Saturday, 5 January 2013

Susceptance

sus·cep·tance

-s

ns)

n. Electronics

The imaginary part of the complex representation of admittance.

Susceptance (symbolized B ) is an expression of the ease with which alternating current ( AC) passes through a capacitance or inductance .

http://whatis.techtarget.com/definition/susceptance

susceptance

DEFINITION

In some respects, susceptance is like an AC counterpart of direct current ( DC ) conductance . But the two phenomena are different in important ways, and they can vary independently of each other. Conductance and susceptance combine to form admittance , which is defined in terms of two-dimensional quantities known as complex number s.

When AC passes through a component that contains a finite, nonzero susceptance, energy is alternately stored in, and released from, a magnetic field or an electric field. In the case of a magnetic field, the susceptance is inductive. In the case of an electric field, the susceptance is capacitive. Inductive susceptance is assigned negative imaginary number values, and capacitive susceptance is assigned positive imaginary number values.

As the inductance of a component increases, its susceptance becomes smaller negatively (that is, it approaches zero from the negative side) in imaginary terms, assuming the frequency is held constant. As the frequency increases for a given value of inductance, the same thing happens. If L is the inductance in henries ( H ) and f is the frequency in hertz ( Hz), then the susceptance - jB _L , in imaginary-number siemens , is given by:

- jB _L = - j (6.2832 fL ) ^-1

where 6.2832 is approximately equal to 2 times pi , a constant representing the number ofradian s in a full AC cycle, and j represents the unit imaginary number (the positive square root of -1).

As the capacitance of a component increases, its susceptance becomes larger positively in imaginary terms, assuming the frequency is held constant. As the frequency increases for a given value of capacitance, the same thing happens. If C is the capacitance in farads ( F ) and f is the frequency in Hz, then the susceptance +jB _C , in imaginary-number ohms, is given by:

+jX _C = + j (6.2832 fC )

http://en.wikipedia.org/wiki/Susceptance

Susceptance

From Wikipedia, the free encyclopedia

In electrical engineering, susceptance (B) is the imaginary part of admittance. The inverse of admittance is impedance and the real part of admittance is conductance. In SI units, susceptance is measured in siemens. Oliver Heaviside first defined this property, which he called permittance, in June 1887^{[citation needed]}.

[edit]Formula

The general equation defining admittance is given by

$Y = G + j B \,$

where

Y is the admittance, measured in siemens (a.k.a. mho, the inverse of ohm).

G is the conductance, measured in siemens.

j is the imaginary unit, and

B is the susceptance, measured in siemens.

Rearranging yields

$B = \frac{Y - G} {j}$ .

But since

$\frac{1}{j} =\frac{j}{j \cdot j} = \frac{j}{-1} = -j$ ,

we obtain

$B = -j \cdot (Y -G)$ .

The admittance (Y) is the inverse of the impedance (Z)

$Y = \frac {1} {Z} = \frac {1} {R + j X} = \left( \frac {R} {R^2+X^2} \right) + j \left( \frac{-X} {R^2+X^2} \right) \,$

$B = Im(Y) = \left( \frac{-X} {R^2+X^2} \right) = \frac{-X}{|Z|^2}$

where

$Z = R + j X \,$

Z is the impedance, measured in ohms

R is the resistance, measured in ohms

X is the reactance, measured in ohms.

Note: The susceptance is the imaginary part of the admittance.

The magnitude of admittance is given by:

$\left | Y \right | = \sqrt {G^2 + B^2} \,$

Reactance

http://www.kpsec.freeuk.com/imped.htm

Reactance, X

Reactance (symbol X) is a measure of the opposition of capacitance and inductance to current. Reactance varies with the frequency of the electrical signal. Reactance is measured in ohms, symbol

There are two types of reactance: capacitive reactance (Xc) and inductive reactance (X_L).

