Wednesday, January 25, 2012

What is Gradient ? - Basic understanding

What is Gradient ?

Gradient is not just a slope of of a certain function. It is an operator to change a scalar field to a vector field. A scalar field has no direction like temperature, density, energy, and Electric Potential of Voltage. It seems simple as you do not have to analyze it (a scalar field) in terms of direction, just it exists there or anywhere around us. But it is sometime inconvenient when you want to analyze it or simply want to know more about it .

Wiki (26-Jan-2012) introduces gradient as

"
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

"

It is more like a definition, not an explanation to those who do not know gradient well.

Wiki further introduces some well quoted examples as below:

"
Consider a room in which the temperature is given by a scalar field, $T$ , so at each point $(x, y, z)$ the temperature is $T (x, y, z)$ . (We will assume that the temperature does not change over time.) At each point in the room, the gradient of $T$ at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

"

This is the same as the above brief explanation. No enough explanation to those who do not know gradient well. Please note that time is not involved and only in terms of location (x,y,z) in this explanation.

Wiki continues

"
Consider a surface whose height above sea level at a point $(x, y)$ is $H (x, y)$ . The gradient of $H$ at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.
"

Again, the same thing, but slightly different from the case of temperature of a room.

Temperature of a room - Temperature (scalar field) in 3D (room) --- (A)
Height (scalar) to or from the see level. The see level is regarded as constant --- (B)

(B) is confusing when we compare with the case of (A), in which a scalar quantity of the temperature can be measured at any point and may differ in 3D space (x, y, z). $T (x, y, z).$ While in (B) the height is the quantity in z-direction.

$(A) T (x, y, z) may have some or any value (say 20 deg C or 68 deg F, or simply 1, 2, 3 ...)$
$(B)$ $H (x, y) has some value of height (say 100m or 328 feet, or simply$ $1, 2, 3 ...)$

$But the problem is, can we write in the following ways ?$

$(A) T (x, y, z) = 20 (68)$
$(B) H (x, y) = 100 (328)$

$Yes but this is one type. We can use any value as$

$(A) T (x, y, z) = c (constant)$
$(B) H (x, y) = c (constant)$
$(A) is a bit difficult to visualize to express 'c'.$

$We can use a certain color in 3D space where 'c' occupies. When 'c' changes we use different colors or different tones of gray.$
$While (B) is easier to visualize as we can express it$ $H (x, y) = z or$ $H (x, y, z) = z$

$Level line and Level surface$
$This is an important concept to capture the concept of gradient.$

$(B) H (x, y) = z or$ $H (x, y, z) = z (z = constant)$
$We can plot the points which show a certain constant z value as height above the x-y plane in the 3D space. It will be a line or curve. We also can plot these on$ $the$ $2D x-y plane though there is no height.$ $It will be a line or curve called <Level line> showing the places in x-y plane having the selected same level (value) in the direction of z (height).$ $z (height) can take any value but should remain constant in the function. This is the point.$

$(A) T (x, y, z) = c (constant)$

$We can apply a similar thing to this equation. Now z in a variable of the function, not a constant. The result we will get will be a surface in x-y-z 3D space showing the selected same level (value) of < T > and mostly a curved surface in 3D. Now I think that you can visualize this.$

How about the following cases ?

A (x, 0) = c or A (x, c) = c (constant)   This means x = c
A (0, y) = c or A (c, y) = c (constant)   This means y = c

We can plots <level points>, which produces a straight line on the x-y plane with x being constant for any y value, a vertical straight line, parallel with the y-axis; and the one with y being constant for any x value, a horizontal straight line, parallel with the x-axis. If c = 0, then the line is either y-axis or x-axis itself.

So we can find some relations between dimensions (Ds).

