Theorem
Thévenin's
Theorem
Any
linear voltage network which may be viewed from two terminals can be
replaced by a voltage-source equivalent circuit comprising a single voltage
source E and a single series impedance Z. The voltage E is the open-circuit
voltage between the two terminals and the impedance Z is the impedance of
the network viewed from the terminals with all voltage sources replaced by
their internal impedances.
Norton's
Theorem
Any
linear current network which may be viewed from two terminals can be
replaced by a current-source equivalent circuit comprising a single current
source I and a single shunt admittance Y. The current I is the short-circuit
current between the two terminals and the admittance Y is the admittance of
the network viewed from the terminals with all current sources replaced by
their internal admittances.
Thévenin
and Norton Equivalence
The
open circuit, short circuit and load conditions of the Thévenin model are:
V_{oc} = E
I_{sc} = E / Z
V_{load} = E - I_{load}Z
I_{load} = E / (Z + Z_{load})
The open
circuit, short circuit and load conditions of the Norton model are:
V_{oc} = I / Y
I_{sc} = I
V_{load} = I / (Y + Y_{load})
I_{load} = I - V_{load}Y
Thévenin model from Norton model
Voltage
= Current / Admittance |
E
= I / Y |
Norton model from Thévenin model
Current
= Voltage / Impedance |
I
= E / Z |
When
performing network reduction for a Thévenin or Norton model, note that:
- nodes with zero voltage difference may be short-circuited with no effect
on the network current distribution,
- branches carrying zero current may be open-circuited with no effect on the
network voltage distribution.
Superposition
Theorem
In
a linear network with multiple voltage sources, the current in any branch is
the sum of the currents which would flow in that branch due to each voltage
source acting alone with all other voltage sources replaced by their
internal impedances.
Reciprocity
Theorem
If
a voltage source E acting in one branch of a network causes a current I to
flow in another branch of the network, then the same voltage source E acting
in the second branch would cause an identical current I to flow in the first
branch.
Compensation
Theorem
If
the impedance Z of a branch in a network in which a current I flows is
changed by a finite amount
dZ,
then the change in the currents in all other branches of the network may be
calculated by inserting a voltage source of -IdZ
into that branch with all other voltage sources replaced by their internal
impedances.
Millman's
Theorem (Parallel Generator Theorem)
If
any number of admittances Y_{1}, Y_{2}, Y_{3}, ...
meet at a common point P, and the voltages from another point N to the free
ends of these admittances are E_{1}, E_{2}, E_{3},
... then the voltage between points P and N is:
V_{PN} = (E_{1}Y_{1} + E_{2}Y_{2} +
E_{3}Y_{3} + ...) / (Y_{1} + Y_{2} + Y_{3}
+ ...)
V_{PN} =
SEY
/ SY
The short-circuit currents available between points P and N due to each
of the voltages E_{1}, E_{2}, E_{3}, ... acting
through the respective admitances Y_{1}, Y_{2}, Y_{3},
... are E_{1}Y_{1}, E_{2}Y_{2}, E_{3}Y_{3},
... so the voltage between points P and N may be expressed as:
V_{PN} =
SI_{sc} /
SY