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In [[electrochemistry]], the '''Nernst equation''' is an equation that relates the [[reduction potential]] of a [[half-cell]] (or the total [[voltage]] ([[electromotive force]]) of the [[electrochemical cell|full cell]]) at any point in time to the [[standard electrode potential]], [[Thermodynamic temperature|temperature]], [[Thermodynamic activity|activity]], and [[reaction quotient]] of the underlying reactions and species used. When the reaction quotient is equal to the equilibrium constant of the reaction for a given temperature, i.e. when the concentration of species are at their equilibrium values, the Nernst equation gives the equilibrium voltage of the half-cell (or the full cell), which is zero; at equilibrium, Q=K, &Delta;G=0, and therefore, E=0. It is named after the German physical chemist who first formulated it, [[Walther Nernst]].<ref name="isbn0-8412-1572-3">{{cite book |author=Orna, Mary Virginia; Stock, John |title=Electrochemistry, past and present |publisher=American Chemical Society |location=Columbus, Ohio |year=1989 |pages= |isbn=0-8412-1572-3 |oclc= 19124885|doi= |accessdate=}}</ref><ref name=Wahl2005>{{Cite journal | last = Wahl | year = 2005 | title = A Short History of Electrochemistry | journal = Galvanotechtnik | volume = 96 | issue = 8 | pages = 1820–1828 }}</ref>
In [[electrical engineering]], the '''maximum power transfer theorem''' states that, to obtain ''maximum'' external power from a source with a finite [[internal resistance]], the resistance of the load must equal the resistance of the source as viewed from its output terminals. [[Moritz von Jacobi]] published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".<ref>{{citation |url=http://books.google.com/?id=dKVbT-ZmdDwC&pg=PA406 |title=Dynamo-Electric Machinery; A Manual for Students of Electrotechnics |author=Thompson Phillips |publisher=BiblioBazaar, LLC |isbn=978-1-110-35104-6 |date=2009-05-30}}</ref>


The Nernst equation gives a formula that relates the numerical values of the [[concentration gradient]] to the [[electric gradient]] that balances it. For example, if a concentration gradient is established by dissolving KCl in half of a divided vessel that is full of H<sub>2</sub>O, and then a membrane permeable to K<sup>+</sup> ions is introduced between the two halves, then after a relaxation period, an equilibrium situation arises where the chemical concentration gradient, which at first causes ions to move from the region of high concentration to the region of low concentration, is exactly balanced by an electrical gradient that opposes the movement of charge.
The [[theorem]] results in maximum ''[[Electric power|power]]'' transfer, and not maximum ''[[efficiency]]''. If the [[Electrical resistance|resistance]] of the load is made larger than the resistance of the source, then efficiency is higher, since a higher percentage of the source power is transferred to the load, but the ''magnitude'' of the load power is lower since the total circuit resistance goes up.


==Expression==
If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced.
The two (ultimately equivalent) equations for these two cases (half-cell, full cell) are as follows:
 
:<math>
The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does ''not'' say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer is always zero, regardless of the value of the load resistance.
E_\text{red} = E^{\ominus}_\text{red} + \frac{RT}{zF} \ln\frac{a_\text{Ox}}{a_\text{Red}}
</math> &nbsp;&nbsp; (half-cell reduction potential for <math>\text{Ox} + z e^- = \text{Red}</math>)
:<math>
E_\text{cell} = E^{\ominus}_\text{cell} - \frac{RT}{zF} \ln Q_r
</math> &nbsp;&nbsp; (total cell potential)
where
*{{math|''E''<sub>red</sub>}} is the half-cell [[reduction potential]] at the temperature of interest
*{{math|''E''<sup><s>o</s></sup><sub>red</sub>}} is the [[standard electrode potential|''standard'' half-cell reduction potential]]
*{{math|''E''<sub>cell</sub>}} is the cell potential ([[electromotive force]]) at the temperature of interest
*{{math|''E''<sup><s>o</s></sup><sub>cell</sub>}} is the ''standard'' cell potential
*{{mvar|R}} is the [[universal gas constant]]: {{math|''R'' {{=}} 8.314&thinsp;472(15) J&thinsp;K<sup>&minus;1</sup>&thinsp;mol<sup>&minus;1</sup>}}
*{{mvar|T}} is the [[absolute temperature]]
*{{mvar|a}} is the chemical [[activity (chemistry)|activity]] for the relevant species, where {{math|''a''<sub>Red</sub>}} is the [[reductant]] and {{math|''a''<sub>Ox</sub>}} is the [[oxidant]]. {{math|''a''<sub>X</sub>&nbsp;{{=}} ''γ''<sub>X</sub>''c''<sub>X</sub>}}, where {{math|''γ''<sub>X</sub>}} is the [[activity coefficient]] of species X. (Since activity coefficients tend to unity at low concentrations, activities in the Nernst equation are frequently replaced by simple concentrations.)
*{{mvar|F}} is the [[Faraday constant]], the number of [[coulomb]]s per [[mole (unit)|mole]] of electrons: {{math|''F'' {{=}} 9.648&thinsp;533&thinsp;99(24)×10<sup>4</sup> C&thinsp;mol<sup>&minus;1</sup>}}
*{{mvar|z}} is the number of moles of [[electron]]s transferred in the cell reaction or [[half-reaction]]
*{{math|''Q''<sub>r</sub>}} is the [[reaction quotient]].