The total reactance (X) is the difference between the two: X = X_L - Xc

Capacitive reactance, Xc

  Xc =      1      where:   Xc = reactance in ohms ()
f    = frequency in hertz (Hz)
C   = capacitance in farads (F)

2fC

Xc is large at low frequencies and small at high frequencies.
For steady DC which is zero frequency, Xc is infinite (total opposition),
hence the rule that capacitors pass AC but block DC.For example: a 1µF capacitor has a reactance of 3.2k for a 50Hz signal,
but when the frequency is higher at 10kHz its reactance is only 16.
Inductive reactance, X_L

  X_L = 2fL   where:   X_L = reactance in ohms ()
f    = frequency in hertz (Hz)
L   = inductance in henrys (H)

X_L is small at low frequencies and large at high frequencies.
For steady DC (frequency zero), X_L is zero (no opposition),
hence the rule that inductors pass DC but block high frequency AC.For example: a 1mH inductor has a reactance of only 0.3 for a 50Hz signal,
but when the frequency is higher at 10kHz its reactance is 63.

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/signals/complex/react.html

Reactance

It is fairly simple to understand what happens when we apply a signal to a resistance. This is because resistance obeys Ohm's Law. However, the behaviour of Capacitors and Inductors is rather more complicated. Fortunately, we can use complex numbers and the concept of Reactance to help us understand and analyse what happens when we apply signals to circuits which contain capacitors and/or resistors.

Reactance of a Capacitor.

Consider first what happens when we apply sinewave voltage to a capacitor.

We can represent the voltage applied to the capacitor as a complex quantity

although only the real part of this is 'visible', of course.

The current flowing 'through' the capacitor (i.e. the rate at which we have to remove charge from one plate and put it onto the other) is proportional to how quickly we are changing the voltage at any instant. We can (as in the 'First 11' section of this guide) therefore use a form of differential calculus to analyse what is happening. Here, instead, we'll just say that the Capacitor's behaviour can be defined in terms of a Reactance, X_C, whose value is

Where C is the capacitance value (in Farads) and f is the frequency of the wave we're applying.

To work out the current we can now use this in exactly the same way as we'd use the resistance of a resistor. Hence we can say that the current will be given by

Looking at this we can see that the 'j' which made it 'imaginary' has meant that the real (i.e. 'visible') part of the current is the Sin function, when the real part of the applied voltage is the Cos function. Hence we have a current which is out of step (by 90 degrees) with the applied voltage. This is the result we'd expect from using differential calculus since we want the peaks of the current waveform to occur when the voltage is changing most swiftly (when it is passing though zero). Note also that the magnitude of the current depends on the signal frequency. This is what we'd expect as a higher frequency means the voltage will have to change more rapidly.

Reactance of an Inductor

We can define the behaviour of an inductor in a similar way. In this case it is the rate of change of the current which is proportional to the applied voltage. So when we apply a voltage V_L which is exactly the same as the V_C we used earlier we get the behaviour shown in the diagram below.

To describe this behaviour can say that the reactance of the inductor, X_L, will be

where L is the inductance value in Henries.

As a result we get a current of

Comparing this result with the capacitor we can see that in both cases an input cosine voltage waveform produces a sinewave current variation, but the signs of the currents produced are different. Because of this the current through the capacitor is said to lead the applied voltage, and that through the inductor is said to lag the applied voltage. We can see why these terms are used by looking at the waveforms in the above diagrams which show the sorts of patterns we'd see using an oscilloscope. The difference arises because the capacitor reactance contains a '1/j' whereas the inductor reactance has a 'j'. As a result, when we divide the voltage by the reactance this means we get a 'jj = -1' for the sine term when using a capacitor and a 'j/j = 1' for the sine term when using an inductor. The terms, 'lead' and 'lag' are used generally to indicate the 'sign' of the reactance of a circuit (i.e. capacitive or inductive).

Impedance

http://www.kpsec.freeuk.com/imped.htm

Impedance

Impedance, Z =	V
	I

Resistance, R =	V
	I

V = voltage in volts (V)
I = current in amps (A)
Z = impedance in ohms (

)
R = resistance in ohms (

)

Impedance (symbol Z) is a measure of the overall opposition of a circuit to current, in other words: how much the circuit impedes the flow of current. It is like resistance, but it also takes into account the effects of capacitance and inductance. Impedance is measured in ohms, symbol

Impedance is more complex than resistance because the effects of capacitance and inductance vary with the frequency of the current passing through the circuit and this means impedance varies with frequency! The effect of resistance is constant regardless of frequency.

The term 'impedance' is often used (quite correctly) for simple circuits which have no capacitance or inductance - for example to refer to their 'input impedance' or 'output impedance'. This can seem confusing if you are learning electronics, but for these simple circuits you can assume that it is just another word for resistance.

