Contents |
Notation | ||||||
The symbol font
is used for some notation and formulae. If the
Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. |
||||||
E G I R P |
voltage source conductance current resistance power |
[volts, V] [siemens, S] [amps, A] [ohms, W] [watts] |
V X Y Z |
voltage drop reactance admittance impedance |
[volts, V] [ohms, W] [siemens, S] [ohms, W] |
Impedance = Voltage / Current | Z = E / I |
Similarly, when a voltage E is applied across an impedance Z, the resulting current I through
the impedance is equal to the voltage E divided by the impedance Z.
Current = Voltage / Impedance | I = E / Z |
Similarly, when a current I is passed through an impedance Z, the resulting voltage drop V
across the impedance is equal to the current I multiplied by the impedance Z.
Voltage = Current * Impedance | V = IZ |
Alternatively, using admittance Y which is the reciprocal of impedance Z:
Voltage = Current / Admittance | V = I / Y |
Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
SI = 0
Kirchhoff's Voltage Law
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage
drops in that circuit:
SE = SIZ
Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
SE - SIZ = 0
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.
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.
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 Impedance = 1 / Admittance |
E = I / Y Z = Y^{ -1} |
Norton model from Thévenin model
Current = Voltage / Impedance Admittance = 1 / Impedance |
I = E / Z Y = Z^{ -1} |
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.
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.
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.
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.
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
When a current I is passed through a resistance R, the resulting power P
dissipated in the resistance is equal to the square of the current I multiplied by the
resistance R:
P = I^{2}R
By substitution using Ohm's Law for the corresponding voltage drop V (= IR) across the
resistance:
P = V^{2} / R = VI = I^{2}R
Note that power is zero for an open-circuit (zero current) and for a short-circuit (zero voltage).
Voltage Source
When a load resistance R_{T} is connected to a voltage source E_{S}
with series resistance R_{S}, maximum power transfer to the load occurs when
R_{T} is equal to R_{S}.
Under maximum power transfer conditions, the load resistance R_{T}, load voltage
V_{T}, load current I_{T} and load power P_{T} are:
R_{T} = R_{S}
V_{T} = E_{S} / 2
I_{T} = V_{T} / R_{T} = E_{S} / 2R_{S}
P_{T} = V_{T}^{2} / R_{T}
= E_{S}^{2} / 4R_{S}
Current Source
When a load conductance G_{T} is connected to a current source I_{S}
with shunt conductance G_{S}, maximum power transfer to the load occurs when
G_{T} is equal to G_{S}.
Under maximum power transfer conditions, the load conductance G_{T}, load current
I_{T}, load voltage V_{T} and load power P_{T} are:
G_{T} = G_{S}
I_{T} = I_{S} / 2
V_{T} = I_{T} / G_{T} = I_{S} / 2G_{S}
P_{T} = I_{T}^{2} / G_{T}
= I_{S}^{2} / 4G_{S}
Complex Impedances
When a load impedance Z_{T} (comprising variable resistance R_{T} and
variable reactance X_{T}) is connected to an alternating voltage source E_{S}
with series impedance Z_{S} (comprising resistance R_{S} and reactance
X_{S}), maximum power transfer to the load occurs when Z_{T} is equal to
Z_{S}^{*} (the complex conjugate of Z_{S}) such that
R_{T} and R_{S} are equal and X_{T} and X_{S}
are equal in magnitude but of opposite sign (one inductive and the other capacitive).
When a load impedance Z_{T} (comprising variable resistance R_{T} and
constant reactance X_{T}) is connected to an alternating voltage source E_{S}
with series impedance Z_{S} (comprising resistance R_{S} and reactance
X_{S}), maximum power transfer to the load occurs when R_{T} is equal to the
magnitude of the impedance comprising Z_{S} in series with X_{T}:
R_{T} = |Z_{S} + X_{T}|
= (R_{S}^{2} + (X_{S} + X_{T})^{2})^{½}
Note that if X_{T} is zero, maximum power transfer occurs when R_{T} is equal
to the magnitude of Z_{S}:
R_{T} = |Z_{S}| = (R_{S}^{2} + X_{S}^{2})^{½}
When a load impedance Z_{T} with variable magnitude and constant phase angle (constant power
factor) is connected to an alternating voltage source E_{S} with series impedance
Z_{S}, maximum power transfer to the load occurs when the magnitude of Z_{T}
is equal to the magnitude of Z_{S}:
(R_{T}^{2} + X_{T}^{2})^{½} = |Z_{T}|
= |Z_{S}| = (R_{S}^{2} + X_{S}^{2})^{½}
Similarly, using admittances:
Y_{AB} = Y_{AN}Y_{BN} / (Y_{AN} + Y_{BN} + Y_{CN})
Y_{BC} = Y_{BN}Y_{CN} / (Y_{AN} + Y_{BN} + Y_{CN})
Y_{CA} = Y_{CN}Y_{AN} / (Y_{AN} + Y_{BN} + Y_{CN})
In general terms:
Z_{delta} = (sum of Z_{star} pair products)
/ (opposite Z_{star})
Y_{delta} = (adjacent Y_{star} pair product)
/ (sum of Y_{star})
Similarly, using admittances:
Y_{AN} = Y_{CA} + Y_{AB} + (Y_{CA}Y_{AB} / Y_{BC})
= (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB})
/ Y_{BC}
Y_{BN} = Y_{AB} + Y_{BC} + (Y_{AB}Y_{BC} / Y_{CA})
= (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB})
/ Y_{CA}
Y_{CN} = Y_{BC} + Y_{CA} + (Y_{BC}Y_{CA} / Y_{AB})
= (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB})
/ Y_{AB}
In general terms:
Z_{star} = (adjacent Z_{delta} pair product)
/ (sum of Z_{delta})
Y_{star} = (sum of Y_{delta} pair products)
/ (opposite Y_{delta})
Updated 09 June 2008 Copyright ©1998-2008 BOWest Pty Ltd |
All content subject to disclaimer and copyright notices |
Home | Staff | Library | Links | Feedback | Email |