0D (point) --> 1D (line) A (x, 0) = c or A (x, c) = c (constant)
$We can plot$ $Level points in 1D space (on a straight line).$
$Notes:$
$1) x can be either separate spots. But cannot be continuous (which is a line).$
$2) We can say that x depends on c, actually x = c.$

1D --> 2D   $H (x, y) = c or$ $H (x, y, c) = c$ $(z = constant)$
$We can plot$ $Level lines (usually curves) in 2D space (on a plane).$
$Notes:$
$1) H generally indicates a function of x and y here with a constant c.$
$2) We can say that x and y values depend on c, actually H(x,y) = c. But c is constant (can be changed but not a variable element of the function). So c restricts the freedom of variables x and y.$

$Example$

$xy = 1 or H(x,y) - 1 = 0$

x =1, then y =1
x= 2, then y= 1/2
x= 3, then y= 1/3

$These are on the line of y = 1/x.$

$x+y = 0 or F(x,y) = 0$

x =1, then y =-1
x= 2, then y= -2
x= 3, then y= -3

These are on the line of y = -x

Seems stupidly simple but important relations as you can see later.

2D --> 3D $T (x, y, z) = c (constant)$
$We can plot$ $Level surfaces$ $(usually a curved surface) in 3D space.$

Notes

1) $Like above c is constant (can be changed but not a variable element of the function). So c restricts the freedom of variables x, y and z.$

$Derivative$

Wiki shows the calculation formulas of gradient.

Expression in 3-dimensional rectangular coordinates

The form of the gradient depends on the coordinate system used. In Cartesian coordinates, the above expression expands to

$\nabla f(x, y, z) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$ --- (1)

which is often written using the standard unit vectors $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$ as

$\frac{\partial f}{\partial x} {\mathbf{\hat i}} + \frac{\partial f}{\partial y} \hat{\mathbf{j}} + \frac{\partial f}{\partial z} \hat{\mathbf{k}}$ ---- (2)

(1) is a partial differentiation of a function f in the three mutually perpendicular directions, x, y and z - ∂f/∂x, ∂f/∂y and ∂f/∂y. Is this a vector function showing a related vector field? It does not seem so as no vector shown. It can be a vector (function) as it shows a derivative of f (scalar function) in each direction of x, y, z. Meanwhile (2) seems more like a vector as three unit vectors are shown and please note <+> sign, which means Grad f is a addition of each partial differentiation of a function f with respect to x, y, and z and multiplied by a i, j, and k unit vector respectively. Grad f (x, y, z) is a addition of of ∂f/∂x, ∂f/∂y and ∂f/∂y.

However, even if you already know the partial differentiation it is difficult to get the idea of the following underlined italic part intuitively.

"
the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

"

Simply why the greatest ?

Some more explanations are needed.

1. What is differentiation? or What is derivative?

Spatially, the derivative is a rate change of a function with respect to a certain direction.

y = 1, the derivative is 0. Though 'x' is not shown in this equation but it is usually regarded as the rate change of y of this function with respect to 'unit x' is 0.
where 'unit x' is the direction in the x-axis and the value is one (1).

Besides 1, any other constant like 1/2, 2,10, the derivative of

y = 1/2, 2, 10 etc

all 0 (zeer). This should be noticed in that derivative of a function having a independent constant the constant portion disappeared. So a constant of the scalar field function does not affect the derivative.
y = x, the derivative is 1. The rate change of y of this function with respect to 'unit x' is 1, constant, no change at any x and the y values on the line of y = x (straight line).

y= x², the derivative is 2x. This time the rate change of 'y' value to 'unit x' of this function is not a number (constant) but a function "y = 2x" with respect to 'unit x'. So

when x = 0, y = 0   --> the derivative at (0, 0) is 0. (y' = 2 x 0 = 0)
when x = 1, y = 1 --> the derivative at (1, 1) is 2. (y' = 2 x 1 = 2)
when x = 2, y = 4 --> the derivative at (2, 4) is 4. (y = 2 x 2 = 4)
when x = 10, y = 100 --> the derivative at (10, 100) is 20. (y' = 2 x 10 = 20)

The above numbers 0, 2, 4, 20 are the tangent value of the tangent line at each point.

How about the first part - direction - the direction of the greatest rate of increase of the scalar field ? How to get this direction? By differentiation (or getting derivative) as well. This is the point of understanding gradient.

The 2nd part of the definition of gradient " (the) magnitude is that rate of increase (or decrease as well or more generally, just change)" is referring to this. So differentiation (partial differentiation in case of gradient) of a scalar function makes sense to get this value.

