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### Re: What's on youw CTRL+V?

Posted: April 1st, 2019, 5:47 pm
Due to a conflict with scouts on April 28 and Graduation on March 11th, I have changed Ferret Face's Eagle Ceremony to July 38th at 5:00 pm. I hope everyone can make this date. Invites will go out soon. Thanks. Jolene

{Names and dates were changed to protect the "innocent"...}

### Re: What's on your CTRL+V?

Posted: April 9th, 2019, 12:40 pm
did someone order a gnome

### Re: What's on your CTRL+V?

Posted: April 10th, 2019, 10:44 am
RC Differentiator
The passive RC differentiator is a series connected RC network that produces an output signal which corresponds to the mathematical process of differentiation.

For a passive RC differentiator circuit, the input is connected to a capacitor while the output voltage is taken from across a resistance being the exact opposite to the RC Integrator Circuit.

A passive RC differentiator is nothing more than a capacitance in series with a resistance, that is a frequency dependant device which has reactance in series with a fixed resistance (the opposite to an integrator). Just like the integrator circuit, the output voltage depends on the circuits RC time constant and input frequency.

Thus at low input frequencies the reactance, Xc of the capacitor is high blocking any d.c. voltage or slowly varying input signals. While at high input frequencies the capacitors reactance is low allowing rapidly varying pulses to pass directly from the input to the output.

This is because the ratio of the capacitive reactance (Xc) to resistance (R) is different for different frequencies and the lower the frequency the less output. So for a given time constant, as the frequency of the input pulses increases, the output pulses more and more resemble the input pulses in shape.

We saw this effect in our tutorial about Passive High Pass Filters and if the input signal is a sine wave, an rc differentiator will simply act as a simple high pass filter (HPF) with a cut-off or corner frequency that corresponds to the RC time constant (tau, τ) of the series network.

Thus when fed with a pure sine wave an RC differentiator circuit acts as a simple passive high pass filter due to the standard capacitive reactance formula of Xc = 1/(2πƒC).

But a simple RC network can also be configured to perform differentiation of the input signal. We know from previous tutorials that the current through a capacitor is a complex exponential given by: iC = C(dVc/dt). The rate at which the capacitor charges (or discharges) is directly proportional to the amount of resistance and capacitance giving the time constant of the circuit. Thus the time constant of a RC differentiator circuit is the time interval that equals the product of R and C. Consider the basic RC series circuit below.

RC Differentiator Circuit
rc differentiator circuit

For an RC differentiator circuit, the input signal is applied to one side of the capacitor with the output taken across the resistor, then VOUT equals VR. As the capacitor is a frequency dependant element, the amount of charge that is established across the plates is equal to the time domain integral of the current. That is it takes a certain amount of time for the capacitor to fully charge as the capacitor can not charge instantaneously only charge exponentially.

We saw in our tutorial about RC Integrators that when a single step voltage pulse is applied to the input of an RC integrator, the output becomes a sawtooth waveform if the RC time constant is long enough. The RC differentiator will also change the input waveform but in a different way to the integrator.

Resistor Voltage
resistor voltage
We said previously that for the RC differentiator, the output is equal to the voltage across the resistor, that is: VOUT equals VR and being a resistance, the output voltage can change instantaneously.

However, the voltage across the capacitor can not change instantly but depends on the value of the capacitance, C as it tries to store an electrical charge, Q across its plates. Then the current flowing into the capacitor, that is it depends on the rate of change of the charge across its plates. Thus the capacitor current is not proportional to the voltage but to its time variation giving: i = dQ/dt.

As the amount of charge across the capacitors plates is equal to Q = C x Vc, that is capacitance times voltage, we can derive the equation for the capacitors current as:

Capacitor Current
rc differentiator capacitor current

Therefore the capacitor current can be written as:

capacitor current

As VOUT equals VR where VR according to ohms law is equal too: iR x R. The current that flows through the capacitor must also flow through the resistance as they are both connected together in series. Thus:

rc differentiator output voltage

Thus the standard equation given for an RC differentiator circuit is:

RC Differentiator Formula
rc differentiator formula

Then we can see that the output voltage, VOUT is the derivative of the input voltage, VIN which is weighted by the constant of RC. Where RC represents the time constant, τ of the series circuit.