At room temperature (25&nbsp;°C), {{math|''RT''/''F''}} may be treated like a constant and replaced by 25.693&nbsp;mV for cells.
The theorem can be extended to [[alternating current]] circuits that include [[Electrical reactance|reactance]], and states that maximum power transfer occurs when the load [[Electrical impedance|impedance]] is equal to the [[complex conjugate]] of the source impedance.


The Nernst equation is frequently expressed in terms of base 10 [[logarithms]] (''i.e.'', [[common logarithm]]s) rather than [[natural logarithms]], in which case it is written, ''for a cell at 25 °C'':
== Maximizing power transfer versus power efficiency ==
The theorem was originally misunderstood (notably by [[James Prescott Joule|Joule]]) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either [[Thomas Edison|Edison]] or his colleague [[Francis Robbins Upton]], who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a [[dynamo]]) could be made close to zero.  Using this new understanding, they obtained an efficiency of about 90%, and proved that the [[electric motor]] was a practical alternative to the [[heat engine]].


:<math>
[[File:Source and load circuit.png]] [[File:Maximum Power Transfer Graph.svg]]
E = E^0 - \frac{0.05916\mbox{ V}}{z} \log_{10}\frac{a_\text{Ox}}{a_\text{Red}}.
</math>


The Nernst equation is used in [[physiology]] for finding the [[electric potential]] of a [[cell membrane]] with respect to one type of [[ion]].
The condition of maximum power transfer does not result in maximum [[Electrical efficiency|efficiency]]. If we define the efficiency {{math|''η''}} as the ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculate from the above circuit diagram that


==Nernst potential==
:<math>\eta = \frac{R_\mathrm{load}}{R_\mathrm{load} + R_\mathrm{source}} = \frac{1}{1 + R_\mathrm{source} / R_\mathrm{load}}.</math>
{{main|Reversal potential}}


The Nernst equation has a physiological application when used to calculate the potential of an ion of charge ''z'' across a membrane. This potential is determined using the concentration of the ion both inside and outside the cell:
Consider three particular cases:


:<math>E = \frac{R T}{z F} \ln\frac{[\text{ion outside cell}]}{[\text{ion inside cell}]} = 2.3026\frac{R T}{z F} \log_{10}\frac{[\text{ion outside cell}]}{[\text{ion inside cell}]}.</math>
* If <math>R_\mathrm{load} = R_\mathrm{source}</math>, then <math>\eta = 0.5,</math>
* If <math>R_\mathrm{load} = \infty</math> or <math>R_\mathrm{source} = 0,</math> then <math>\eta = 1,</math>
* If <math>R_\mathrm{load} = 0</math>, then <math>\eta = 0.</math>


When the membrane is in [[thermodynamic equilibrium]] (i.e., no net flux of ions), the [[membrane potential]] must be equal to the Nernst potential. However, in physiology, due to active [[Na+/K+-ATPase|ion pumps]], the inside and outside of a cell are not in equilibrium. In this case, the [[resting potential]] can be determined from the [[Goldman equation]]:
The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity, though the total power level tends towards zero. Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a [[short circuit]] is zero.


<math>E_{m} = \frac{RT}{F} \ln{ \left( \frac{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\mathrm{out} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\mathrm{in}}{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\mathrm{in} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\mathrm{out}} \right) }</math>
==Impedance matching==
{{main|impedance matching}}
A related concept is reflectionless [[impedance matching]].  In [[radio]], [[transmission line]]s, and other [[electronics]], there is often a requirement to match the [[source impedance]] (such as a transmitter) to the [[load impedance]] (such as an [[antenna (electronics)|antenna]]) to avoid reflections in the [[transmission line]].