We have seen the two types of equations above.

y = x
y= x²

y is dependent on x. But we can change them to

x = y
x = +/- √ y

Now x depends on y. But these are actually the same as far as the x-y relation is concerned.

$H (x, y) = c (constant)$
$T (x, y, z) = c (constant)$

$These two are different. x, y, z are independent and controlled by the function of H or T to make the equation correct.$

1) y = x
2) y= x²

We already had derivative rate change of y with respect to x.

1)-a) y' = 1
2)-a) y' = 2x

Another way, we change 1) and 2)

1)-b) y - x = 0
2)-b) y - x² = 0

Then differentiate with respect to x

1)-c) dy/dx - dx/dx = d0/dx
dy/dx - 1= 0
dy/dx = 1 (same result as y' = 1)

2)-c) dy/dx - dx²/dx = d0/dx
    dy/dx - 2x = 0
          dy/dx = 2x (same result as y' = 2x)

The above two are just another presentation of differentiating with respect to x. But we can apply this method to

$H (x, y) = c (constant)$
$T (x, y, z) = c (constant)$

H' $(dx/dx, dy/dx) = dc/dx = 0$ $with respect to x$
H' $(dx/dy, dy/dy) = dc/d = 0$ $with respect to y$

$T' (dx /dx, d y/dx, dz/ dx) = dc/dx = 0$ $with respect to x$
$T' (dx /dy, d y/dy, dz/ dy) = dc/dy= 0$ $with respect to y$
$T' (dx /dz, d y/dz, dz/ dz) = dc/dz= 0$ $with respect to z$

This procedure is wrong. We do not have functions of H and T and x, y and z are values, not equations.

$H (x, y) = c (constant)$
$We applied this to the following very simple equation.$

$y = x$

Then change this to

H (x, y) = y - x = 0
$Which means$
$The function H of x and y is$

$y - x = 0$
$We can write$

$H (x, y) = 0$
$Then let's apply the gradient operation (scalar to vector)$
$grad$ $H (x, y) = ($ $- dx/dx$ $) i + (dy/dy) j = -1 i + 1$ $j (=0)$

$What does this mean? We seem to be getting closer to <$ $the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase>, at least the direction and some magnitude, the greatest is another issue.$

$<$ $-1 i + 1$ $j > is a vector with the direction and we can calculate its magnitude.$

$Direction: j / - i or - (j / i)$

       \      l
       \    l
           \ l ^
<-- i     \l $l j$
$----------------------$
              l \
              l \
              l    \
              l      \

Magnitude: $\sqrt 2$

The direction is normal to the line of y = x or <x - y = 0>. Let' try one more.

$H (x, y) = y -$ $x 2 (=0)$
$which means$
$The function H of x and y is$

$y - x 2 = 0$

$We can write$

$H (x, y) = 0$

Applying the $gradient operation (scalar to vector)$
$grad$ $H (x, y) = - ($ $dx 2 /dx$ $) i + (dy/dy) j = -2x i + 1$ $j$ $(=0)$

$This is a function of x. The value of x can change.$
$At (0 ,0) : -2 x 0$ $i + 1$ $j =$ $0$ $i + 1$ $j = +$ $1 j Magnitude 1 -- (a)$

$At (1, 1) : -2$ $i + 1 j$ $Magnitude$ $\sqrt 5 -- (b)$

$At (-1, 1) : -2 x (-1)$ $i + 1 j = 2$ $i + 1$ $j$ $Magnitude \sqrt 5 -- (c)$
$At (2, 4): -2 x 2$ $i + 1 j$ $= -4$ $i + 1$ $j$ $Magnitude \sqrt 17 -- (d)$
$At (-2, 4) : -2 x (-2)$ $j$ $= 4$ $i + 1$ $j$ $Magnitude \sqrt 17 -- (e)$

