Equivalent Devices

Important

Equivalent devices are governed by equivalent equations

Device equivalence

Series connection

Two one-ports are in series if they have exactly one common terminal and no other device is connected to it.

Resistors

R_{e q} = R_{1} + R_{2} + \dots + R_{n} = \sum_{i}^{n} R_{i}

u = u_{1} + \dots + u_{n} = i (R_{1} + \dots + R_{n})

Inductors

L_{e q} = L_{1} + \dots L_{n}

Capacitors

C_{e q} = {(\frac{1}{C_{1}} + \dots + \frac{1}{C_{n}})}^{- 1}

u^{'} = u_{1}^{'} + \dots u_{n}^{'} = i {(\frac{1}{C_{1}} + \dots + \frac{1}{C_{n}})}^{- 1}

i = {(\frac{1}{C_{1}} + \dots + \frac{1}{C_{n}})}^{- 1} u^{'}

Voltage sources

E_{e q} = E_{1} + \dots + E_{n}

Current sources

J_{e q} = J

Parallel connection

Two or more one-ports are in parallel if they share distinct nodes that are terminals for all of there one-ports

Resistors

![[rM drawing 2026-01-11-20.12.27##### As part of a bigger circuit

As independent circuit

R_{e q} = {(\frac{1}{R_{1}} + \dots + \frac{1}{R_{n}})}^{- 1}

i = i_{1} + \dots + i_{n} = u (\frac{1}{R_{1}} + \dots + \frac{1}{R_{n}})

u = i {(\frac{1}{R_{1}} + \dots + \frac{1}{R_{n}})}^{- 1}

Inductors

L_{e q} = {(\frac{1}{L_{1}} + \dots \frac{1}{L_{n}})}^{- 1}

Capacitors

C_{e q} = C_{1} + \dots + C_{n}

Current sources

J_{e q} = J_{1} + \dots + J_{n}

Voltage sources

E_{e q} = E

Thévenin and Norton equivalences

Thévenin's equivalent

U = E_{T} - R_{T} I

Norton's equivalent

I = J_{n} - U G_{n} = J_{n} - \frac{U}{R_{n}}

Thévenin's and Norton's equivalents equivalence

Thévenin's and Norton's equivalents are equivalent iff

R_{T} = R_{T} and E_{T} = J_{N} R_{N}

U = E_{T} - R_{T} I compare to I = J_{N} - \frac{U}{R_{N}} \to U = J_{N} R_{N} - R_{N} I

Example

△-☆ and ☆-△ transformations

Equivalent if:
$R_{A} = \frac{R_{1} R_{2} + R_{1} R_{3} + R_{2} R_{3}}{R_{1}}$
$R_{B} = \frac{R_{1} R_{2} + R_{1} R_{3} + R_{2} R_{3}}{R_{2}}$
$R_{C} = \frac{R_{1} R_{2} + R_{1} R_{3} + R_{2} R_{3}}{R_{3}}$
In other words:
$R_{1} = \frac{R_{B} R_{C}}{R_{A} + R_{B} + R_{C}}$
$R_{2} = \frac{R_{A} R_{C}}{R_{A} + R_{B} + R_{C}}$
$R_{3} = \frac{R_{A} R_{B}}{R_{A} + R_{B} + R_{C}}$