Single Pulse RC Differentiator
When a single step voltage pulse is firstly applied to the input of an RC differentiator, the capacitor “appears” initially as a short circuit to the fast changing signal. This is because the slope dv/dt of the positive-going edge of a square wave is very large (ideally infinite), thus at the instant the signal appears, all the input voltage passes through to the output appearing across the resistor.

charging capacitor voltage
After the initial positive-going edge of the input signal has passed and the peak value of the input is constant, the capacitor starts to charge up in its normal way via the resistor in response to the input pulse at a rate determined by the RC time constant, τ = RC.

As the capacitor charges up, the voltage across the resistor, and thus the output decreases in an exponentially way until the capacitor becomes fully charged after a time constant of 5RC (5T), resulting in zero output across the resistor. Thus the voltage across the fully charged capacitor equals the value of the input pulse as: VC = VIN and this condition holds true so long as the magnitude of the input pulse does not change.

If now the input pulse changes and returns to zero, the rate of change of the negative-going edge of the pulse pass through the capacitor to the output as the capacitor can not respond to this high dv/dt change. The result is a negative going spike at the output.

discharging capacitor voltage
After the initial negative-going edge of the input signal, the capacitor recovers and starts to discharge normally and the output voltage across the resistor, and therefore the output, starts to increases exponentially as the capacitor discharges.

Thus whenever the input signal is changing rapidly, a voltage spike is produced at the output with the polarity of this voltage spike depending on whether the input is changing in a positive or a negative direction, as a positive spike is produced with the positive-going edge of the input signal, and a negative spike produced as a result of the negative-going input signal.

Thus the RC differentiator output is effectively a graph of the rate of change of the input signal which has no resemblance to the square wave input wave, but consists of narrow positive and negative spikes as the input pulse changes value.

By varying the time period, T of the square wave input pulses with respect to the fixed RC time constant of the series combination, the shape of the output pulses will change as shown.

RC Differentiator Output Waveforms
rc differentiator output waveforms

Then we can see that the shape of the output waveform depends on the ratio of the pulse width to the RC time constant. When RC is much larger (greater than 10RC) than the pulse width the output waveform resembles the square wave of the input signal. When RC is much smaller (less than 0.1RC) than the pulse width, the output waveform takes the form of very sharp and narrow spikes as shown above.

So by varying the time constant of the circuit from 10RC to 0.1RC we can produce a range of different wave shapes. Generally a smaller time constant is always used in RC differentiator circuits to provide good sharp pulses at the output across R. Thus the differential of a square wave pulse (high dv/dt step input) is an infinitesimally short spike resulting in an RC differentiator circuit.

Lets assume a square wave waveform has a period, T of 20mS giving a pulse width of 10mS (20mS divided by 2). For the spike to discharge down to 37% of its initial value, the pulse width must equal the RC time constant, that is RC = 10mS. If we choose a value for the capacitor, C of 1uF, then R equals 10kΩ.

For the output to resemble the input, we need RC to be ten times (10RC) the value of the pulse width, so for a capacitor value of say, 1uF, this would give a resistor value of: 100kΩ. Likewise, for the output to resemble a sharp pulse, we need RC to be one tenth (0.1RC) of the pulse width, so for the same capacitor value of 1uF, this would give a resistor value of: 1kΩ, and so on.

RC Differentiator Example
rc differentiator example

So by having an RC value of one tenth the pulse width (and in our example above this is 0.1 x 10mS = 1mS) or lower we can produce the required spikes at the output, and the lower the RC time constant for a given pulse width, the sharper the spikes. Thus the exact shape of the output waveform depends on the value of the RC time constant.

RC Differentiator Summary
We have seen here in this RC Differentiator tutorial that the input signal is applied to one side of a capacitor and the the output is taken across the resistor. A differentiator circuit is used to produce trigger or spiked typed pulses for timing circuit applications.

When a square wave step input is applied to this RC circuit, it produces a completely different wave shape at the output. The shape of the output waveform depending on the periodic time, T (an therefore the frequency, ƒ) of the input square wave and on the circuit’s RC time constant value.

