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electrical_engineering_and_electronics_2:block11 [2026/04/11 07:13] – created mexleadminelectrical_engineering_and_electronics_2:block11 [2026/05/18 03:07] (current) – mexleadmin
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     - MOSFET (structure, comparison with bipolar transistor)     - MOSFET (structure, comparison with bipolar transistor)
     - Optional: Transistor as an amplifier     - Optional: Transistor as an amplifier
 +
 +
 +====== Block 11 — Semiconductor Fundamentals and Diodes ======
 +
 +===== Learning objectives =====
 +<callout>
 +After this 90-minute block, you can
 +
 +  * distinguish conductors, semiconductors, and insulators using the band model.
 +  * explain intrinsic conduction, electron conduction, and hole conduction.
 +  * explain how n-doping and p-doping change the number of mobile charge carriers.
 +  * describe the formation of a pn junction and the depletion region.
 +  * decide whether a diode is forward-biased or reverse-biased from \(u_{\rm AK}\).
 +  * compare the ideal diode model, the constant-voltage model, and the piecewise-linear diode model.
 +  * use the diode equation
 +\[
 +\begin{align*}
 +i_{\rm D}=I_{\rm S}(T)\left({\rm e}^{\frac{u_{\rm AK}}{mU_{\rm T}}}-1\right)
 +\end{align*}
 +\]
 +at a qualitative level.
 +  * calculate simple diode operating points with a series resistor.
 +  * identify basic diode types such as universal diodes, Z-diodes, and LEDs.
 +</callout>
 +
 +===== 90-minute plan =====
 +
 +  * **Warm-up (10 min):**
 +    * Why does a diode conduct in one direction but not in the other?
 +    * Recall from EEE1: voltage, current direction, power, and resistors.
 +    * Recall from EEE2: transient overvoltages at inductive loads will later need diode protection.
 +
 +  * **Core concepts (55 min):**
 +    * Conductors, semiconductors, insulators, and the band gap.
 +    * Intrinsic conduction, electron conduction, hole conduction.
 +    * Doping: n-type and p-type material.
 +    * pn junction, depletion region, diffusion voltage.
 +    * Diode operation in forward and reverse direction.
 +    * Ideal and real diode characteristics.
 +    * Practical diode models for circuit calculations.
 +
 +  * **Practice (20 min):**
 +    * Determine diode polarity and conduction state.
 +    * Calculate current with a constant-voltage diode model.
 +    * Estimate differential resistance at a given operating point.
 +    * Compare ideal and real diode assumptions.
 +
 +  * **Wrap-up (5 min):**
 +    * Key messages: pn junction, forward/reverse bias, current limiting, diode models.
 +    * Preview: rectifiers, smoothing, protection circuits, LEDs, and Z-diode stabilizers in [[block12|Block 12]].
 +
 +===== Conceptual overview =====
 +<callout icon="fa fa-lightbulb-o" color="blue">
 +  * A semiconductor is neither a good conductor nor a perfect insulator. Its conductivity can be controlled by material, temperature, light, and doping.
 +  * A diode is a pn junction with two terminals:
 +    * **anode A** on the p-side,
 +    * **cathode K** on the n-side.
 +  * In forward direction, the external voltage reduces the depletion region and current can flow.
 +  * In reverse direction, the depletion region becomes wider and only a very small leakage current flows, until breakdown occurs.
 +  * A diode is nonlinear. It is not a resistor.
 +  * In circuits, diode current must usually be limited by another component, often a resistor.
 +</callout>
 +
 +<panel type="info" title="Scope of this block">
 +This block explains **why** diodes behave as they do and how we model them.
 +
 +Diode applications such as
 +
 +  * rectifiers,
 +  * smoothing capacitors,
 +  * freewheeling diodes,
 +  * input protection circuits,
 +  * LED circuits,
 +  * Z-diode voltage stabilizers
 +
 +are continued in [[block12|Block 12]].
 +</panel>
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 +
 +===== Core content =====
 +
 +==== Conductors, semiconductors, and insulators ====
 +
 +Materials differ strongly in their specific resistance \(\rho\).