$At each point (0, 0), (1, 1), (2, 4), (-2, 4) of <$ $y - x 2 = 0> we can get the direction and calculate the magnitude of of the vector <grad H (x, y)>. Let's check them if they are correct or not.$
Now we refer to the concept of tangent and the tangent line. Tangent line at any point of y= x² shows the slope of y= x² at this point (the value of y with respect to x = 1 (unit)) of y= x²). For instance,

when x = 0 (then y = 0) the tangent or the slope of the tangent line at (0, 0) = 0 and
the tangent line is y = 0.   --- (a')

When x = 1 (then y = 1), the tangent at (1, 1) is 2 and the tangent line is y = 2x - 1.    --- (b')

When x = -1 (then y = 1), the tangent at (-1, 1) is -2 and the tangent line is y = -2x - 1.    --- (c')

When x = 2 (then y = 4), the tangent at (2, 4) is 4 and the tangent line is y = 4x - 4.    --- (d')

When x = -2 (then y = 4), the tangent at (-2, 4) is -4 and the tangent line is y = -4x - 4.    --- (e')

(a) and (a'), (b) and (b'), (c) and (c'), (d) and (d'), and (e) and (e') are apparently closely related.

(a) and (a')

At (0, 0)

The tangent line is y = 0 while the direction of the gradient vector is $+ 1$ $j .$

             l
             l ^
             l l   $j$
------------------- y= 0

(b) and (b')

At (1, 1)
The tangent line is y = 2x - 1 while the direction of the gradient vector $-2$ $i + 1 j is perpendicular to$
y = 2x - 1.

At (-1, 1)
The tangent line is y = -2x - 1 while the direction of the gradient vector $2$ $i + 1 j is perpendicular to$
y = -2x - 1.

At (2, 4)
The tangent line is y = 4x - 4 while the direction of the gradient vector $-4$ $i + 1$ $j$ $is perpendicular to$
y = 4x - 4.

At (-2, 4)
The tangent line is y = -4x - 4 while the direction of the gradient vector $4$ $i + 1$ $j$ $is perpendicular to$
y = -4x - 4.

What does this (perpendicular) relation mean ? Also <Simply why the greatest ?>

Continue to the next post.

sptt

Friday, January 20, 2012

What is Electric Current? -2

What is Electric Current? - Continued

Wiki (20-Jan-2012) explains Electric Current as

"
For a steady flow of charge through a surface, the current I (in amperes) can be calculated with the following equation:

$I = {Q \over t} \, ,$

where Q is the electric charge transferred through the surface over a time t. If Q and t are measured in coulombs and seconds respectively, I is in amperes.
More generally, electric current can be represented as the rate at which charge flows through a given surface as:

$I = \frac{\mathrm{d}Q}{\mathrm{d}t} \, .$

"

This is very common explanation. But how many people understand the meaning of this? It seems that we must memorize these equations without understanding what these mean or even without being able to visualize it.

Without understanding the first equation

$I = {Q \over t} \, ,$

it is further more difficult to understand its generalization.

$I = \frac{\mathrm{d}Q}{\mathrm{d}t} \, .$

This explanation is like one for those who already know well the meaning of Electric Current.

One more

Hyperphiscs' (20-Jan-2012) explanation

"

Electric Current

Electric current is the rate of charge flow past a given point in an electric circuit, measured in Coulombs/second which is named Amperes.

"

Again how many people understand the meaning of "the rate of charge of flow past a given point in an electric circuit" ? This time "past a given point" not "through a surface".

Wiki explains Electric Current in another way (aspect) in Drift Speed (the same site as the above).

"
Drift speed

The mobile charged particles within a conductor move constantly in random directions, like the particles of a gas. In order for there to be a net flow of charge, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the opposite direction of the electric field. The speed at which they drift can be calculated from the equation:

$I=nAvQ \, ,$ or v = I/nAQ

where

I

is the electric current

n

is number of charged particles per unit volume (or charge carrier density)

A

is the cross-sectional area of the conductor

v

is the drift velocity, and

Q

is the charge on each particle.