When the periodic time of the input waveform is similar too, or shorter than, (higher frequency) the circuits RC time constant, the output waveform resembles the input waveform, that is a square wave profile. When the periodic time of the input waveform is much longer than, (lower frequency) the circuits RC time constant, the output waveform resembles narrow positive and negative spikes.

The positive spike at the output is produced by the leading-edge of the input square wave, while the negative spike at the output is produced by the falling-edge of the input square wave. Then the output of an RC differentiator circuit depends on the rate of change of the input voltage as the effect is very similar to the mathematical function of differentiation.

### Re: What's on your CTRL+V?

Posted: April 10th, 2019, 12:59 pm
Oh my....

### Re: What's on your CTRL+V?

Posted: April 11th, 2019, 3:54 pm
An organ pipe 2.7 m long is open at one end and closed at the other end.

### Re: What's on your CTRL+V?

Posted: April 11th, 2019, 4:35 pm
An organ pipe 2.7 m long is open at one end and closed at the other end.
Lemme guess, you're studying frequencies, overtones, and harmonics?

### Re: What's on your CTRL+V?

Posted: April 11th, 2019, 4:36 pm
1. Between 1945 and 1970, virtually all European colonies achieved independence. Discuss the changes within Europe that contributed to this development. (2002)

(yay, AP Euro History is much fun .... )

### Re: What's on your CTRL+V?

Posted: April 11th, 2019, 9:23 pm
An organ pipe 2.7 m long is open at one end and closed at the other end.
Lemme guess, you're studying frequencies, overtones, and harmonics?
yeep that was some homework from yesterday lol

### Re: What's on your CTRL+V?

Posted: April 12th, 2019, 1:54 am
Being able to interpret how topographic features fit together in the landscape gives the user a much deeper understanding of the lay of the land and how to identify where they are and how to move through it.

On topographic maps, contours represent the shape of the land. Contour lines fit together in many different ways, and they form shapes which can be recognised by the user.

Features of the landscape that are useful to know are:

Elevation and steepness
Hills, valleys, depression, gullies, ridges
ELEVATION AND SLOPE
UNDERSTANDING HOW ELEVATION AND SLOPE ARE DEPICTED ON TOPOGRAPHIC MAPS
Elevation and slope are the two elements that determine how landforms physically appear and connect.

Elevation
The ‘contour interval’ – the elevation between contours – is the vertical distance between adjacent contour lines. On 1:25,000 maps usually used by bushwalkers, contours are either 10 or 20 m apart.
elevation
For measuring between contour lines see here.

Slope (Steepness)
The rate of rise or fall of a terrain feature is known as its slope. The speed at which a bushwalking group can move is affected by the slope of the ground or terrain features. This slope can be determined from the map by studying the contour lines—the closer the contour lines, the steeper the slope; the farther apart the contour lines, the gentler the slope. Totally flat ground has no contour lines.

Four types of slopes that concern bushwalkers are gentle, steep, concave, and convex.

Gentle: Contour lines showing a uniform, gentle slope will be evenly spaced and wide apart. Easy walking.
gentle-full
Steep: Contour lines showing a uniform, steep slope on a map will be evenly spaced, but close together. Very challenging, or impossible walking (i.e. contour lines may be so close that they create an impassable cliff line).
steep-full
Concave: Contour lines showing a concave slope on a map will be closely spaced at the top of the terrain feature and widely spaced at the bottom. Bushwalkers going up the slope will find the terrain increasingly steep and challenging.
concave-full
Convex: Contour lines showing a convex slope on a map will be widely spaced at the top and closely spaced at the bottom. Bushwalkers going down the slope cannot observe most of the slope or the terrain at the bottom, so extra care must be taken when route finding.
convex-full
COMMON TERRAIN FEATURES
UNDERSTANDING HOW COMMON TERRAIN FEATURES ARE DEPICTED ON TOPOGRAPHIC MAPS
All terrain features are derived from a complex landmass known as a ridgeline, not to be confused with a ridge.

The US Army states that “A ridgeline is a line of high ground, usually with changes in elevation along its top and low ground on all sides from which a total of 10 natural or constructed terrain features are classified”.[1] By comparison, a ridge is a sloping line of high ground.