Four electrical quantities determine the impedance (Z) of a circuit:
resistance (R), capacitance (C), inductance (L) and frequency (f).

Impedance can be split into two parts:

Resistance R (the part which is constant regardless of frequency)
Reactance X (the part which varies with frequency due to capacitance and inductance)

For further details please see the section on Reactance below.

The capacitance and inductance cause a phase shift* between the current and voltage which means that the resistance and reactance cannot be simply added up to give impedance. Instead they must be added as vectors with reactance at right angles to resistance as shown in the diagram.

* Phase shift means that the current and voltage are out of step with each other. Think of charging a capacitor. When the voltage across the capacitor is zero, the current is at a maximum; when the capacitor has charged and the voltage is at a maximum, the current is at a minimum. The charging and discharging occur continually with AC and the current reaches its maximum shortly before the voltage reaches its maximum: so we say the current leads the voltage.

Input Impedance Z_IN

Input impedance (Z_IN) is the impedance 'seen' by anything connected to the input of a circuit or device (such as an amplifer). It is the combined effect of all the resistance, capacitance and inductance connected to the input inside the circuit or device.It is normal to use the term 'input impedance' even for simple cases where there is only resistance and the term 'input resistance' could be used instead. In fact it is usually reasonable to assume that an input impedance is just resistance providing the input signal has a low frequency (less than 1kHz say).
The effects of capacitance and inductance vary with frequency, so if these are present the input impedance will vary with frequency. The effects of capacitance and inductance are generally most significant at high frequencies.
Usually input impedances should be high, at least ten times the output impedance of the circuit (or component) supplying a signal to the input. This ensures that the input will not 'overload' the source of the signal and reduce the strength (voltage) of the signal by a substantial amount.

Output Impedance Z_OUT

The equivalent circuit of any output

The output of any circuit or device is equivalent to an output impedance (Z_OUT) in series with a perfect voltage source (V_SOURCE). This is called the equivalent circuit and it repesents the combined effect of all the voltage sources, resistance, capacitance and inductance connected to the output inside the circuit or device. Note that V_SOURCE is usually not the same as the supply voltage Vs.It is normal to use the term 'output impedance' even for simple cases where there is only resistance and the term 'output resistance' could be used instead. In fact it is usually reasonable to assume that an output impedance is just resistance providing the output signal has a low frequency (less than 1kHz say).
The effects of capacitance and inductance vary with frequency, so if these are present the output impedance will vary with frequency. The effects of capacitance and inductance are generally most significant at high frequencies.

Usually output impedances should be low, less than a tenth of the load impedance connected to the output. If an output impedance is too high it will be unable to supply a sufficiently strong signal to the load because most of the signal's voltage will be 'lost' inside the circuit driving current through the output impedance Z_OUT. The load could be a single component or the input impedance of another circuit.

The load can be a single component or
the input impedance of another circuit

Low output impedance, Z_OUT << Z_LOAD
Most of V_SOURCE appears across the load, very little voltage is 'lost' driving the output current through the output impedance. Usually this is the best arrangement.
Matched impedances, Z_OUT = Z_LOAD
Half of V_SOURCE appears across the load, the other half is 'lost' driving the output current through the output impedance. This arrangement is useful in some situations (such as an amplifier driving a loudspeaker) because it delivers maximum power to the load. Note that an equal amount of power is wasted driving the output current through Z_OUT, an efficiency of 50%.
High output impedance, Z_OUT >> Z_LOAD
Only a small portion of V_SOURCE appears across the load, most is 'lost' driving the output current through the output impedance. This arrangement is unsatisfactory.

http://dictionary.reference.com/browse/impedance

noun

Electricity . the total opposition to alternating current by anelectric circuit, equal to the square root of the sum of thesquares of the resistance and reactance of the circuit andusually expressed in ohms. Symbol: Z

Also called mechanical impedance. Physics. the ratio of theforce on a system undergoing simple harmonic motion to thevelocity of the particles in the system.

something that impedes; an obstacle or hindrance.

Origin:
1886; impede + -ance; term introduced by O. Heaviside

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html

Impedance

While Ohm's Law applies directly to resistors in DC or in AC circuits, the form of the current-voltage relationship in AC circuits in general is modified to the form:

where I and V are the rms or "effective" values. The quantity Z is called impedance. For a pure resistor, Z = R. Because the phase affects the impedance and because the contributions of capacitors and inductors differ in phase from resistive components by 90 degrees, a process like vector addition (phasors) is used to develop expressions for impedance. More general is the complex impedancemethod.