*<math>E_{m}</math> = The membrane potential (in [[volt]]s, equivalent to [[joule]]s per [[coulomb]])
== Calculus-based proof for purely resistive circuits ==
*<math>P_\mathrm{ion}</math> = the permeability for that ion (in meters per second)
(See Cartwright<ref>{{citation
*<math>[ion]_\mathrm{out}</math> = the extracellular concentration of that ion (in [[Mole (unit)|moles]] per cubic meter, to match the other  [[SI]] units, though the units strictly don't matter, as the ion concentration terms become a dimensionless ratio)
|last=Cartwright
*<math>[ion]_\mathrm{in}</math> = the intracellular concentration of that ion (in moles per cubic meter)
|first=Kenneth V
*<math>R</math> = The [[ideal gas constant]] (joules per [[kelvin]] per mole)
|title=Non-Calculus Derivation of the Maximum Power Transfer Theorem
*<math>T</math> = The temperature in [[kelvin]]
|journal=Technology Interface
*<math>F</math> = [[Faraday constant|Faraday's constant]] (coulombs per mole)
|volume=8
|issue=2
|pages=19 pages
|date=Spring 2008


The potential across the cell membrane that exactly opposes net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. As seen above, the magnitude of the Nernst potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio the greater the tendency for the ion to diffuse in one direction, and therefore the greater the Nernst potential required to prevent the diffusion.
|url=http://technologyinterface.nmsu.edu/Spring08/
}}</ref> for a non-calculus-based proof)
[[File:maxpowertheorem.png|left|Circuit diagram]]


A similar expression exists that includes r (the absolute value of the transport ratio).  This takes transporters with unequal exchanges into account.  See: [[Sodium-Potassium Pump]] where the transport ratio would be 2/3.  The other variables are the same as above.  The following example includes two ions: Potassium (K<sup>+</sup>) and sodium (Na<sup>+</sup>).  Chloride is assumed to be in equilibrium.
In the diagram opposite, power is being transferred from the source, with voltage {{math|''V''}} and fixed [[source resistance]] {{math|''R''<sub>S</sub>}}, to a load with resistance {{math|''R''<sub>L</sub>}}, resulting in a current {{math|''I''}}.  By [[Ohm's law]], {{math|''I''}} is simply the source voltage divided by the total circuit resistance:


<math>V_{m} = \frac{RT}{F} \ln{ \left( \frac{ rP_{K}[K]_{o} + P_{Na}[Na]_{o}}{ rP_{K}[K]_{i} + P_{Na}[Na]_{i}} \right) }</math>
:<math>I = \frac{V}{R_\mathrm{S} + R_\mathrm{L}}.</math>


When Chloride (Cl<sup>&minus;</sup>) is taken into account, its part is flipped to account for the negative charge.
The power {{math|''P''<sub>L</sub>}} dissipated in the load is the square of the current multiplied by the resistance:
<math>V_{m} = \frac{RT}{F} \ln{ \left( \frac{ P_{K}[K]_{o} + P_{Na}[Na]_{o} + P_{Cl}[Cl]_{i}}{ P_{K}[K]_{i} + P_{Na}[Na]_{i} + P_{Cl}[Cl]_{o}} \right) }</math>


==Derivation==
:<math>P_\mathrm{L} = I^2 R_\mathrm{L} = \left(\frac{V}{R_\mathrm{S} + R_\mathrm{L}}\right)^2 R_\mathrm{L} = \frac{V^2}{R_\mathrm{S}^2 / R_\mathrm{L} + 2 R_\mathrm{S} + R_\mathrm{L}}.</math>


===Using Boltzmann factors===
The value of {{math|''R''<sub>L</sub>}} for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of {{math|''R''<sub>L</sub>}} for which the denominator


For simplicity, we will consider a solution of redox-active molecules that undergo a one-electron reversible reaction
:<math>R_\mathrm{S}^2 / R_\mathrm{L} + 2 R_\mathrm{S} + R_\mathrm{L}</math>


:<math>\text{Ox} + e^- \rightleftharpoons \text{Red}\,</math>
is a minimum.  The result will be the same in either case.  Differentiating the denominator with respect to {{math|''R''<sub>L</sub>}}:


and that have a standard potential of zero. The [[chemical potential]] <math>\mu_c</math> of this solution is the difference between the energy barriers for taking electrons from and for giving electrons to the [[Cyclic voltammetry|working electrode]] that is setting the solution's [[electrochemical potential]].
:<math>
\frac{d}{d R_\mathrm{L}} \left(R_\mathrm{S}^2 / R_\mathrm{L} + 2R_\mathrm{S} + R_\mathrm{L}\right) = -R_\mathrm{S}^2 / R_\mathrm{L}^2 + 1.</math>


