Series-Parallel Circuits & Network Theorems Notes

Last reviewed 16 Sept 2026 · 6 min read

Series and parallel circuits

Feature	Series	Parallel
Current	Same through all elements	Divides among branches
Voltage	Divides: $V = V_{1} + V_{2} + \dots$	Same across all branches
Equivalent resistance	$R_{e q} = \sum R_{i}$ (greater than largest)	$\frac{1}{R _{e q}} = \sum \frac{1}{R _{i}}$ (less than smallest)
Failure of one element	Whole circuit opens	Others continue working
Use	Decorative lamp strings, current-limiting	Domestic wiring (appliances at same voltage)

Series–parallel circuits are reduced step by step, combining series and parallel groups.

Network terminology

Term	Meaning
Node	Junction of two or more elements
Branch	Element or elements between two nodes
Loop	Closed path
Mesh	Loop containing no other loop inside
Active element	Supplies energy (sources)
Passive element	Absorbs/stores energy (R, L, C)
Linear element	Parameters constant with voltage/current (obeys superposition)
Bilateral element	Same behaviour in both directions (R, L, C; not diodes)
Lumped network	Elements physically separable

Source transformation

A voltage source $E$ in series with $R$ ⇔ current source $I = E / R$ in parallel with $R$ (equivalent at the terminals). Used to simplify circuits before applying other methods.

Superposition theorem

In a linear bilateral network with several independent sources, the current (or voltage) in any element equals the algebraic sum of the currents (or voltages) produced by each source acting alone, with other sources replaced by their internal resistances:

Voltage sources → short circuit (if ideal).
Current sources → open circuit (if ideal).

Limitation: not applicable to power directly (power is non-linear, $I^{2} R$ ), nor to non-linear elements.

Thevenin's theorem

Any linear bilateral two-terminal network can be replaced by an equivalent voltage source $V_{t h}$ in series with a resistance $R_{t h}$ .

FormulaThevenin equivalent

$V_{t h}$ = open-circuit voltage across the terminals (load removed)
$R_{t h}$ = resistance seen from the terminals with all independent sources replaced by internal resistances (voltage sources shorted, current sources opened)
Load current: $I_{L} = \frac{V _{t h}}{R _{t h} + R _{L}}$

Useful when the load changes — only one calculation of the equivalent is needed.

Norton's theorem

Any linear bilateral two-terminal network can be replaced by an equivalent current source $I_{N}$ in parallel with a resistance $R_{N}$ .

FormulaNorton equivalent

$I_{N}$ = short-circuit current through the terminals
$R_{N} = R_{t h}$
Relation: $V_{t h} = I_{N} R_{N}$
Load current: $I_{L} = I_{N} \frac{R _{N}}{R _{N} + R _{L}}$

Maximum power transfer theorem

FormulaMaximum power transfer (DC)

Maximum power is delivered to a load when

R_{L} = R_{t h}

P_{ma x} = \frac{V _{t h}^{2}}{4 R _{t h}}

Efficiency at maximum power transfer = 50% (half the power is lost in $R_{t h}$ ).

AC circuits: maximum power when load impedance is the complex conjugate of source impedance, $Z_{L} = Z_{t h}^{*}$ (if only $∣ Z_{L} ∣$ can vary: $∣ Z_{L} ∣ = ∣ Z_{t h} ∣$ ).

Used in communication/electronic circuits (matching); power systems operate far from this condition because 50% efficiency is unacceptable.

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