Maximum power transfer theorem

http://en.wikipedia.org/wiki/Maximum_power_transfer_theorem

Maximum power transfer theorem

From Wikipedia, the free encyclopedia

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 themagnitude 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 AC circuits that include reactance, and states that maximum power transfer occurs when the loadimpedance is equal to the complex conjugate of the source impedance.

[hide]

[edit]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 $\eta$ 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

$\eta = {R_\mathrm{load} \over {R_\mathrm{load} + R_\mathrm{source} } } = { 1 \over { 1 + { R_\mathrm{source} \over R_\mathrm{load} } } }. \,\!$

Consider three particular cases:

If $R_\mathrm{load}=R_\mathrm{source}\,\!$ , then $\eta=0.5,\,\!$
If $R_\mathrm{load}=\infty\,\!$ or $R_\mathrm{source} = 0,\,\!$ then $\eta=1,\,\!$
If $R_\mathrm{load}=0\,\!$ , then $\eta=0.\,\!$

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 can be made close to zero. When the load resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is zero) so the efficiency is zero.

[edit]Impedance matching

Main article: impedance matching

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.

[edit]Calculus-based proof for purely resistive circuits

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

In the diagram opposite, power is being transferred from the source, with voltage $V \,\!$ and fixed source resistance $R_\mathrm{S} \,\!$ , to a load with resistance $R_\mathrm{L} \,\!$ , resulting in a current $I \,\!$ . By Ohm's law, $I \,\!$ is simply the source voltage divided by the total circuit resistance:

$I = {V \over R_\mathrm{S} + R_\mathrm{L}}. \,\!$

The power $P_\mathrm{L} \,\!$ dissipated in the load is the square of the current multiplied by the resistance:

$P_\mathrm{L} = I^2 R_\mathrm{L} = {{\left( {V \over {R_\mathrm{S} + R_\mathrm{L}}} \right) }^2} R_\mathrm{L} = {{V^2} \over {R_\mathrm{S}^2 / R_\mathrm{L} + 2R_\mathrm{S} + R_\mathrm{L}}}. \,\!$

The value of $R_\mathrm{L} \,\!$ for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of $R_\mathrm{L} \,\!$ for which the denominator

$R_\mathrm{S}^2 / R_\mathrm{L} + 2R_\mathrm{S} + R_\mathrm{L} \,\!$

is a minimum. The result will be the same in either case. Differentiating the denominator with respect to $R_\mathrm{L} \,\!$ :

${d\over{dR_\mathrm{L}}} \left( {R_\mathrm{S}^2 / R_\mathrm{L} + 2R_\mathrm{S} + R_\mathrm{L}} \right) = -R_\mathrm{S}^2 / R_\mathrm{L}^2+1. \,\!$

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

${R_\mathrm{S}^2 / R_\mathrm{L}^2} = 1 \,\!$

$R_\mathrm{L} = \pm R_\mathrm{S}. \,\!$

In practical resistive circuits, $R_\mathrm{S} \,\!$ and $R_\mathrm{L} \,\!$ 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:

${{d^2} \over {dR_\mathrm{L}^2}} \left( {R_\mathrm{S}^2 / R_\mathrm{L} + 2 R_\mathrm{S} + R_\mathrm{L}} \right) = {2 R_\mathrm{S}^2} / {R_\mathrm{L}^3}. \,\!$

This is always positive for positive values of $R_\mathrm{S} \,\!$ and $R_\mathrm{L} \,\!$ , showing that the denominator is a minimum, and the power is therefore a maximum, when

$R_\mathrm{S} = R_\mathrm{L}. \,\!$

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.

[edit]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.