Major terrain features include hills, saddles, gullies, ridges, and depressions, and they each have characteristic contour lines that make it easy to pick them out in the landscape.

Hills, peaks, knolls, mountains: A hill, peak, knoll or mountain is an area of high ground. From a hilltop, the ground slopes down in all directions. A hill is shown on a map by contour lines forming concentric circles. The inside of the smallest closed circle is the hilltop.
Hill = an area of high ground; generally, a smaller and rounder than a mountain, and less steep.
Knoll = small, rounded natural hill.
Mountain = a very tall hill, generally with a minimum size of 600m, but varies around the world.
Peak = a mountain with a pointed top.
Munro = a Scottish mountain taller than 3,000 feet (914 m).
hill-full
Saddle: A saddle is a dip or low point between two areas of higher ground. A saddle is not necessarily the lower ground between two hilltops; it may be simply a dip or break along a level ridge crest. When standing in a saddle, there is high ground in two opposite directions and lower ground in the other two directions. A saddle typically looks like an hourglass.
Gully: a gully is a stretched-out groove in the land, usually formed by a watercourse, and has high ground on three sides. Depending on its size and location water sometimes flows through it, from high to low. Contour lines forming a gully are either U-shaped or V-shaped. To determine the direction water is flowing, look at the contour lines. The closed end of the contour line (U or V) always points upstream or toward high ground. A valley is a large gully, often very flat, wide and open with a large watercourse running through it.
gully-full
Ridge: a ridge is a sloping line of high ground. When standing on the centerline of a ridge, there is usually low ground in three directions and high ground in one direction with varying degrees of slope. When crossing a ridge at right angles, there is a steep climb to the crest and then a steep descent to the base. When moving along the path of the ridge, depending on the geographic location, there may be either an almost unnoticeable slope or a very visible incline. Contour lines forming a ridge tend to be U-shaped or V-shaped. The closed end of the contour line points away from high ground.
ridge-full
On a map, a ridge is depicted as two contour lines (often of the same contour) running side by side at the same elevation for some distance. When the lines diverge, the ridge is either flattening out to a high plateau or continues to rise with additional contour lines. When the lines converge, the ridge is falling in elevation, creating a spur.

Closed contour loops represent hills or bumps along the ridgeline.
ridgeline_annotated
Spur: A spur is a short, continuous sloping line of higher ground, normally jutting out from the side of a ridge. A spur is often formed by two roughly parallel streams cutting draws down the side of a ridge. The ground will slope down in three directions and up in one. Contour lines on a map depict a spur with the U or V pointing away from high ground.
spur-full
Depression: A depression is a low point in the ground or a sinkhole. It could be described as an area of low ground surrounded by higher ground in all directions, or simply a hole in the ground. Usually, only depressions that are equal to or greater than the contour interval will be shown. On maps, depressions are represented by closed contour lines that have tick marks pointing toward low ground.
pit-full
Cliff: A cliff is a vertical or near vertical feature; it is an abrupt change of the land. When a slope is so steep that the contour lines converge into one “carrying” contour of contours, this last contour line sometimes has tick marks pointing toward low ground (image below). Cliffs are also shown by contour lines very close together and, in some instances, touching each other.
cliff-full
Topographic maps cannot always be used to identify cliffs, however, particularly on those with 20m contour intervals, and hence some steep areas require careful negotiation.
TIPS, TRICKS AND COMMON MISTAKES
SOME TIPS, TRICKS AND COMMON MISTAKES TO AVOID WHEN READING TOPOGRAPHIC MAPS
The real art of map reading comes with interpreting how individual landscape features fit together in the terrain: saddles connect ridges to knolls to cliffs; gullies form into rivers and valleys. Interpreting how contour lines fit together helps understand the lay of the land and be able to navigate through it.

