
- Determine output resistance by deactivating independent sources while preserving the original port.
- Derive the complete one-port relation V=VOC−IRout and identify the corresponding Thévenin and Norton parameters.
- Verify the original network, Thévenin form and Norton form against the same finite load.
- Explain the distinction between local source transformation and the general Thévenin/Norton theorem, including the role of test sources when dependent sources are present.
Can nodal analysis, superposition, source transformation and terminal-resistance reasoning all describe the same multi-source network without contradiction?
They must. Each method reorganizes the same linear circuit equations. This final chapter uses one network to make the equivalence explicit and to build the conceptual bridge to the general Thévenin and Norton theorems.
05.1Define the one-port before simplifying it
Figure 5.1 — One multi-source one-port viewed from VOUT–ground
The two sources and three resistors form the internal network. No external element is connected to the VOUT–ground port here, so the figure is the open-circuit one-port.
The internal network contains a 12 V source through 3 kΩ, a 6 V source through 6 kΩ, and a 2 kΩ shunt to ground. The external port is VOUT relative to ground.
Once the port is chosen, an external load is allowed to interact with the network only through two variables: port voltage V and port current I. Define I positive when the network delivers current out of VOUT toward the external load.
Every valid reduction in this chapter must preserve the same V–I relation at those two terminals.
05.2Open-circuit voltage by direct nodal analysis
With no external load, I=0. KCL at VOUT gives
As established in chapter 03,
This is the voltage an ideal infinite-resistance measurement would observe at the port. It is the first terminal parameter of the equivalent one-port.
05.3The same 5 V emerges from superposition
Keep the 12 V source active and set the 6 V source to 0 V. The contribution is
Then keep the 6 V source active and set the 12 V source to 0 V:
The open-circuit voltage therefore has a clear source-by-source interpretation, but it remains one terminal property of the complete network.
05.4Local source transformations collapse the network directly into Norton form
Transform the left branch:
Transform the second branch:
Both current-source arrows inject toward the positive VOUT node, so their signed values add:
All three resistances are now parallel:
The complete original network has therefore reduced, by local transformations alone, to
05.5Output resistance gives the same 1 kΩ independently
For this resistor network with only independent sources, set both independent voltage sources to 0 V. Each ideal voltage source becomes a short. Looking into the unchanged VOUT–ground port, the three resistors appear in parallel:
This is the same resistance found by source transformation.
Combining VOC=5 V with Rout=1 kΩ immediately gives the current-source value
which matches the Norton current obtained by transforming and adding the two source branches.
05.6Derive the complete terminal law of the original network
Now allow an arbitrary external current I to leave the port toward a load. Current injected into VOUT from the two source branches is
Current leaving VOUT internally through the 2 kΩ resistor and externally through the port is
KCL gives
Collect constant and V-dependent terms:
Therefore
This one equation is the complete external behavior. It is simultaneously the voltage-source form
with VTH=5 V and RTH=1 kΩ, and the current-source form obtained from IN=VTH/RTH=5 mA with RN=1 kΩ.
05.7One finite load verifies all three descriptions
Connect RL=4 kΩ.
Original network:
Figure 5.2 — Equivalent voltage-source form tested with 4 kΩ
Viewed from the same port, the multi-source network is replaced by an ideal 5 V source in series with 1 kΩ. The test load is 4 kΩ.
Equivalent voltage-source form:
Figure 5.3 — Equivalent current-source form tested with the same load
The same terminal relation can be represented by an ideal 5 mA source in parallel with 1 kΩ. The external load remains 4 kΩ.
Equivalent current-source form:
The same external load point appears three times because all three descriptions impose the same terminal relation.
05.8What Thévenin and Norton add beyond local source transformation
Source transformation is a local algebraic replacement: when a source-plus-resistance pair has the appropriate structure, replace that pair by another form with the same port law.