"

In this,

I is the total quantity of charges have gone through

the cross-sectional area of the conductor during the time (t) with a velocity of

v (m/t), which means the

total quantity of charges in a volume created by the t

he cross-sectional area (

A) and the distance

v x t ((m/t) x t). This is easier to understand because this is a fact or an observation of a fact - generalization is not require. Generalization processes

1)

First generalization Hyperphisics says

" the rate of charge of flow past a given point in an electric circuit". To get this we must divide the both side by / m³

$I=nAvQ \, ,$

I / m³= nAvQ / m³= nQAv / m³
As A in ＜m²＞ and v in ＜m/t＞ in unit
nQAv / m³= nQA ＜m²＞v ＜m/t＞ / m³= nQ/t
nQ is the total Q (Charges) and change it Qtotal. Then we can get I /m³ = Qtotal/t
But what does the right hand side I /m³ means? Not I but I /m³.
This is the first generalization. I /m³is the density of I (total current passed), or we can say an average of I ( the flow of charges past a given point). We also call it the rate of charge of flow past a given point or the density of current in terms of time (the time dimension in stead of the spatial dimensions as a point is zero dimension at least mathematically).

Similarly we can think of the average I through a surface by dividing the both sides of the equation by m².
I / m²= nAvQ /m²= nQAv / m²
nQAv / m²= nQA＜m²＞v ＜m/t＞ / m²= nQm/t = (also = nQ

v

)

This time the right hand side is not nQ/t but nQm/t or nQ

v

(where

v

is velocity).

We do not elaborate this here.
nQAv / m²= nQm/t
As far as the rate or the average are concerned (average over area versus (nQ/t) x distance)
nQAv = nQ/t ---- Generalized I = nQ/t = Qtotal/t

Again similarly we can think of the average I through a line (or maybe more precisely on or in a line) by dividing the both side of the equation by m.
I / m= nAvQ /m= nQAv / m
nQAv / m= nQA＜m²＞v ＜m/t＞ / m= nQm²/t
This time the right hand side is nQm²/t. What does m²/t means? We do not elaborate it here.
nQAv / m= nQm²/t
As far as the rate or the average are concerned (average over line versus (nQ/t) x area)
nQAv = nQm²/t ---- Generalized I = nQ/t = Qtotal/t

The above calculations seem redundant. " I = Q/t " is a definition.

2)

Second generalization

As we quoted above, Wiki says

More generally, electric current can be represented as the rate at which charge flows through a given surface as:

$I = \frac{\mathrm{d}Q}{\mathrm{d}t} \, .$

$I = {Q \over t} \, ,$ a steady flow of charge through a surface

we must generalize this under a more generalized a changing flow. Steady state is included in changing state. Steady state is a special case of changing state when the rate of change is zero. So

$I = \frac{\mathrm{d}Q}{\mathrm{d}t} \, .$

and in a more general way, not "charge flows through a given surface" (Wiki) but " the rate of charge of flow past a given point" (Hyperphysics). Quantity at an absolute point could be averaged by spatial dimensions but could be averaged by the time dimension (t).

This reminds me of Divergence (Theory) but Divergence does treat a flow not involve time - so a kind of static flow, then leads to the density of something in a flow or rather flux form.

sptt

Thursday, January 19, 2012

What is Electric Current?

What is Electric Current actually ?

The name "current" indicates that Electric Current is a flow of something. Not a flow of current. A flow of electrons, ions or some other very tiny things or liquid like water? In Electronics it is a flow of electric charges. Charge is not a thing, but a property measurable and related with force (electrical force). Charges are said to be "carried" by electrons, ions or some other tiny things. In Electronics, however, current is more precisely defined as the quantity of charges passed through an area, or more generally through a point (any point in a volume, or area or line) per time. Or more generally by including of the changing quantity with time, it is a rate change of the quantity of charges which go through a point with time.

The following article (form physicsclassroom.com) explains Electric Current in a similar way.

http://www.physicsclassroom.com/class/circuits/u9l2c.cfm

Electric Current

If the two requirements of an electric circuit are met, then charge will flow through the external circuit. It is said that there is a current - a flow of charge. Using the word current in this context is to simply use it to say that something is happening in the wires - charge is moving. Yet current is a physical quantity that can be measured and expressed numerically. As a physical quantity, current is the rate at which charge flows past a point on a circuit. As depicted in the diagram below, the current in a circuit can be determined if the quantity of charge Q passing through a cross section of a wire in a time t can be measured. The current is simply the ratio of the quantity of charge and time.

http://www.physicsclassroom.com/class/circuits/u9l2c.cfm

The article says

As a physical quantity, current is the rate at which charge flows past a point on a circuit.