[edit]Proof

In this diagram, AC power is being transferred from the source, with phasor magnitude voltage $|V_\mathrm{S}|$ (peak voltage) and fixed source impedance $Z_\mathrm{S}$ , to a load with impedance $Z_\mathrm{L}$ , resulting in a phasor magnitude current $|I|$ . $|I|$ is simply the source voltage divided by the total circuit impedance:

$|I| = { |V_\mathrm{S}| \over |Z_\mathrm{S} + Z_\mathrm{L}| }.$

The average power $P_\mathrm{L}$ dissipated in the load is the square of the current multiplied by the resistive portion (the real part) $R_\mathrm{L}$ of the load impedance:

$\begin{align} P_\mathrm{L} & = I_\mathrm{rms}^2 R_\mathrm{L} = {1 \over 2} |I|^2 R_\mathrm{L} = {1 \over 2} \left( {|V_\mathrm{S}| \over |Z_\mathrm{S} + Z_\mathrm{L}|} \right)^2 R_\mathrm{L} \\ & = {1 \over 2}{ |V_\mathrm{S}|^2 R_\mathrm{L} \over (R_\mathrm{S} + R_\mathrm{L})^2 + (X_\mathrm{S} + X_\mathrm{L})^2}, \end{align}$

where the resistance $R_\mathrm{S}$ and reactance $X_\mathrm{S}$ are the real and imaginary parts of $Z_\mathrm{S}$ , and $X_\mathrm{L}$ is the imaginary part of $Z_\mathrm{L}$ .

To determine the values of $R_\mathrm{L}$ and $X_\mathrm{L}$ (since $V_\mathrm{S}$ , $R_\mathrm{S}$ , and $X_\mathrm{S}$ are fixed) for which this expression is a maximum, we first find, for each fixed positive value of $R_\mathrm{L}$ , the value of the reactive term $X_\mathrm{L}$ for which the denominator

$(R_\mathrm{S} + R_\mathrm{L})^2 + (X_\mathrm{S} + X_\mathrm{L})^2 \,$

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

$X_\mathrm{L} = -X_\mathrm{S}.\,$

The power equation is now reduced to:

$P_\mathrm{L} = {1 \over 2}{{|V_\mathrm{S}|^2 R_\mathrm{L}}\over{(R_\mathrm{S} + R_\mathrm{L})^2}}\,\!$

and it remains to find the value of $R_\mathrm{L}$ which maximizes this expression. However, this maximization problem has exactly the same form as in the purely resistive case, and the maximizing condition $R_\mathrm{L} = R_\mathrm{S}$ can be found in the same way.

The combination of conditions

$R_\mathrm{L} = R_\mathrm{S}\,\!$
$X_\mathrm{L} = -X_\mathrm{S}\,\!$

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

$Z_\mathrm{L} = Z_\mathrm{S}^*.$

http://www.tina.com/English/tina/course/11maxim/maxim

MAXIMUM POWER TRANSFER THEOREM

Definition

Sometimes in engineering we are asked to design a circuit that will transfer the maximum power to a load from a given source. According to the maximum power transfer theorem, a load will receive maximum power from a source when its resistance (R_L) is equal to the internal resistance (R_I) of the source. If the source circuit is already in the form of a Thevenin or Norton equivalent circuit (a voltage or current source with an internal resistance), then the solution is simple. If the circuit is not in the form of a Thevenin or Norton equivalent circuit, we must first use Thevenin’s orNorton’s theorem to obtain the equivalent circuit.

Here’s how to arrange for the maximum power transfer.

1. Find the internal resistance, R_I. This is the resistance one finds by looking back into the two load terminals of the source with no load connected. As we have shown in the Thevenin’s Theorem and Norton’s Theorem chapters, the easiest method is to replace voltage sources by short circuits and current sources by open circuits, then find the total resistance between the two load terminals.

2. Find the open circuit voltage (U_T) or the short circuit current (I_N) of the source between the two load terminals, with no load connected.

Once we have found R_I,we know the optimal load resistance
(R_Lopt = R_I). Finally, the maximum power can be found

In addition to the maximum power, we might want to know another important quantity: the efficiency. Efficiency is defined by the ratio of the power received by the load to the total power supplied by the source. For the Thevenin equivalent:

and for the Norton equivalent:

Using TINA’s Interpreter, it is easy to draw P, P/P_max, and h as a function of R_L. The next graph shows P/Pmax, the power on R_L divided by the maximum power, P_max, as a function of R_L (for a circuit with internal resistance R_I=50).

Now let’s see the efficiency h as a function of R_L.

The circuit and the TINA Interpreter program to draw the diagrams above are shown below. Note that we we also used the editing tools of TINA’s Diagram window to add some text and the dotted line.