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_band_model|Band model for conductors, semiconductors, and insulators.></imgcaption>
 +{{drawio>block11_band_model_overview.svg}}
 +</panel>
 +</WRAP>
 +
 +In the band model, two energy ranges are especially important:
 +
 +  * the **valence band**, where electrons are bound,
 +  * the **conduction band**, where electrons can move through the crystal.
 +
 +The energy gap between them is called the **band gap** \(E_{\rm g}\).
 +
 +<tabcaption tab_band_gap|Qualitative band model>
 +
 +^ Material type ^ Band model ^ Electrical behavior ^
 +| conductor | conduction band available or overlapping | many mobile charge carriers |
 +| semiconductor | small band gap, typically a few \({\rm eV}\) | conductivity can be controlled |
 +| insulator | large band gap | almost no mobile charge carriers |
 +
 +<panel type="info" title="Physical picture">
 +A semiconductor can be imagined as a parking garage with two floors.
 +
 +  * The lower floor is almost full: the valence band.
 +  * The upper floor allows movement: the conduction band.
 +  * The band gap is the energy needed to move an electron to the upper floor.
 +
 +Doping adds useful “parking spots” or “missing spots” so that charge transport becomes much easier.
 +</panel>
 +
 +==== Intrinsic conduction, electrons, and holes ====
 +
 +In a pure semiconductor, some electrons can gain enough energy to leave their bonds. Then
 +
 +  * the electron becomes mobile in the conduction band,
 +  * a missing electron remains in the valence band,
 +  * this missing electron behaves like a positive mobile charge carrier.
 +
 +The missing electron is called a **hole**.
 +
 +<callout>
 +There are two types of mobile charge carriers in semiconductors:
 +
 +  * **electrons** with negative charge,
 +  * **holes** with positive effective charge.
 +</callout>
 +
 +<panel type="info" title="Analogy: empty seat in a lecture hall">
 +Imagine a fully occupied row of seats.  
 +If one student moves to the right into an empty seat, the empty seat appears to move to the left.
 +
 +The empty seat is not a real object, but it behaves as if it moves.  
 +A hole in a semiconductor is similar: it is a missing electron, but it behaves like a positive moving charge carrier.
 +</panel>
 +
 +==== Doping: n-type and p-type semiconductors ====
 +
 +Doping means adding a very small amount of foreign atoms to the semiconductor crystal.
 +
 +<WRAP group>
 +<WRAP column half>
 +<panel type="default">
 +<imgcaption fig_n_doping|N-doping: donor atoms provide additional mobile electrons.></imgcaption>
 +{{:circuit_design:ndoping.svg?500}}
 +</panel>
 +</WRAP>
 +
 +<WRAP column half>
 +<panel type="default">
 +<imgcaption fig_p_doping|P-doping: acceptor atoms create additional holes.></imgcaption>
 +{{:circuit_design:pdoping.svg?500}}
 +</panel>
 +</WRAP>
 +</WRAP>
 +
 +<tabcaption tab_doping|Doping of silicon>
 +
 +^ Doping type ^ Typical dopant atoms ^ Main mobile charge carriers ^ Name of dopant ^
 +| n-type | phosphorus, arsenic, antimony | electrons | donors |
 +| p-type | boron, aluminium, indium | holes | acceptors |
 +
 +<callout>
 +Doping does **not** mean that the semiconductor becomes strongly charged as a whole.  
 +The crystal is still approximately electrically neutral.  
 +Doping mainly changes how many mobile charge carriers are available.
 +</callout>
 +
 +==== The pn junction ====
 +
 +A diode is formed when p-doped and n-doped regions meet.
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_pn_junction|Diode symbol and pn junction with anode A and cathode K.></imgcaption>
 +{{:circuit_design:pnjunction.svg?650}}
 +</panel>
 +</WRAP>
 +
 +At the junction:
 +
 +  * electrons diffuse from the n-side into the p-side,
 +  * holes diffuse from the p-side into the n-side,
 +  * electrons and holes recombine,
 +  * a region with almost no mobile charge carriers forms.