The ratio of oxidized to reduced molecules, [Ox]/[Red], is equivalent to the probability of being oxidized (giving electrons) over the probability of being reduced (taking electrons), which we can write in terms of the [[Boltzmann factor]] for these processes:
For a maximum or minimum, the first derivative is zero, so


:<math>
:<math>
\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}
R_\mathrm{S}^2 / R_\mathrm{L}^2 = 1</math>
= \frac{\exp \left(-[\mbox{barrier for losing an electron}]/kT\right)}
 
{\exp \left(-[\mbox{barrier for gaining an electron}]/kT\right)}
or
= \exp \left(\mu_c / kT \right).
 
</math>
:<math>R_\mathrm{L} = \pm R_\mathrm{S}.</math>
 
In practical resistive circuits, {{math|''R''<sub>S</sub>}} and {{math|''R''<sub>L</sub>}} are both positive, so the positive sign in the above is the correct solution. To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:
 
:<math>{{d^2} \over {dR_\mathrm{L}^2}} \left( {R_\mathrm{S}^2 / R_\mathrm{L} + 2 R_\mathrm{S} + R_\mathrm{L}} \right) = {2 R_\mathrm{S}^2} / {R_\mathrm{L}^3}. \,\!</math>
 
This is always positive for positive values of <math>R_\mathrm{S} \,\!</math> and <math>R_\mathrm{L} \,\!</math>, showing that the denominator is a minimum, and the power is therefore a maximum, when
 
:<math>R_\mathrm{S} = R_\mathrm{L}. \,\!</math>
 
A note of caution is in order here.  This last statement, as written, implies to many people that for a given load, the source resistance must be set equal to the load resistance for maximum power transfer.  However, this equation only applies if the source resistance cannot be adjusted, e.g., with antennas (see the first line in the proof stating "fixed source resistance"). For any given load resistance a source resistance of zero is the way to transfer maximum power to the load. As an example, a 100 volt source with an internal resistance of 10 ohms connected to a 10 ohm load will deliver 250 watts to that load. Make the source resistance zero ohms and the load power jumps to 1000 watts.
 
==In reactive circuits==
The theorem also applies where the source and/or load are not totally resistive. This invokes a refinement of the maximum power theorem, which says that any reactive components of source and load should be of equal magnitude but opposite phase.  (''See below for a derivation.'')  This means that the source and load impedances should be ''[[complex conjugate]]s'' of each other.  In the case of purely resistive circuits, the two concepts are identical.  However, physically realizable sources and loads are not usually totally resistive, having some inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.
 
If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle.  The resultant circuit is nothing other than a resonant [[LC circuit]] in which the energy continues to oscillate to and fro.  This is called [[reactive power]]. [[Power factor correction]] (where an inductive reactance is used to "balance out" a capacitive one), is essentially the same  idea as complex conjugate impedance matching although it is done for entirely different reasons.
 
For a fixed reactive ''source'', the maximum power theorem maximizes the real power (P) delivered to the load by complex conjugate matching the load to the source.
 
For a fixed reactive ''load'', power factor correction minimizes the [[apparent power]] (S) (and unnecessary current) conducted by the transmission lines, while maintaining the same amount of real power transfer.  This is done by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.


Taking the natural logarithm of both sides gives
=== Proof ===


:<math>
[[File:Source and load boxes.png|left|source and load impedance diagram]]
\mu_c = kT \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
</math>


If <math>\mu_c \ne 0</math> at [Ox]/[Red] = 1, we need to add in this additional
In this diagram, [[AC power]] is being transferred from the source, with [[phasor (electronics)|phasor]] magnitude voltage <math>|V_\mathrm{S}|</math> (peak voltage) and fixed [[source impedance]] <math>Z_\mathrm{S}</math>, to a load with impedance <math>Z_\mathrm{L}</math>, resulting in a phasor magnitude current <math>|I|</math>.  <math>|I|</math> is simply the source voltage divided by the total circuit impedance:
constant:


:<math>
:<math>
\mu_c = \mu_c^0 + kT \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
|I| = { |V_\mathrm{S}| \over |Z_\mathrm{S} + Z_\mathrm{L}| }.
</math>
</math>