Now let’s explore the efficiency (h) for the case of maximum power transfer, where R_L = R_Th.

The efficiency is:

which when given as a percentage is only 50%. This is acceptable for some applications in electronics and telecommunication, such as amplifiers, radio receivers or transmitters However, 50% efficiency is not acceptable for batteries, power supplies, and certainly not for power plants.

Another undesirable consequence of arranging a load to achieve maximum power transfer is the 50% voltage drop on the internal resistance. A 50% drop in source voltage can be a real problem. What is needed, in fact, is a nearly constant load voltage. This calls for systems where the internal resistance of the source is much lower than the load resistance. Imagine a 10 GW power plant operating at or close to maximum power transfer. This would mean that half of the energy generated by the plant would be dissipated in the transmission lines and in the generators (which would probably burn out). It would also result in load voltages that would randomly fluctuate between 100% and 200% of the nominal value as consumer power usage varied.

To illustrate the application of the maximum power transfer theorem, let’s find the optimum value of the resistor R_L to receive maximum power in the circuit below.

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

We get the maximum power if R_L= R₁, so R_L = 1 kohm. The maximum power:

A similar problem, but with a current source:

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

Find the maximum power of the resistor R_L .

We get the maximum power if R_L = R₁ = 8 ohm. The maximum power:

The following problem is more complex, so first we must reduce it to a simpler circuit.

Find R_I to achieve maximum power transfer, and calculate this maximum power.

Click/tap the circuit above to analyze on-line or click this link to Save under Windows
First find the Norton equivalent using TINA.

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

Finally the maximum power:

{Solution by TINA's Interpreter}
O1:=Replus(R4,(R1+Replus(R2,R3)))/(R+Replus(R4,(R1+Replus(R2,R3))));
IN:=Vs*O1*Replus(R2,R3)/(R1+Replus(R2,R3))/R3;
RN:=R3+Replus(R2,(R1+Replus(R,R4)));
Pmax:=sqr(IN)/4*RN;
IN=[250u]
RN=[80k]
Pmax=[1.25m]

We can also solve this problem using one of TINA’s most interesting features, the Optimization analysis mode.

To set up for an Optimization, use the Analysis menu or the icons at the top right of the screen and select Optimization Target. Click on the Power meter to open its dialog box and select Maximum. Next, select Control Object, click on R_I,and set the limits within which the optimum value should be searched.

To carry out the optimization in TINA v6 and above, simply use the Analysis/Optimization/DC Optimization command from the Analysis menu.

In older versions of TINA, you can set this mode from the menu, Analysis/Mode/Optimization, and then execute a DC Analysis.

After running Optimization for the problem above, the following screen appears:

After Optimization, the value of RI is automatically updated to the value found. If we next run an interactive DC analysis by pressing the DC button, the maximum power is displayed as shown in the following figure.

Jana

Saturday, 5 January 2013

Susceptance

susceptance

Susceptance

[edit]Formula

Reactance

Reactance, X

Reactance

Reactance of a Capacitor.

Reactance of an Inductor

Impedance

Impedance

Input Impedance Z_IN

Output Impedance Z_OUT

im·ped·ance

Impedance

Maximum power transfer theorem

Maximum power transfer theorem

Contents

[edit]Maximizing power transfer versus power efficiency

[edit]Impedance matching

[edit]Calculus-based proof for purely resistive circuits

[edit]In reactive circuits

[edit]Proof

MAXIMUM POWER TRANSFER THEOREM

Xc =	1	where:	Xc = reactance in ohms () f = frequency in hertz (Hz) C = capacitance in farads (F)
	2fC

Saturday, 5 January 2013

Susceptance

susceptance

Susceptance

[edit]Formula

Reactance

Reactance, X

Reactance

Reactance of a Capacitor.

Reactance of an Inductor

Impedance

Impedance

Input Impedance ZIN

Output Impedance ZOUT

im·ped·ance

Impedance

Maximum power transfer theorem

Maximum power transfer theorem

Contents

[edit]Maximizing power transfer versus power efficiency

[edit]Impedance matching

[edit]Calculus-based proof for purely resistive circuits

[edit]In reactive circuits

[edit]Proof

MAXIMUM POWER TRANSFER THEOREM

Input Impedance Z_IN

Output Impedance Z_OUT