But it does not elaborate charge flows past a point and changes to the quantity of charge Q passing through a cross section of a wire in a time t by using a diagram. In terms of the rate, this seems the same thing, either past a point or through a cross section.

A small cross section can be regarded as a point? - Yes and No.

Yes - A limit of an area (volume and line as well) is a point.

No - A point is zero dimension while an area is two (2) dimensions.

Yes and No

When the quantity of charge Q passing through a cross section in a time t divided by the area, we can get an average of the quantity of charge Q passing through a cross section. And this average can be regarded as the charge Q passing through a point.
sptt

Sunday, January 15, 2012

Mystery of "V x I" - 2 or What Electric Energy is?

This is a continuation of the previous post titled Mystery of "Power = V x I" with some additional information and thinking.

Mystery of "Power = V x I" was posted before as

"
V (Joule/Coulomb) x I (Coulomb/second) = Power (Joule/sec = Energy/sec). So after multiplication of V by I, Coulomb (unit of electric charge) disappears like a magic and only Energy and time remain. This explains well that Current is not consumed in a load such as a resistor and a light bulb - Kirchhoff's Current law but Energy is consumed, changed to some other form of Energy - conservation of Energy.

"

Question - Why Charges disappear after I x V (= Power) and Energy

remains ?

Assumption 1 - Charge only carries Energy.

One example

Electricity and the Electron

Next Page: Series and Parallel Connections
Also see: Circuit Symbols and Circuit Diagrams

What is electricity?

Electricity is the flow of charge around a circuit carrying energy from the battery (or power supply) to components such as lamps and motors.

Electricity can flow only if there is a complete circuit from the battery through wires to components and back to the battery again.

The diagram shows a simple circuit of a battery, wires, a switch and a lamp. The switch works by breaking the circuit.

With the switch open the circuit is broken - so electricity cannot flow and the lamp is off.

With the switch closed the circuit is complete - allowing electricity to flow and the lamp is on. The electricity is carrying energy from the battery to the lamp.

We can see, hear or feel the effects of electricity flowing such as a lamp lighting, a bell ringing, or a motor turning - but we cannot see the electricity itself, so which way is it flowing?

-----

This (Electricity is the flow of charge around a circuit carrying energy) explains "Current is not consumed in a load such as a resistor and a light bulb - Kirchhoff's Current law" but does not explain how current (charges) carry energy. No equation shows the relation between electricity and energy. Charges seem to be carriers of energy. No definitions on what electricity is, what electric energy is.

Assumption 2 - Charge creates Energy by moving (mechanical Energy).

Current is a flow (motion) of charges.

$E_{kinetic} = \tfrac 1 2 m v^2 \,\!$

To move charges potential energy from some energy source like a battery (chemical energy) , capacitor (stored energy (1/2) C V² ) and inductor (stored energy (1/2) L I² ) is required and used.

An electron has a charge (−1.602176565(35)×10⁻¹⁹ C) as well as a mass (9.10938291(40)×10⁻³¹ kg). It flows so have velocity. How fast? changing direction?

1) DC current in a conductor or somewhere in a conductive material in a circuit is not very fast and may not necessarily move straight (not 1D but 3D)

2) AC current moves back and forth in a circuit and changes direction slowly or fast with different frequencies.

3) More generally electron move dynamically according to the electromagnetic field conditions.

4) Electron spins which requires mechanical energy.

5) Electron is not a rigid body. $E_{kinetic} = \tfrac 1 2 m v^2 \,\!$ cannot be applied to electron.

6) Electron motion or behavior or status in terms of energy is believed be more precisely described by quantum properties.

sptt

Friday, January 6, 2012

One very easy way to understand Solar Cell key function

I have found a very good way to understand the basic key function (or phenomenon) of a Solar Cell. I have read some documents on the function (or phenomenon) of a Solar Cell in some books and some websites (some are good, some are not good or even wrong at least confusing). But I did not encounter my method so far.