 +
 +This region is called the **depletion region** or **space-charge region**.
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_pn_depletion|Formation of the depletion region at a pn junction.></imgcaption>
 +{{:circuit_design:evolutionofpnjunction.svg?650}}
 +</panel>
 +</WRAP>
 +
 +The depletion region behaves like an internal barrier.  
 +Without an external voltage, it prevents a large current.
 +
 +<panel type="info" title="Analogy: a door with a spring">
 +The depletion region is like a spring-loaded door.
 +
 +  * In one direction, you push against the spring and can open the door.
 +  * In the other direction, the spring pushes the door more firmly closed.
 +
 +The diode behaves similarly: one polarity reduces the barrier, the other polarity increases it.
 +</panel>
 +
 +==== Forward and reverse operation ====
 +
 +We define the diode voltage
 +
 +\[
 +\begin{align*}
 +u_{\rm AK}=u_{\rm A}-u_{\rm K}.
 +\end{align*}
 +\]
 +
 +  * \(u_{\rm AK}>0\): anode is more positive than cathode.
 +  * \(u_{\rm AK}<0\): anode is more negative than cathode.
 +
 +<tabcaption tab_diode_bias|Diode operation depending on \(u_{\rm AK}\)>
 +
 +^ Condition ^ Name ^ Effect on depletion region ^ Current ^
 +| \(u_{\rm AK}>0\) | forward bias | depletion region becomes smaller | large current possible |
 +| \(u_{\rm AK}<0\) | reverse bias | depletion region becomes larger | only small leakage current, until breakdown |
 +
 +<callout type="info" icon="true">
 +**Mnemonic**
 +
 +\[
 +\begin{align*}
 +\text{Positive Anode, Negative Is Cathode}
 +\end{align*}
 +\]
 +
 +This helps to remember the forward direction of a diode.
 +</callout>
 +
 +==== Ideal diode model ====
 +
 +The simplest model is the ideal diode.
 +
 +\[
 +\begin{align*}
 +\text{forward direction: } u_{\rm AK}=0,\quad i_{\rm D}>0
 +\end{align*}
 +\]
 +
 +\[
 +\begin{align*}
 +\text{reverse direction: } i_{\rm D}=0,\quad u_{\rm AK}<0
 +\end{align*}
 +\]
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_ideal_diode_characteristic|Ideal diode characteristic.></imgcaption>
 +{{drawio>block11_ideal_diode_characteristic.svg}}
 +</panel>
 +</WRAP>
 +
 +<panel type="info" title="Engineering meaning">
 +The ideal diode is useful for a first decision:
 +
 +  * Is the diode conducting?
 +  * Is the diode blocking?
 +  * Which path can current take?
 +
 +It is too simple for accurate voltage and current calculations.
 +</panel>
 +
 +==== Real diode characteristic ====
 +
 +A real diode has an exponential current-voltage characteristic.
 +
 +\[
 +\begin{align*}
 +\boxed{
 +i_{\rm D}
 +=
 +{\color{red}{I_{\rm S}(T)}}
 +\left(
 +{\rm e}^{\frac{{\color{blue}{u_{\rm AK}}}}{{\color{green}{mU_{\rm T}}}}}
 +-1
 +\right)
 +}
 +\end{align*}
 +\]
 +
 +with
 +
 +\[
 +\begin{align*}
 +U_{\rm T}=\frac{kT}{e}.
 +\end{align*}
 +\]
 +
 +<tabcaption tab_diode_equation_symbols|Symbols in the diode equation>
 +
 +^ Symbol ^ Meaning ^
 +| \(I_{\rm S}(T)\) | reverse saturation current, strongly temperature-dependent |
 +| \(m\) | emission coefficient, typically \(1\ldots 2\) |
 +| \(U_{\rm T}\) | thermal voltage |
 +| \(k\) | Boltzmann constant |
 +| \(e\) | elementary charge |
 +| \(T\) | absolute temperature in \({\rm K}\) |
 +
 +At room temperature, \(U_{\rm T}\) is approximately
 +
 +\[
 +\begin{align*}
 +U_{\rm T}\approx 26~{\rm mV}.