Dividing the equation by {{mvar|e}} to convert from chemical potentials to electrode potentials, and remembering that {{math|''k''/''e'' {{=}} ''R/F''}},<ref>{{math|''R''{{=}}''N<sub>A</sub>k''}}; see [[gas constant]]<br>{{math|''F''{{=}}''eN<sub>A</sub>''}}; see [[Faraday constant]]</ref> we obtain the Nernst equation for the one-electron process
The average power <math>P_\mathrm{L}</math> dissipated in the load is the square of the current multiplied by the resistive portion (the real part) <math>R_\mathrm{L}</math> of the load impedance:
<math>\mathrm{Ox} + e^- \rightarrow \mathrm{Red}</math>:


:<math>
: <math>
\begin{align}
\begin{align}
E &= E^0 + \frac{kT}{e} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
P_\mathrm{L} & = I_\mathrm{rms}^2 R_\mathrm{L} = {1 \over 2} |I|^2 R_\mathrm{L} = {1 \over 2} \left( {|V_\mathrm{S}| \over |Z_\mathrm{S} + Z_\mathrm{L}|} \right)^2 R_\mathrm{L} \\
  &= E^0 - \frac{RT}{F} \ln \frac{[\mathrm{Ox}]}{[\mathrm{Red}]}.
& = {1 \over 2}{ |V_\mathrm{S}|^2 R_\mathrm{L} \over (R_\mathrm{S} + R_\mathrm{L})^2 + (X_\mathrm{S} + X_\mathrm{L})^2},
\end{align}
\end{align}
</math>
</math>


===Using thermodynamics (chemical potential)===
where the resistance <math>R_\mathrm{S}</math> and reactance <math>X_\mathrm{S}</math> are the real and imaginary parts of <math>Z_\mathrm{S}</math>, and <math>X_\mathrm{L}</math> is the imaginary part of <math>Z_\mathrm{L}</math>.


Quantities here are given per molecule, not per mole,
To determine the values of <math>R_\mathrm{L}</math> and <math>X_\mathrm{L}</math> (since <math>V_\mathrm{S}</math>, <math>R_\mathrm{S}</math>, and <math>X_\mathrm{S}</math> are fixed) for which this expression is a maximum, we first find, for each fixed positive value of <math>R_\mathrm{L}</math>, the value of the reactive term <math>X_\mathrm{L}</math> for which the denominator
and so [[Boltzmann constant]] ''k'' and the electron charge ''e'' are used
instead of the gas constant ''R'' and Faraday's constant ''F''. To convert
to the molar quantities given in most chemistry textbooks, it is simply
necessary to multiply by Avogadro's number: <math>R = kN_A</math> and
<math>F = eN_A</math>.


The entropy of a molecule is defined as
:<math>
:<math>
S \ \stackrel{\mathrm{def}}{=}\  k \ln \Omega,
(R_\mathrm{S} + R_\mathrm{L})^2 + (X_\mathrm{S} + X_\mathrm{L})^2 \,
</math>
where <math>\Omega</math> is the number of states available to the molecule.
The number of states must vary linearly with the volume ''V'' of the
system, which is inversely proportional to the concentration ''c'', so
we can also write the entropy as
:<math>
S = k\ln \ (\mathrm{constant}\times V) = -k\ln \ (\mathrm{constant}\times c).
</math>
The change in entropy from some state 1 to another state 2 is therefore
:<math>
\Delta S = S_2 - S_1 = - k \ln \frac{c_2}{c_1},
</math>
so that the entropy of state 2 is
:<math>
S_2 = S_1 - k \ln \frac{c_2}{c_1}.
</math>
If state 1 is at standard conditions, in which <math>c_1</math> is unity (e.g.,
1 atm or 1 M), it will merely cancel the units of <math>c_2</math>. We can, therefore,
write the entropy of an arbitrary molecule ''A'' as
:<math>
S(A) = S^0(A) - k \ln [A], \,
</math>
where <math>S^0</math> is the entropy at standard conditions and [''A''] denotes the
concentration of ''A''.
The change in entropy for a reaction
:<math>
aA + bB \rightarrow yY + zZ
</math>
is then given by
:<math>
\Delta S_\mathrm{rxn} = [yS(Y) + zS(Z)] - [aS(A) +  bS(B)]
= \Delta S^0_\mathrm{rxn} - k \ln \frac{[Y]^y [Z]^z}{[A]^a [B]^b}.
</math>
We define the ratio in the last term as the [[reaction quotient]]:
:<math>
Q = \frac{\prod_j a_j^{\nu_j}}{\prod_i a_i^{\nu_i}} \approx \frac{[Z]^z [Y]^y}{[A]^a [B]^b}.
</math>
</math>