The Thévenin/Norton theorem is broader. It says that a suitable linear two-terminal network can be represented at its port by a voltage-source/series-resistance form or an equivalent current-source/parallel-resistance form even when the original network is not visually reducible by a chain of obvious source transformations.
That broader theorem motivates systematic extraction methods:
- Find open-circuit voltage VTH.
- Find port resistance RTH by deactivating independent sources when allowed, or by applying a test source when dependent sources remain.
- Alternatively find short-circuit current IN and use the consistent port relation.
CT-01 develops those methods, loading consequences, measurement procedures and design uses in depth. DC-05 supplies the conceptual prerequisites so CT-01 is no longer a theorem to memorize.
05.9Dependent sources and multiport caveats
A linear one-port may contain dependent sources. When independent sources are deactivated, dependent sources remain active, so the resistance seen at the port is generally obtained with a test voltage or test current:
If a circuit contains only dependent sources and no independent excitation, both open-circuit voltage and short-circuit current may be zero. The ratio 0/0 does not determine the port resistance; the test-source method does.
Also ask whether the object is truly a one-port. If a dependent source is controlled by an external signal entering through additional terminals, the device is effectively a multiport or controlled one-port. A fixed independent Thévenin pair cannot generally erase that extra control dependence. The equivalent model must preserve every externally accessible control/port relation relevant to the problem.
05.10Mastery synthesis and the DC-foundations bridge
- For the chapter network, derive ISC directly by shorting VOUT rather than using VOC/Rout.
- Reverse the 6 V source polarity. Find the new Norton current, open-circuit voltage and unchanged output resistance.
- Connect RL=1 kΩ to the original network. Find V and IL from the original KCL and from the 5 V + 1 kΩ equivalent.
- Remove the internal 2 kΩ resistor. Find the new VOC, Rout and Norton current.
- Add a linear dependent downward current source id=(0.5 mS)V at the port of the original network. Find VOC and Rout using the circuit equations/test-source reasoning.
- Why is VOC/ISC not a safe standalone resistance formula for a network containing only dependent sources and no independent excitation?
- Explain why the transformation chain in section 05.4 is a convenient proof for this network but is not itself the general Thévenin theorem.
- A learner says: “Once I have the Thévenin equivalent, I can compute the original internal resistor powers from it.” Diagnose the mistake.
Solutions
1. With VOUT=0, the 12 V branch injects 4 mA and the 6 V branch injects 1 mA; the internal 2 kΩ branch carries zero. ISC=5 mA.
2. The second transformed current is −1 mA, so IN=3 mA. Rout remains 1 kΩ and VOC=(3 mA)(1 kΩ)=3 V.
3. Equivalent form gives V=5·1/(1+1)=2.5 V and IL=2.5 mA. Direct KCL gives the same result.
4. Without 2 kΩ, total conductance is 1/3k+1/6k=0.5 mS, so Rout=2 kΩ. Source injection remains 5 mA, hence VOC=10 V and IN=5 mA.
5. The original passive port conductance is 1 mS; adding the dependent 0.5 mS branch gives 1.5 mS. The independent source injection remains 5 mA, so VOC=5 mA/1.5 mS=10/3 V≈3.333 V. With independent voltage sources set to zero, a test voltage sees 1.5 mS, hence Rout=2/3 kΩ≈666.7 Ω.
6. Both VOC and ISC can be zero even while the port has a finite or active small-signal resistance. The ratio becomes indeterminate 0/0; apply a test source.
7. Local transformations happen to reduce every branch here. The theorem is broader because it guarantees a terminal equivalent for a suitable linear one-port even when no obvious sequence of local transformations exists.
8. The equivalent preserves port behavior, not internal branch observables. Internal currents and powers can differ radically between equivalent representations, as chapter 04 demonstrated.
DC-05 closes the DC foundations chain: element laws → Kirchhoff constraints → series/parallel reduction → nodal/mesh solution → independent/dependent sources → linearity → superposition → terminal equivalence. CT-01 can now be read as a systematic one-port theory rather than an isolated trick.