 +\end{align*}
 +\]
 +
 +Typical values at \(25^\circ{\rm C}\):
 +
 +<tabcaption tab_typical_diode_values|Typical diode values>
 +
 +^ Diode material ^ Approximate threshold voltage \(U_{\rm TO}\) ^ Reverse saturation current \(I_{\rm S}\) ^
 +| silicon | \(\approx 0.7~{\rm V}\) | some \({\rm pA}\) |
 +| germanium | \(\approx 0.3~{\rm V}\) | some \(\mu{\rm A}\) |
 +
 +<callout type="warning" icon="true">
 +The value \(0.7~{\rm V}\) for a silicon diode is not a physical constant.  
 +It is a useful approximation for typical currents in small signal and basic power circuits.
 +</callout>
 +
 +==== Practical diode models for circuit calculation ====
 +
 +For hand calculations we usually do not use the full exponential equation.
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_diode_models|Comparison of ideal, constant-voltage, and piecewise-linear diode models.></imgcaption>
 +{{drawio>block11_diode_models.svg}}
 +</panel>
 +</WRAP>
 +
 +<tabcaption tab_diode_models|Diode models for circuit calculations>
 +
 +^ Model ^ Forward direction ^ Reverse direction ^ Use ^
 +| ideal diode | \(u_{\rm AK}=0\) | \(i_{\rm D}=0\) | switching logic, first estimate |
 +| constant-voltage model | \(u_{\rm AK}\approx U_{\rm TO}\) | \(i_{\rm D}\approx 0\) | quick current calculations |
 +| piecewise-linear model | \(u_{\rm AK}\approx U_{\rm TO}+r_{\rm F}i_{\rm D}\) | \(i_{\rm D}\approx 0\) | more accurate operating point |
 +
 +The differential forward resistance is
 +
 +\[
 +\begin{align*}
 +r_{\rm F}
 +=
 +\frac{\Delta U_{\rm F}}{\Delta I_{\rm F}}.
 +\end{align*}
 +\]
 +
 +For large forward voltages compared with \(U_{\rm T}\), the diode equation leads approximately to
 +
 +\[
 +\begin{align*}
 +r_{\rm D}
 +=
 +\frac{{\rm d}u_{\rm D}}{{\rm d}i_{\rm D}}
 +\approx
 +\frac{mU_{\rm T}}{I_{\rm D}}.
 +\end{align*}
 +\]
 +
 +<callout type="info" icon="true">
 +**Unit check**
 +
 +\[
 +\begin{align*}
 +[r_{\rm D}]
 +=
 +\frac{[U_{\rm T}]}{[I_{\rm D}]}
 +=
 +\frac{{\rm V}}{{\rm A}}
 +=
 +\Omega.
 +\end{align*}
 +\]
 +</callout>
 +
 +==== Operating point with a series resistor ====
 +
 +A diode must usually be operated with a current-limiting element.
 +
 +For the circuit
 +
 +\[
 +\begin{align*}
 +U_{\rm E}
 +\rightarrow R
 +\rightarrow D
 +\end{align*}
 +\]
 +
 +the loop equation is
 +
 +\[
 +\begin{align*}
 +U_{\rm E}
 +=
 +U_R+U_{\rm D}.
 +\end{align*}
 +\]
 +
 +With the constant-voltage model,
 +
 +\[
 +\begin{align*}
 +U_{\rm D}\approx U_{\rm TO}.
 +\end{align*}
 +\]
 +
 +Therefore
 +
 +\[
 +\begin{align*}
 +I_{\rm D}
 +\approx
 +\frac{U_{\rm E}-U_{\rm TO}}{R}.
 +\end{align*}
 +\]
 +
 +<callout type="danger" icon="true">
 +Never connect a normal diode or LED directly to an ideal voltage source in forward direction.  
 +The diode current must be limited.