The big picture
1 - hill, 2 - valley, 3 - ridge, 4 - saddle, 5 - depression, 6 - gully, 7 - spur, 8 - cliff, 9 - cut, 10 - fill
1 – hill, 2 – valley, 3 – ridge, 4 – saddle, 5 – depression, 6 – gully, 7 – spur, 8 – cliff, 9 – cut, 10 – fill

This image describes a landscape by contours. In words:
Running east to west across the complex landmass is a ridgeline. A ridgeline is a line of high ground, usually with changes in elevation along its top and low ground on all sides. The changes in elevation are the three hilltops and two saddles along the ridgeline. From the top of each hill, there is lower ground in all directions. The saddles have lower ground in two directions and high ground in the opposite two directions. The contour lines of each saddle form half an hourglass shape. Because of the difference in size of the higher ground on the two opposite sides of a saddle, a full hourglass shape of a saddle may not be apparent.

There are four prominent ridges. A ridge is on each end of the ridgeline, and two ridges extend south from the ridgeline. All of the ridges have lower ground in three directions and higher ground in one direction. The closed ends of the U’s formed by the contour lines point away from higher ground.

To the south lies a valley; the valley slopes downward from east to west. Note that the U of the contour line points to the east, indicating higher ground in that direction and lower ground to the west. Another look at the valley shows high ground to the north and south of the valley.

Just east of the valley is a depression. Looking from the bottom of the depression, there is higher ground in all directions.

Several spurs extend south from the ridgeline. They, like ridges, have lower ground in three directions and higher ground in one direction. Their contour line U’s point away from higher ground. Between the ridges and spurs are draws. They, like valleys, have higher ground in three directions and lower ground in one direction. Their contour line U’s and V’s point toward the higher ground.

Two contour lines on the north side of the centre hill are touching or almost touching. They have ticks indicating a vertical or nearly vertical slope or a cliff.

The road cutting through the eastern ridge depicts cuts and fills. The breaks in the contour lines indicate cuts, and the ticks pointing away from the road bed on each side of the road show fills.

Common mistakes
Here are some tips and tricks to identify between standard features.

Spur vs. gully: Contour lines on a map depict a spur with the U or V pointing away from the high ground; for a gully, the closed end of the contour line (U or V) always points upstream or toward high ground.
Spur
Spur

Gully
Gully

Knoll vs. depression: for knolls, contour lines form concentric circles, and there is lower ground all around, whereas depressions have closed contour lines with tick marks pointing toward the low ground.
Knoll
Knoll

Depression
Depression

Saddle vs. ridge: When standing in a saddle, there is high ground in two opposite directions and lower ground in the other two directions. When standing on the centerline of a ridge, there is usually low ground in three directions and high ground in one direction with varying degrees of slope. Be careful not to confuse ‘ridge’ with ‘ridgeline’ here: a ridgeline is a line of high ground, which can rise and fall through saddle features.
ridgeline_annotated2
Practicing
Map reading takes practice. One of the easiest ways to do this is to become aware of the shape of the surrounding land at all times, even when driving and walking through an urban area. Most navigation and map reading is about matching up the form of the land with that on the map. Practice recognising and naming key features (knoll, hill, spur, ridge, cliff, valley, etc.). Take maps on all bushwalks and follow the route on the map, even if it’s well signposted. Look at the map regularly and match it with the surrounding landscape.

Some bushwalkers enjoy taking part in Rogaine competitions to improve their navigation. The Bushwalkers Wilderness Rescue Squad also runs an annual navigation competition in NSW.

SUBTLE FEATURES
RECOGNISING SUBTLE FEATURES ON TOPOGRAPHIC MAPS
There are subtleties to map reading that take time to develop. Some common things to watch out for include:

Implied knoll: A bump or small hill that is too small to generate it’s own closed loop contour. Occurs in places where two ridge lines diverge and converge or on the top of a hill where the contour lines are furthest apart.
implied_knoll
Implied saddle: The opposite of an implied knoll. An implied saddle is a saddle that is not formed enough to have two parallel contours cross the ridgeline and connect.
Minor gully: A dent in the side of a slope or ridge often too high for a proper water course to form.
subtlegully2
Minor spur or outcrop: a bump on the side of a ridge or slope that’s too localised or flat to form a proper spur.
minor_spur

### Re: What's on your CTRL+V?

Posted: April 12th, 2019, 4:48 am
RC Differentiator
The passive RC differentiator is a series connected RC network that produces an output signal which corresponds to the mathematical process of differentiation.

