Source Transformation: Thevenin ↔ Norton

Every linear circuit reduces, from the perspective of a single port, to either a voltage source in series with a resistance or a current source in parallel with a resistance. Knowing that these two descriptions are mathematically interchangeable — and being able to swap between them in one line of algebra — is what makes multi-source, multi-loop networks tractable by hand. Source transformation is the tool that lets you collapse a messy sub-network into whichever form (Thevenin or Norton) is more convenient for the next step of the analysis.

The Equivalence

A Thevenin source (Vth in series with Rth) and a Norton source (Ino in parallel with Rno) are equivalent at a pair of terminals if and only if they produce the same V-I relationship at those terminals for every possible load. The conversion is:

Vth = Ino · Rno        Rth = Rno
Ino = Vth / Rth        Rno = Rth

The series/parallel resistance is identical in both forms — only the source type and its value change. This isn't a coincidence; it falls directly out of writing the terminal V-I equation both ways and matching coefficients.

Derivation. For the Thevenin form, applying KVL from the terminals looking into the source:

V = Vth − I·Rth

For the Norton form, applying KCL at the top node (I flowing out to the load, Ino flowing in, Rno carrying the rest):

I = Ino − V/Rno   →   V = Ino·Rno − I·Rno

Comparing the two V(I) lines term by term: the slope (−Rth vs. −Rno) must match, so Rth = Rno. The intercept (Vth vs. Ino·Rno) must match, so Vth = Ino·Rno. Same line, two parametrizations.

Why the Resistance Doesn't Change

A useful mental check: to find Rth/Rno, you zero out all independent sources (short voltage sources, open current sources) and look into the terminals — this is a property of the network, not of which source type you're using to represent it. Converting between Thevenin and Norton never touches this resistance; it only redistributes the "drive" between a series voltage and a parallel current. If a transformation changes your resistance value, you've made an arithmetic error.

Worked Example

Consider a source network feeding a load: a 12 V source in series with a 4 Ω resistor, driving a 6 Ω load.

Step 1 — Thevenin form (given):

Vth = 12 V,  Rth = 4 Ω

Step 2 — Convert to Norton:

Ino = Vth / Rth = 12 / 4 = 3 A
Rno = Rth = 4 Ω

So the equivalent is a 3 A current source in parallel with 4 Ω.

Step 3 — Verify with the load attached (6 Ω), both forms must give the same terminal voltage and load current.

Thevenin form: series circuit, 12 V through 4 Ω + 6 Ω = 10 Ω total.

I_load = 12 / 10 = 1.2 A
V_load = I_load · 6 = 7.2 V

Norton form: 3 A splits between Rno = 4 Ω and RL = 6 Ω (current divider):

I_load = Ino · Rno/(Rno + RL) = 3 · 4/10 = 1.2 A
V_load = I_load · 6 = 7.2 V

Both forms agree exactly: 1.2 A through the load, 7.2 V across it. The transformation preserved the external behavior, as it must.

Check via power balance (Thevenin side): power delivered by the source = 12 V × 1.2 A = 14.4 W. Power dissipated: I²·Rth = 1.2² × 4 = 5.76 W in the internal resistor, I²·RL = 1.2² × 6 = 8.64 W in the load. Sum = 5.76 + 8.64 = 14.4 W. Balances.

Converting the Other Direction

Given a Norton equivalent — say Ino = 2 mA, Rno = 5 kΩ — the Thevenin form is:

Vth = Ino · Rno = 2 mA × 5 kΩ = 10 V
Rth = Rno = 5 kΩ

This direction is exactly as mechanical: multiply current by the parallel resistance to get the open-circuit voltage.

Practical Use: Collapsing Sub-Networks

Source transformation earns its keep in circuits with multiple sources scattered across different branches, where straight node or mesh analysis would require tracking every source simultaneously. The typical workflow:

1. Isolate a two-terminal sub-network containing one source and its series/parallel resistance.
2. Transform it to whichever form (V-series or I-parallel) lets it merge with an adjacent element of the same type — series voltage sources add directly once resistances are in series; parallel current sources add directly once resistances are in parallel.
3. Repeat, walking the transformation across the circuit, combining sources and resistances at each step, until only one source and one resistance remain facing the load or node of interest.

This is often faster than setting up a full KCL/KVL system for circuits with 2–3 sources, and it's the standard technique for simplifying the source network before applying Thevenin's or Norton's theorem to the whole circuit as seen from a particular pair of terminals.

Limits and Caveats

• Dependent sources: the transformation still holds algebraically (Vth = Ino·Rth), but Rth/Rno for a network containing dependent sources must be found by the test-source method (apply a 1 A or 1 V test excitation at the terminals with independent sources zeroed) rather than by simple series/parallel reduction, since dependent sources cannot simply be zeroed.
• Ideal source restriction: an ideal voltage source with zero series resistance has no Norton equivalent (Ino would be infinite) — the transformation requires a genuine series or parallel resistance to pivot on. Likewise an ideal current source with infinite parallel resistance has no finite Thevenin form.
• Only valid at the stated terminal pair: the equivalence guarantees identical external V-I behavior, not identical internal power dissipation or internal node voltages of the original network — those are only preserved for the elements outside the transformed sub-network.
• Linearity requirement: the sub-network being transformed must be linear (only independent/dependent sources and linear R, L, C, or their impedances) — this is the same restriction that underlies Thevenin's and Norton's theorems themselves.

Key Takeaways

• A Thevenin source (Vth, Rth) and Norton source (Ino, Rno) are equivalent at a terminal pair when Vth = Ino·Rth and Rth = Rno — the resistance never changes, only the source type.
• The equivalence follows directly from matching the terminal V-I line's slope and intercept in both representations.
• Always verify a transformation by attaching a load (or arbitrary test resistance) and confirming both forms give the same load voltage and current — and ideally that power balances too.
• Source transformation is most useful for collapsing multi-source sub-networks step by step before full nodal/mesh analysis or before applying Thevenin/Norton to the complete circuit.
• Ideal sources with zero series resistance (voltage) or infinite parallel resistance (current) have no finite equivalent in the other form; dependent-source networks need the test-source method to find Rth/Rno.