 +</callout>
 +
 +==== Z-diodes and LEDs as diode types ====
 +
 +A Z-diode is operated in reverse breakdown. In its operating range, the diode voltage is approximately constant:
 +
 +\[
 +\begin{align*}
 +u_{\rm Z}\approx U_{\rm Z}.
 +\end{align*}
 +\]
 +
 +The piecewise-linear model is
 +
 +\[
 +\begin{align*}
 +u_{\rm Z}
 +\approx
 +U_{\rm Z}+r_{\rm Z}i_{\rm Z}.
 +\end{align*}
 +\]
 +
 +<panel type="info" title="Z-diode preview">
 +Z-diodes are useful for voltage limitation and voltage stabilization.  
 +The practical circuits are treated in [[block12|Block 12]].
 +</panel>
 +
 +An LED is a diode that emits light in forward direction. The required forward voltage depends on the semiconductor material and therefore on the color.
 +
 +<tabcaption tab_led_forward_voltage|Typical LED forward voltages>
 +
 +^ LED color ^ Typical \(U_{\rm TO}\) ^
 +| infrared | \(\approx 1.3~{\rm V}\) |
 +| red | \(\approx 1.6~{\rm V}\) |
 +| yellow | \(\approx 1.7~{\rm V}\) |
 +| green | \(\approx 1.8~{\rm V}\) |
 +| blue | \(\approx 3.2~{\rm V}\) |
 +
 +<callout type="warning" icon="true">
 +LEDs usually tolerate only small reverse voltages.  
 +Do not operate an LED in reverse direction unless the datasheet explicitly allows it.
 +</callout>
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 +
 +===== Exercises =====
 +
 +#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: doping and charge carriers
 +#@TaskText_HTML@#
 +
 +Complete the table.
 +
 +^ Doping type ^ Typical dopant atom ^ Main mobile charge carrier ^ Dopant name ^
 +| n-type | ? | ? | ? |
 +| p-type | ? | ? | ? |
 +
 +#@ResultBegin_HTML~ExerciseDoping~@#
 +
 +^ Doping type ^ Typical dopant atom ^ Main mobile charge carrier ^ Dopant name ^
 +| n-type | phosphorus, arsenic, or antimony | electrons | donor |
 +| p-type | boron, aluminium, or indium | holes | acceptor |
 +
 +N-type material has additional mobile electrons.  
 +P-type material has additional mobile holes.
 +
 +The semiconductor as a whole remains approximately electrically neutral.
 +
 +#@ResultEnd_HTML@#
 +#@TaskEnd_HTML@#
 +
 +
 +#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: diode polarity
 +#@TaskText_HTML@#
 +
 +A diode has the anode voltage
 +
 +\[
 +\begin{align*}
 +U_{\rm A}=4.8~{\rm V}
 +\end{align*}
 +\]
 +
 +and the cathode voltage
 +
 +\[
 +\begin{align*}
 +U_{\rm K}=4.1~{\rm V}.
 +\end{align*}
 +\]
 +
 +  * Calculate \(u_{\rm AK}\).
 +  * Is the diode forward-biased or reverse-biased?
 +  * For a silicon diode, is a noticeable current likely?
 +
 +#@ResultBegin_HTML~ExercisePolarity~@#
 +
 +\[
 +\begin{align*}
 +u_{\rm AK}
 +=
 +U_{\rm A}-U_{\rm K}
 +=
 +4.8~{\rm V}-4.1~{\rm V}
 +=
 +0.7~{\rm V}.
 +\end{align*}
 +\]
 +
 +Since
 +
 +\[
 +\begin{align*}
 +u_{\rm AK}>0,
 +\end{align*}
 +\]
 +
 +the diode is forward-biased.
 +
 +For a silicon diode, \(0.7~{\rm V}\) is a typical forward voltage in the mA range.  
 +Therefore a noticeable current is likely.
 +
 +#@ResultEnd_HTML@#
 +#@TaskEnd_HTML@#
 +
 +
 +#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: current with the constant-voltage model
 +#@TaskText_HTML@#
 +
 +A silicon diode is connected in series with a resistor.