For a passive RC differentiator circuit, the input is connected to a capacitor while the output voltage is taken from across a resistance being the exact opposite to the RC Integrator Circuit.

A passive RC differentiator is nothing more than a capacitance in series with a resistance, that is a frequency dependant device which has reactance in series with a fixed resistance (the opposite to an integrator). Just like the integrator circuit, the output voltage depends on the circuits RC time constant and input frequency.

Thus at low input frequencies the reactance, Xc of the capacitor is high blocking any d.c. voltage or slowly varying input signals. While at high input frequencies the capacitors reactance is low allowing rapidly varying pulses to pass directly from the input to the output.

This is because the ratio of the capacitive reactance (Xc) to resistance (R) is different for different frequencies and the lower the frequency the less output. So for a given time constant, as the frequency of the input pulses increases, the output pulses more and more resemble the input pulses in shape.

We saw this effect in our tutorial about Passive High Pass Filters and if the input signal is a sine wave, an rc differentiator will simply act as a simple high pass filter (HPF) with a cut-off or corner frequency that corresponds to the RC time constant (tau, τ) of the series network.

Thus when fed with a pure sine wave an RC differentiator circuit acts as a simple passive high pass filter due to the standard capacitive reactance formula of Xc = 1/(2πƒC).

But a simple RC network can also be configured to perform differentiation of the input signal. We know from previous tutorials that the current through a capacitor is a complex exponential given by: iC = C(dVc/dt). The rate at which the capacitor charges (or discharges) is directly proportional to the amount of resistance and capacitance giving the time constant of the circuit. Thus the time constant of a RC differentiator circuit is the time interval that equals the product of R and C. Consider the basic RC series circuit below.

RC Differentiator Circuit
rc differentiator circuit

For an RC differentiator circuit, the input signal is applied to one side of the capacitor with the output taken across the resistor, then VOUT equals VR. As the capacitor is a frequency dependant element, the amount of charge that is established across the plates is equal to the time domain integral of the current. That is it takes a certain amount of time for the capacitor to fully charge as the capacitor can not charge instantaneously only charge exponentially.

We saw in our tutorial about RC Integrators that when a single step voltage pulse is applied to the input of an RC integrator, the output becomes a sawtooth waveform if the RC time constant is long enough. The RC differentiator will also change the input waveform but in a different way to the integrator.

Resistor Voltage
resistor voltage
We said previously that for the RC differentiator, the output is equal to the voltage across the resistor, that is: VOUT equals VR and being a resistance, the output voltage can change instantaneously.

However, the voltage across the capacitor can not change instantly but depends on the value of the capacitance, C as it tries to store an electrical charge, Q across its plates. Then the current flowing into the capacitor, that is it depends on the rate of change of the charge across its plates. Thus the capacitor current is not proportional to the voltage but to its time variation giving: i = dQ/dt.

As the amount of charge across the capacitors plates is equal to Q = C x Vc, that is capacitance times voltage, we can derive the equation for the capacitors current as:

Capacitor Current
rc differentiator capacitor current

Therefore the capacitor current can be written as:

capacitor current

As VOUT equals VR where VR according to ohms law is equal too: iR x R. The current that flows through the capacitor must also flow through the resistance as they are both connected together in series. Thus:

rc differentiator output voltage

Thus the standard equation given for an RC differentiator circuit is:

RC Differentiator Formula
rc differentiator formula

Then we can see that the output voltage, VOUT is the derivative of the input voltage, VIN which is weighted by the constant of RC. Where RC represents the time constant, τ of the series circuit.

Single Pulse RC Differentiator
When a single step voltage pulse is firstly applied to the input of an RC differentiator, the capacitor “appears” initially as a short circuit to the fast changing signal. This is because the slope dv/dt of the positive-going edge of a square wave is very large (ideally infinite), thus at the instant the signal appears, all the input voltage passes through to the output appearing across the resistor.

charging capacitor voltage
After the initial positive-going edge of the input signal has passed and the peak value of the input is constant, the capacitor starts to charge up in its normal way via the resistor in response to the input pulse at a rate determined by the RC time constant, τ = RC.

