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The construction of a diode involves joining an anode and a cathode two pieces of material with different electrical charges. The remaining supply voltage is dropped across the resistor which is there to limit the current through the diode so the power rating is not exceeded.

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It would appear to mefrom the charts on that page that the forward voltage drop would change from about 07 volts at the 20ma current to about 085volt at the 150ma current level.

Voltage drop across a diode. Diodes are devices that pass electric current in one direction only and therefore may be thought of as a kind of one way valve for electrons. Without this voltage the diode would not meet its threshold voltage needed to conduct current. A silicon diode will have a voltage drop of approximately 07v across its leads.
If the diode was in backwards ideally no current would flow through it so it would be open and the there would be no voltage drop. Because the load is in parallel with the zener diode the voltage drop across the load is the same as the zener voltage which is 68 v. It is evident that the notion of a constant forward voltage drop on these diodes is a considerable oversimplification and somewhat misleading.
A diode is a device that drops a specific voltage across its terminals. It would be like measuring before and after the diode with nothing being there. It looks like this diode is in a circuit with something else as well and the voltage drop is 2 volts.
The voltage drop across a forward biased diode varies only a little with the current and is a function of temperature. This effect can be used as a temperature sensor or as a voltage reference. It is necessary that a diode get this voltage so that the diode can conduct.
For silicon diodes which are generally the most common the diode forward voltage drop is about 07 volts v. Forward voltage for a diode varies only slightly for changes in forward current and temperature. When voltage is applied across a diode in such a way that the diode prohibits current the diode is said to be reverse biased.
Also diodes high resistance to current flowing in the reverse direction suddenly drops to a low resistance when the reverse voltage across the diode reaches a value called the breakdown voltage. The voltage dropped across a conducting forward biased diode is called the forward voltage. Although a diode voltage drop can refer to either the diodes forward or reverse voltage drop it typically describes the forward voltage drop.

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