 +
 +\[
 +\begin{align*}
 +U_{\rm E}=5.0~{\rm V},
 +\qquad
 +R=1.0~{\rm k}\Omega.
 +\end{align*}
 +\]
 +
 +Use the constant-voltage model
 +
 +\[
 +\begin{align*}
 +U_{\rm D}\approx 0.7~{\rm V}.
 +\end{align*}
 +\]
 +
 +Calculate the diode current \(I_{\rm D}\).
 +
 +#@ResultBegin_HTML~ExerciseSeriesResistor~@#
 +
 +The voltage across the resistor is
 +
 +\[
 +\begin{align*}
 +U_R
 +=
 +U_{\rm E}-U_{\rm D}
 +=
 +5.0~{\rm V}-0.7~{\rm V}
 +=
 +4.3~{\rm V}.
 +\end{align*}
 +\]
 +
 +Therefore
 +
 +\[
 +\begin{align*}
 +I_{\rm D}
 +=
 +\frac{U_R}{R}
 +=
 +\frac{4.3~{\rm V}}{1.0~{\rm k}\Omega}
 +=
 +4.3~{\rm mA}.
 +\end{align*}
 +\]
 +
 +#@ResultEnd_HTML@#
 +#@TaskEnd_HTML@#
 +
 +
 +#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: differential diode resistance
 +#@TaskText_HTML@#
 +
 +A diode operates at
 +
 +\[
 +\begin{align*}
 +I_{\rm D}=10~{\rm mA}.
 +\end{align*}
 +\]
 +
 +Assume
 +
 +\[
 +\begin{align*}
 +m=1,
 +\qquad
 +U_{\rm T}=26~{\rm mV}.
 +\end{align*}
 +\]
 +
 +Estimate the differential diode resistance
 +
 +\[
 +\begin{align*}
 +r_{\rm D}\approx \frac{mU_{\rm T}}{I_{\rm D}}.
 +\end{align*}
 +\]
 +
 +#@ResultBegin_HTML~ExerciseDifferentialResistance~@#
 +
 +\[
 +\begin{align*}
 +r_{\rm D}
 +&\approx
 +\frac{mU_{\rm T}}{I_{\rm D}}
 +\\
 +&=
 +\frac{1\cdot 26~{\rm mV}}{10~{\rm mA}}
 +\\
 +&=
 +2.6~\Omega.
 +\end{align*}
 +\]
 +
 +This is a small-signal resistance around the operating point.  
 +It is not the same as the large-signal ratio \(\frac{U_{\rm D}}{I_{\rm D}}\).
 +
 +#@ResultEnd_HTML@#
 +#@TaskEnd_HTML@#
 +
 +
 +#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Longer exercise: operating point with a piecewise-linear diode
 +#@TaskText_HTML@#
 +
 +A diode is connected in series with a resistor.
 +
 +\[
 +\begin{align*}
 +U_{\rm E}=12~{\rm V},
 +\qquad
 +R=560~\Omega.
 +\end{align*}
 +\]
 +
 +For the diode, use the piecewise-linear forward model
 +
 +\[
 +\begin{align*}
 +U_{\rm D}
 +=
 +U_{\rm TO}+r_{\rm F}I_{\rm D}
 +\end{align*}
 +\]
 +
 +with
 +
 +\[
 +\begin{align*}
 +U_{\rm TO}=0.65~{\rm V},
 +\qquad
 +r_{\rm F}=5.0~\Omega.
 +\end{align*}
 +\]
 +
 +  * Draw the loop equation.
 +  * Calculate \(I_{\rm D}\).
 +  * Calculate \(U_{\rm D}\).
 +  * Calculate the diode power \(P_{\rm D}\).
 +  * Compare briefly with the constant-voltage model \(U_{\rm D}=0.65~{\rm V}\).
 +
 +#@ResultBegin_HTML~ExercisePiecewiseLinearDiode~@#
 +
 +The loop equation is
 +
 +\[
 +\begin{align*}
 +U_{\rm E}
 +=
 +RI_{\rm D}
 ++
 +U_{\rm D}.