As the capacitor charges up, the voltage across the resistor, and thus the output decreases in an exponentially way until the capacitor becomes fully charged after a time constant of 5RC (5T), resulting in zero output across the resistor. Thus the voltage across the fully charged capacitor equals the value of the input pulse as: VC = VIN and this condition holds true so long as the magnitude of the input pulse does not change.

If now the input pulse changes and returns to zero, the rate of change of the negative-going edge of the pulse pass through the capacitor to the output as the capacitor can not respond to this high dv/dt change. The result is a negative going spike at the output.

discharging capacitor voltage
After the initial negative-going edge of the input signal, the capacitor recovers and starts to discharge normally and the output voltage across the resistor, and therefore the output, starts to increases exponentially as the capacitor discharges.

Thus whenever the input signal is changing rapidly, a voltage spike is produced at the output with the polarity of this voltage spike depending on whether the input is changing in a positive or a negative direction, as a positive spike is produced with the positive-going edge of the input signal, and a negative spike produced as a result of the negative-going input signal.

Thus the RC differentiator output is effectively a graph of the rate of change of the input signal which has no resemblance to the square wave input wave, but consists of narrow positive and negative spikes as the input pulse changes value.

By varying the time period, T of the square wave input pulses with respect to the fixed RC time constant of the series combination, the shape of the output pulses will change as shown.

RC Differentiator Output Waveforms
rc differentiator output waveforms

Then we can see that the shape of the output waveform depends on the ratio of the pulse width to the RC time constant. When RC is much larger (greater than 10RC) than the pulse width the output waveform resembles the square wave of the input signal. When RC is much smaller (less than 0.1RC) than the pulse width, the output waveform takes the form of very sharp and narrow spikes as shown above.

So by varying the time constant of the circuit from 10RC to 0.1RC we can produce a range of different wave shapes. Generally a smaller time constant is always used in RC differentiator circuits to provide good sharp pulses at the output across R. Thus the differential of a square wave pulse (high dv/dt step input) is an infinitesimally short spike resulting in an RC differentiator circuit.

Lets assume a square wave waveform has a period, T of 20mS giving a pulse width of 10mS (20mS divided by 2). For the spike to discharge down to 37% of its initial value, the pulse width must equal the RC time constant, that is RC = 10mS. If we choose a value for the capacitor, C of 1uF, then R equals 10kΩ.

For the output to resemble the input, we need RC to be ten times (10RC) the value of the pulse width, so for a capacitor value of say, 1uF, this would give a resistor value of: 100kΩ. Likewise, for the output to resemble a sharp pulse, we need RC to be one tenth (0.1RC) of the pulse width, so for the same capacitor value of 1uF, this would give a resistor value of: 1kΩ, and so on.

RC Differentiator Example
rc differentiator example

So by having an RC value of one tenth the pulse width (and in our example above this is 0.1 x 10mS = 1mS) or lower we can produce the required spikes at the output, and the lower the RC time constant for a given pulse width, the sharper the spikes. Thus the exact shape of the output waveform depends on the value of the RC time constant.

RC Differentiator Summary
We have seen here in this RC Differentiator tutorial that the input signal is applied to one side of a capacitor and the the output is taken across the resistor. A differentiator circuit is used to produce trigger or spiked typed pulses for timing circuit applications.

When a square wave step input is applied to this RC circuit, it produces a completely different wave shape at the output. The shape of the output waveform depending on the periodic time, T (an therefore the frequency, ƒ) of the input square wave and on the circuit’s RC time constant value.

When the periodic time of the input waveform is similar too, or shorter than, (higher frequency) the circuits RC time constant, the output waveform resembles the input waveform, that is a square wave profile. When the periodic time of the input waveform is much longer than, (lower frequency) the circuits RC time constant, the output waveform resembles narrow positive and negative spikes.

The positive spike at the output is produced by the leading-edge of the input square wave, while the negative spike at the output is produced by the falling-edge of the input square wave. Then the output of an RC differentiator circuit depends on the rate of change of the input voltage as the effect is very similar to the mathematical function of differentiation.
What'd you have an entire Geologic Map/ Road Scholar CS copied for?