where the numerator is a product of reaction product activities, ''a<sub> j</sub>'', each raised to the power of a [[stoichiometric coefficient]], ''ν<sub> j</sub>'', and the denominator is a similar product of reactant activities. All activities refer to a time ''t''. Under certain circumstances (see [[chemical equilibrium]]) each activity term such as <math>a_j^{\nu_j}</math> may be replaced by a concentration term, [''A''].
is a minimum. Since reactances can be negative, this denominator is easily minimized by making
In an electrochemical cell, the cell potential ''E'' is the
 
chemical potential available from redox reactions (<math>E = \mu_c/e</math>).
''E'' is related to
the [[Gibbs free energy|Gibbs energy]] change <math>\Delta G</math> only by a constant:
<math>\Delta G = -nFE</math>, where ''n'' is the number of electrons transferred and <math>F</math> is the Faraday constant.
There is a negative sign because a spontaneous reaction has a negative free energy <math>\Delta G</math> and a positive potential ''E''.
The Gibbs energy is related to the entropy by <math>G = H - TS</math>, where ''H'' is
the enthalpy and ''T'' is the temperature of the system. Using these
relations, we can now write the change in
Gibbs energy,
:<math>
:<math>
\Delta G = \Delta H - T \Delta S = \Delta G^0 + kT \ln Q, \,
X_\mathrm{L} = -X_\mathrm{S}.\,
</math>
</math>
and the cell potential,
:<math>
E = E^0 - \frac{kT}{ne} \ln Q.
</math>
This is the more general form of the Nernst equation.
For the redox reaction
<math>\mathrm{Ox} + ne^- \rightarrow \mathrm{Red},</math>
<math>Q = \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}</math>, and we have:
:<math>
\begin{align}
E &= E^0 - \frac{kT}{ne} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^0 - \frac{RT}{nF} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^0 - \frac{RT}{nF} \ln Q.
\end{align}
</math>
The cell potential at standard conditions <math>E^0</math> is often
replaced by the formal potential <math>E^{0'}</math>, which includes some small
corrections to the logarithm and is the potential that is actually measured
in an electrochemical cell.


==Relation to equilibrium==
The power equation is now reduced to:


At equilibrium, ''E'' = 0 and ''Q'' = ''K''. Therefore
:<math>
:<math>
\begin{align}
P_\mathrm{L} = {1 \over 2}{{|V_\mathrm{S}|^2 R_\mathrm{L}}\over{(R_\mathrm{S} + R_\mathrm{L})^2}}\,\!
0 &= E^o - \frac{RT}{nF} \ln K\\
\ln K &= \frac{nFE^o}{RT}
\end{align}
</math>
</math>


Or at [[Standard conditions for temperature and pressure|standard temperature]],
and it remains to find the value of <math>R_\mathrm{L}</math> which maximizes this expression. However, this maximization problem has exactly the same form as in the purely resistive case, and the maximizing condition <math>R_\mathrm{L} = R_\mathrm{S}</math> can be found in the same way.
:<math>\log_{10} K = \frac{nE^o}{59.2\text{ mV}} \quad\text{at }T = 298 \text{ K}.</math>


We have thus related the [[standard electrode potential]] and the [[equilibrium constant]] of a redox reaction.
The combination of conditions


==Limitations==
*<math>R_\mathrm{L} = R_\mathrm{S}\,\!</math>
*<math>X_\mathrm{L} = -X_\mathrm{S}\,\!</math>


In dilute solutions, the Nernst equation can be expressed directly in terms of concentrations (since activity coefficients are close to unity). But at higher concentrations, the true activities of the ions must be used. This complicates the use of the Nernst equation, since estimation of non-ideal activities of ions generally requires experimental measurements.
can be concisely written with a [[complex conjugate]] (the *) as:


The Nernst equation also only applies when there is no net current flow through the electrode. The activity of ions at the electrode surface changes [[electrochemical kinetics|when there is current flow]], and there are additional [[overpotential]] and resistive loss terms which contribute to the measured potential.
:<math>Z_\mathrm{L} = Z_\mathrm{S}^*.</math>