To check Diode Voltage by a multimeter (my digital multimeter has this function) with the opposite direction of the +/- shown on the back side of the panel at not very bright place. My multimeter shows "----", which means more than the max checkable (about 2VDC). Then gradually give more light to the panel. The multimeter shows "2.1- 1.9V", which shows . And then further increase the light. It shows less and less values 1.8V -- 1.5V -- 1.2V -- and finally 0.00. This indicates the increasing light creates an increasing voltage at the PN junction opposite to the original PN junction voltage (under a little light condition). This is it. Very easy and you can easily guess what light does to the panel (Solar Cell). After I found this fact it was much easier for me to understand what the text books say like "the current in Solar cell is the reverse current." or "Internal Voltage Difference Creation", etc.

sptt

Saturday, August 20, 2011

Thermistors - a full of formulas

The explanations found in the documents provided by NTC Thermistor companies are more or less the same and usually a full of mathematics formulas.

I have used the following materials:

1) Mitsubishi Materials (not available any more)
2) wiki <Thermistor>
3) wiki <Temperature coefficient>
4) EPCOS - NTC Thermistors, General technical information
5) NTC Thermistors (Bowthorpe Thermometrics, etc)
6) NTC Thermistor theory (BetaTHERM Sensors)

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1) Mitsubishi Materials (not available any more)

My additional explanations and comments are made in bold Italic.
■ Resistance - temperature characteristic
The resistance and temperature characteristics of a thermistor can be approximated by equation 1.

(eq1) R=R_o exp {B(1/T -1 /T_o)}

R	: resistance at absolute temperature T(K)
Ro	: resistance at absolute temperature To(K)
B	: B value
*T(K) = t(ºC) + 273.15 The B value for the thermistor characteristics is not fixed, but can vary by as much as 5K/ºC according to the material composition. Therefore equation 1 may yield different results from actual values if applied over a wide temperature range.

By taking the B value in equation 1 as a function of temperature, as shown in equation 2, the difference with the actual value can be minimized.

(eq2) B_T= CT²+ DT + E

C, D, and E are constants.

The B value distribution caused by manufacturing conditions will change the constant E, but will have no effect on constants C or D. This means, when taking into account the distribution of B value, it is enough to do it with the constant E only.

Calculation for constants C, D and E
Using equations 3~6, constants C, D and E can be determined through four temperature and resistance value data points (T₀, R₀). (T₁, R₁). (T₂, R₂) and (T₃, R₃).
With equation 3, B₁, B₂ and B₃, can be determined from the resistance values for To and T₁, T₂, T₃ and then substituted into the equations below.

Sorry, the equations 3~6 are no more available now.

Example

Using a resistance-temperature characteristic chart, the resistance value over the range of 10ºC~30ºC is sought for a thermistor with a resistance of 5kΩ and a B value deflection of 50K at 25ºC.

Process

(1) Determine the constants C, D and E from the resistance-temperature chart.

T_o = 25+273.15, T₁ = 10+273.15, T₂= 20+273.15, T₃= 30+273.15

(2) B_T = CT² + TD + E + 50 ; substitute the value into equation and solve for B_T

(3) R= 5exp {B_T (1/T - 1/298.15)} ; substitute the values into equation and solve for R

*T : 10+273.15 ~ 30+273.15

Results of plotting the resistance temperature ■ Characteristics are shown figure 1 (not available any more)

The vertical axis is the base 10 log measure, i.e. log10R/R25 while the horizontal axis is 1/T (reciprocal of the temperature). This corresponds with (eq1) R=R_o exp {B(1/T-1/T_o)}. I.e. log10R/R25 = B(1/T-1/T25). Please note the linear relation - straight lines in figure 1.

■ Resistance temperature coefficient

The resistance-temperature coefficient (α) is defined as the rate of change of the zero-power resistance associated with a temperature variation of 1ºC at any given temperature.
The relationship between the resistance-temperature coefficient (α) and the B value can be obtained by differentiating equation 1 above.