 +\end{align*}
 +\]
 +
 +Insert the piecewise-linear diode model:
 +
 +\[
 +\begin{align*}
 +U_{\rm E}
 +=
 +RI_{\rm D}
 ++
 +U_{\rm TO}
 ++
 +r_{\rm F}I_{\rm D}.
 +\end{align*}
 +\]
 +
 +Thus
 +
 +\[
 +\begin{align*}
 +I_{\rm D}
 +=
 +\frac{U_{\rm E}-U_{\rm TO}}{R+r_{\rm F}}.
 +\end{align*}
 +\]
 +
 +Insert the values:
 +
 +\[
 +\begin{align*}
 +I_{\rm D}
 +&=
 +\frac{12~{\rm V}-0.65~{\rm V}}{560~\Omega+5.0~\Omega}
 +\\
 +&=
 +\frac{11.35~{\rm V}}{565~\Omega}
 +\\
 +&=
 +20.1~{\rm mA}.
 +\end{align*}
 +\]
 +
 +The diode voltage is
 +
 +\[
 +\begin{align*}
 +U_{\rm D}
 +&=
 +U_{\rm TO}+r_{\rm F}I_{\rm D}
 +\\
 +&=
 +0.65~{\rm V}
 ++
 +5.0~\Omega\cdot 20.1~{\rm mA}
 +\\
 +&=
 +0.65~{\rm V}+0.101~{\rm V}
 +\\
 +&=
 +0.751~{\rm V}.
 +\end{align*}
 +\]
 +
 +The diode power is
 +
 +\[
 +\begin{align*}
 +P_{\rm D}
 +=
 +U_{\rm D}I_{\rm D}
 +=
 +0.751~{\rm V}\cdot 20.1~{\rm mA}
 +=
 +15.1~{\rm mW}.
 +\end{align*}
 +\]
 +
 +With the constant-voltage model,
 +
 +\[
 +\begin{align*}
 +I_{\rm D}
 +=
 +\frac{12~{\rm V}-0.65~{\rm V}}{560~\Omega}
 +=
 +20.3~{\rm mA}.
 +\end{align*}
 +\]
 +
 +The difference is small here because \(r_{\rm F}\ll R\).
 +
 +#@ResultEnd_HTML@#
 +#@TaskEnd_HTML@#
 +
 +
 +===== Common pitfalls =====
 +
 +  * **Thinking a diode is just a resistor:** A diode is nonlinear. The ratio \(U/I\) is not constant.
 +  * **Forgetting current limitation:** A forward-biased diode needs a current-limiting component.
 +  * **Treating \(0.7~{\rm V}\) as exact:** The forward voltage depends on current, temperature, and semiconductor material.
 +  * **Mixing anode and cathode:** Current flows easily from anode to cathode when the diode is forward-biased.
 +  * **Ignoring reverse limits:** Real diodes have maximum reverse voltage. LEDs often tolerate only small reverse voltages.
 +  * **Confusing hole movement with electron movement:** Holes are missing electrons, but they behave like positive mobile charge carriers.
 +  * **Using the exponential diode equation without unit care:** \(U_{\rm T}\) must be in volts and \(T\) in kelvin.
 +
 +===== Embedded resources =====
 +
 +<WRAP group>
 +<WRAP column half>
 +<panel type="info" title="PhET: Semiconductors">
 +Use this simulation to explore doping and the formation of a diode.
 +
 +{{url>https://phet.colorado.edu/en/simulations/semiconductor 700,500 noborder}}
 +</panel>
 +</WRAP>
 +
 +<WRAP column half>
 +<panel type="info" title="Falstad: Diode I/V curve">
 +Use this simulation to compare a resistor characteristic with the nonlinear diode characteristic.
 +
 +{{url>https://www.falstad.com/circuit/e-diodecurve.html 700,500 noborder}}
 +</panel>
 +</WRAP>
 +</WRAP>
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~