At very low concentrations of the potential-determining ions, the potential predicted by Nernst equation approaches toward ±∞. This is physically meaningless because, under such conditions, the [[exchange current density]] becomes very low, and there is no thermodynamic equilibrium necessary for Nernst equation to hold. The electrode is called to be unpoised in such case. Other effects tend to take control of the electrochemical behavior of the system.
==Notes==
===Time dependence of the potential===
<references/>
The expression of time dependence has been established by Karaoglanoff.<ref>{{citation |first=Z. |last=Karaoglanoff |authorlink=Zakhari Karaoglanoff |title=Über Oxydations- und Reduktionsvorgänge bei der Elektrolyse von Eisensaltzlösungen |language=German |trans_title=On Oxidation and Reduction Processes in the Electrolysis of Iron Salt Solutions |journal=Zeitschrift fuer Elektrochemie |volume=12 |issue=1 |pages=5–16 |date=January 1906 |doi=10.1002/bbpc.19060120105 |url=http://onlinelibrary.wiley.com/doi/10.1002/bbpc.19060120105/abstract}}</ref><ref>{{citation |title=Electrochemical Dictionary |editor1-first=Allen J. |editor1-last=Bard |editor2-first=György |editor2-last=Inzelt |editor3-first=Fritz |editor3-last=Scholz |contribution=Karaoglanoff equation |pages=527&ndash;528 |url=http://books.google.com/books?id=4TBWg3dIyKQC&pg=PA527&lpg=PA527&hl=en&f=false |publisher=Springer}}</ref><ref>{{citation |title=Introduction to Polarography and Allied Techniques |first=Kamala |last=Zutshi |page=127–128 |url=http://books.google.com/books?id=WmiaCVH-MEIC&pg=PA127&lpg=PA127&hl=en&f=false |isbn= |year=2008}}</ref><ref>The Journal of Physical Chemistry, Volume 10, p 316. http://books.google.com/books?id=zCMSAAAAIAAJ&pg=PA316&lpg=PA316&hl=en&f=false</ref>


==Significance to related scientific domains==
==References==
The equation has been involved in the scientific controversy involving [[cold fusion]]. The discoverers of cold fusion, Fleischmann and Pons, calculated that a palladium cathode immersed in a heavy water electrolysis cell could achieve up to 10<sup>27</sup> atmospheres of pressure on the surface of the cathode, enough pressure to cause spontaneous nuclear fusion. In reality, only 10,000-20,000 atmospheres were achieved. [[John R. Huizenga]] claimed their original calculation was affected by a misinterpretation of Nernst equation.<ref>{{cite journal |separator=, | last=Huizenga | first=John R. | authorlink=John R. Huizenga | title=Cold Fusion: The Scientific Fiasco of the Century | edition=2 | location=Oxford and New York | publisher=Oxford University Press | year=1993 | pages=33, 47 | isbn=0-19-855817-1 }}</ref> He cited a paper about Pd-Zr alloys.<ref>[http://jes.ecsdl.org/content/136/3/630.abstract J. Y. Huot, et al J. Electroche. Soc 136 631 (1989)]</ref>
*H.W. Jackson (1959) Introduction to Electronic Circuits, Prentice-Hall.


==See also==
== External links ==
*[[Concentration cell]]
* [http://www.ece.rutgers.edu/~orfanidi/ewa/ch12.pdf ''Conjugate matching versus reflectionless matching''] ([[PDF]]) taken from [http://www.ece.rutgers.edu/~orfanidi/ewa/  ''Electromagnetic Waves and Antennas'']
*[[Electrode potential]]
* [http://home.freeuk.net/dunckx/wireless/maxpower1/maxpower1.html ''The Spark Transmitter. 2. Maximising Power, part 1.'']
*[[Galvanic cell]]
* [http://www.du.edu/~jcalvert/tech/jacobi.htm ''Jacobi's theorem''] - unconfirmed claim that theorem was discovered by Moritz Jacobi
*[[Goldman equation]]
* [http://chem.ch.huji.ac.il/~eugeniik/history/jacobi.html] MH Jacobi Biographical notes
*[[Membrane potential]]
* [http://spreadsheets.google.com/pub?key=pANWpjmy2L4xC96TaLF6MEA Google Docs Spreadsheet] calculating max power transfer efficiencies by Sholto Maud and Dino Cevolatti.
*[[Nernst–Planck equation]]
 
==References==
{{Reflist}}


==External links==
[[Category:Circuit theorems]]
* [http://www.nernstgoldman.physiology.arizona.edu/ Nernst/Goldman Equation Simulator]
[[Category:Electrical engineering]]
* [http://www.physiologyweb.com/calculators/nernst_potential_calculator.html Nernst Equation Calculator]
* [http://thevirtualheart.org/GHKindex.html Interactive Nernst/Goldman Java Applet]
* [http://www.doitpoms.ac.uk/tlplib/pourbaix/index.php DoITPoMS Teaching and Learning Package- "The Nernst Equation and Pourbaix Diagrams"]