1/R x dR/dT is a part of the definition of (α).

can be obtained by differentiating equation 1 above. - How?

(eq1) R = R_o exp {B(1/T - 1/T_o)} -----> ln R/R_{o =}ln {B(1/T-1/T_o)}

By differentiating the both sides - with respect to T or simply differentiating the both sides

R/R_{o =}B(1/T - 1/T_o)

(1/T - 1/T_o)_{is simplified to or regard as}d(1/T) or 1/dT, which is -1/T²

R/R_{o =}-B /T²

d/dT (R/R_o)is considered as (1/R) x (dR/dT) since R is Resistance at T and R_omeans Resistance at T_o

According to the definition of (α) --- 1/R x dR/dT

Or simply d/dT (R/R_o) is dR. dR/dT has a unit of Ohm so should be x 1/R to make a ratio (α), which has no unit.

A negative value signifies that the rated zero-power resistance decreases.

■ Heat dissipation constant (JIS C2570-1)
The dissipation constant (δ) indicates the power necessary for increasing the temperature of the thermistor element by 1ºC through self-heating in a heat equilibrium.
Applying a voltage to a thermistor will cause an electric current to flow, leading to a temperature rise in the thermistor. This " intrinsic heating " process is subject to the following relationship among the thermistor temperature T₁, ambient temperature T₂, and consumed power P.

Measuring conditions for all parts in this catalog are as follows:

(1) Room temp is 25ºC

(2) Axial and radial leaded parts were measured in their shipping condition.

■ Maximum power dissipation (JIS C2570-1)
The power rating is the maximum power for a continuous load at the rated temperature.
For parts in this catalog, the value is calculated from the following formula using 25ºC as the ambient temperature.
(formula) Rated power=heat dissipation constant × (maximum operating temperature-25ºC)
In the detail specification, it is likely to write by "Power rating"that is a past name.
■ Permissible operating power
Definition : The power to reach the maximum operating temperature through self heating when using a thermistor for temperature compensation or as a temperature sensor. (No JIS definition exists.) The permissible operating power, when t ºC is the permissible temperature rise, can be calculated using the following formula.

Permissible operating power= t*heat dissipation constant

■ Thermal time constant by ambient temperature change (JIS C2570-1)
A constant expressed as the time for the temperature at the electrodes of a thermistor, with no load applied, to change to 63.2% of the difference between their initial and final temperatures, during a sudden change in the surrounding temperature.

When the surrounding temperature of the thermistor changes from T₁ to T₂, the relation between the elapsed time t and the thermistor's temperature T can then be expressed by the following equation.

T = (T₁- T₂) exp(-t/τ) + T₂
(T₂ -T₁){1 - exp(-t/τ)} + T₁

The constant t is called the heat dissipation constant.

If t = τ, the equation becomes : (T - T₁) / (T₂-T₁) = 0.632

In other words, the above definition states that the thermal time constant is the time it takes for the temperature of the thermistor to change by 63.2% of its initial temperature difference.
The rate of change of the thermistor temperature versus time is shown in table 1. (not available any more)

Measuring conditions for parts in this catalog are as follows:

(1) Part is moved from a 50ºC environment to a still air 25ºC environment until the temperature of the thermistor reaches 34.2ºC.

(2) Axial and radial leaded parts are measured in their shipping form

Please note, the thermal dissipation constant and thermal time constant will vary according to environment and mounting conditions.

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Thermistors - a full of formulas

The vertical axis is the base 10 log measure, i.e. log10R/R25 while the horizontal axis is 1/T (reciprocal of the temperature). This corresponds with (eq1) R=Ro exp {B(1/T-1/To)}. I.e. log10R/R25 = B(1/T-1/T25). Please note the linear relation - straight lines in figure 1.

■ Resistance temperature coefficient

The vertical axis is the base 10 log measure, i.e. log10R/R25 while the horizontal axis is 1/T (reciprocal of the temperature). This corresponds with (eq1) R=R_o exp {B(1/T-1/T_o)}. I.e. log10R/R25 = B(1/T-1/T25). Please note the linear relation - straight lines in figure 1.