[[Category:Electrochemical equations]]
[[de:Leistungsanpassung]]
[[es:Teorema de máxima potencia]]
[[fr:Adaptation d'impédances]]
[[hi:अधिकतम शक्ति का प्रमेय]]
[[pl:Dopasowanie energetyczne]]
[[fi:Maksimitehonsiirtolause]]
[[uk:Теорема максимальної потужності (теорема Якобі)]]

Revision as of 23:17, 9 August 2014

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".[1]

The theorem results in maximum power transfer, and not maximum efficiency. If the resistance of the load is made larger than the resistance of the source, then efficiency is higher, since a higher percentage of the source power is transferred to the load, but the magnitude of the load power is lower since the total circuit resistance goes up.

If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced.

The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer is always zero, regardless of the value of the load resistance.

The theorem can be extended to alternating current circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance.

Maximizing power transfer versus power efficiency

The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine.

The condition of maximum power transfer does not result in maximum efficiency. If we define the efficiency Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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Consider three particular cases:

The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity, though the total power level tends towards zero. Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit is zero.

Impedance matching

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A related concept is reflectionless impedance matching. In radio, transmission lines, and other electronics, there is often a requirement to match the source impedance (such as a transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission line.

Calculus-based proof for purely resistive circuits

(See Cartwright[2] for a non-calculus-based proof)

Circuit diagram
Circuit diagram

In the diagram opposite, power is being transferred from the source, with voltage Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. dissipated in the load is the square of the current multiplied by the resistance:

The value of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. for which the denominator

is a minimum. The result will be the same in either case. Differentiating the denominator with respect to Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.:

For a maximum or minimum, the first derivative is zero, so

or

In practical resistive circuits, Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. are both positive, so the positive sign in the above is the correct solution. To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:

This is always positive for positive values of and , showing that the denominator is a minimum, and the power is therefore a maximum, when

A note of caution is in order here. This last statement, as written, implies to many people that for a given load, the source resistance must be set equal to the load resistance for maximum power transfer. However, this equation only applies if the source resistance cannot be adjusted, e.g., with antennas (see the first line in the proof stating "fixed source resistance"). For any given load resistance a source resistance of zero is the way to transfer maximum power to the load. As an example, a 100 volt source with an internal resistance of 10 ohms connected to a 10 ohm load will deliver 250 watts to that load. Make the source resistance zero ohms and the load power jumps to 1000 watts.

In reactive circuits

The theorem also applies where the source and/or load are not totally resistive. This invokes a refinement of the maximum power theorem, which says that any reactive components of source and load should be of equal magnitude but opposite phase. (See below for a derivation.) This means that the source and load impedances should be complex conjugates of each other. In the case of purely resistive circuits, the two concepts are identical. However, physically realizable sources and loads are not usually totally resistive, having some inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.

If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle. The resultant circuit is nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This is called reactive power. Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), is essentially the same idea as complex conjugate impedance matching although it is done for entirely different reasons.

For a fixed reactive source, the maximum power theorem maximizes the real power (P) delivered to the load by complex conjugate matching the load to the source.

For a fixed reactive load, power factor correction minimizes the apparent power (S) (and unnecessary current) conducted by the transmission lines, while maintaining the same amount of real power transfer. This is done by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.

Proof

source and load impedance diagram
source and load impedance diagram

In this diagram, AC power is being transferred from the source, with phasor magnitude voltage (peak voltage) and fixed source impedance , to a load with impedance , resulting in a phasor magnitude current . is simply the source voltage divided by the total circuit impedance:

The average power dissipated in the load is the square of the current multiplied by the resistive portion (the real part) of the load impedance:

where the resistance and reactance are the real and imaginary parts of , and is the imaginary part of .

To determine the values of and (since , , and are fixed) for which this expression is a maximum, we first find, for each fixed positive value of , the value of the reactive term for which the denominator

is a minimum. Since reactances can be negative, this denominator is easily minimized by making

The power equation is now reduced to:

and it remains to find the value of which maximizes this expression. However, this maximization problem has exactly the same form as in the purely resistive case, and the maximizing condition can be found in the same way.

The combination of conditions

can be concisely written with a complex conjugate (the *) as:

Notes

  1. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  2. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

References

  • H.W. Jackson (1959) Introduction to Electronic Circuits, Prentice-Hall.

External links

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