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| electrical_engineering_and_electronics_2:block12 [2026/05/18 03:22] – mexleadmin | electrical_engineering_and_electronics_2:block12 [2026/06/10 03:06] (current) – mexleadmin | ||
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| Line 5: | Line 5: | ||
| After this 90-minute block, you can | After this 90-minute block, you can | ||
| + | * identify basic diode types such as universal diodes, Z-diodes, and LEDs. | ||
| + | * calculate simple diode operating points with a series resistor. | ||
| * design a simple LED circuit with a series resistor. | * design a simple LED circuit with a series resistor. | ||
| * explain why LEDs and signal diodes need current limitation. | * explain why LEDs and signal diodes need current limitation. | ||
| Line 28: | Line 30: | ||
| * Freewheeling diode for inductive loads. | * Freewheeling diode for inductive loads. | ||
| * Clamp diodes for sensitive inputs. | * Clamp diodes for sensitive inputs. | ||
| - | * Diode rectifiers: M1, M2, B2. | + | * Diode rectifiers: M1, B2. |
| * Capacitor smoothing and ripple. | * Capacitor smoothing and ripple. | ||
| Line 69: | Line 71: | ||
| ===== Core content ===== | ===== Core content ===== | ||
| - | ==== LED with series resistor ==== | + | ==== Operating point with a series resistor ==== |
| - | An LED is operated | + | A diode must usually be operated |
| < | < | ||
| <panel type=" | <panel type=" | ||
| - | < | + | < |
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| </ | </ | ||
| - | For a supply voltage \(U_{\rm E}\), an LED forward voltage \(U_{\rm F}\), and a desired LED current \(I_{\rm F}\), the series resistor | + | For the circuit in <imgref op_point_circuit> |
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ | + | U_{\rm I} |
| - | R_{\rm V} | + | |
| = | = | ||
| - | \frac{U_{\rm | + | U_R+U_{\rm D}. |
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | With the constant-voltage model, | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm D}\approx U_{\rm TO}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Therefore | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{\rm D} | ||
| + | \approx | ||
| + | \frac{U_{\rm I}-U_{\rm TO}}{R}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | <callout type=" | ||
| + | Never connect a normal diode or LED directly to an ideal voltage source in forward direction. | ||
| + | The diode current must be limited. | ||
| + | </ | ||
| + | |||
| + | ==== LED (with series resistor) ==== | ||
| + | |||
| + | An LED is operated in forward direction. It converts part of the electrical energy into light via electron-hole recombination. \\ | ||
| + | The required forward voltage depends on the semiconductor material and therefore on the color. | ||
| + | |||
| + | For a supply voltage \(U_{\rm I}\), an LED forward voltage \(U_{\rm F}\), and a desired LED current \(I_{\rm F}\), the series resistor is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | \boxed{ | ||
| + | R_{\rm V} = \frac{U_{\rm | ||
| } | } | ||
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | The resistor power is | + | < |
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | |||
| + | For circuit design it is important the check the real resistor power with the absolute maximum ratings of the resistors | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | P_R | + | P_R = (U_{\rm |
| - | = | + | |
| - | (U_{\rm | + | |
| - | = | + | |
| - | R_{\rm V}I_{\rm F}^2. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 108: | Line 147: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | P_{\rm LED} | + | P_{\rm LED} = U_{\rm F}I_{\rm F}. |
| - | = | + | |
| - | U_{\rm F}I_{\rm F}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 121: | Line 158: | ||
| < | < | ||
| - | ^ LED color ^ Typical forward voltage \(U_{\rm F}\) ^ Typical current ^ | + | ^ LED color |
| - | | infrared | \(\approx 1.3~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | | + | | infrared |
| - | | red | \(\approx 1.6~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | | + | | red |
| - | | yellow | \(\approx 1.7~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | | + | | yellow |
| - | | green | \(\approx 1.8~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | | + | | green |
| - | | blue / white | \(\approx 3.0\ldots 3.3~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | | + | | blue / white | \(\approx 3.0\ldots 3.3~{\rm V}\) | \(5\ldots 20~{\rm mA}\) | |
| + | </ | ||
| + | \\ | ||
| <panel type=" | <panel type=" | ||
| - | A robot controller often uses a \(24~{\rm V}\) supply, but a status LED may need only \(10~{\rm mA}\) at about \(2~{\rm V}\). | + | A robot controller often uses a \(24~{\rm V}\) supply, but a status LED may need only \(10~{\rm mA}\) at about \(2~{\rm V}\). \\ |
| Most of the voltage must therefore drop across the resistor, not across the LED. | Most of the voltage must therefore drop across the resistor, not across the LED. | ||
| </ | </ | ||
| - | ==== LED operation with AC voltage ==== | + | <callout type=" |
| + | LEDs usually tolerate only small reverse voltages. | ||
| + | Do not operate an LED in reverse direction unless the datasheet explicitly allows it. | ||
| + | </ | ||
| - | LEDs tolerate only small reverse voltages. Therefore, operation directly at AC voltage needs protection. | + | ~~PAGEBREAK~~ ~~CLEARFIX~~ |
| - | < | + | ==== Z-Diodes ==== |
| - | <panel type=" | + | |
| - | < | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | </ | + | |
| - | A second diode can be placed antiparallel to the LED. Then, during | + | If the reverse |
| + | In this region, | ||
| - | < | + | Two physical effects can cause breakdown: |
| - | For low-frequency indicator circuits, a visible flicker may occur if only one half-wave is used. | + | |
| - | For higher quality indicators, rectification or dedicated LED drivers are used. | + | |
| - | </ | + | |
| - | ==== Z-diode voltage limitation and stabilization ==== | + | * **avalanche breakdown: |
| + | * **Zener breakdown: | ||
| - | A Z-diode is operated in reverse breakdown. In its working range, the voltage is approximately constant: | + | For ordinary diodes, breakdown is usually unwanted and can destroy the diode if the current |
| - | + | **Z-diodes** are designed to operate safely in this reverse-breakdown region at a defined voltage | |
| - | \[ | + | |
| - | \begin{align*} | + | |
| - | u_{\rm | + | |
| - | \end{align*} | + | |
| - | \] | + | |
| < | < | ||
| - | <panel type=" | + | {{url>https://www.falstad.com/ |
| - | < | + | |
| - | {{drawio>block12_z_diode_stabilizer.svg}} | + | |
| - | </ | + | |
| </ | </ | ||
| - | The input current | + | The current |
| + | In its operating range, the diode voltage | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | I_{\rm V} | + | u_{\rm Z}\approx |
| - | = | + | |
| - | \frac{U_{\rm E}-U_{\rm Z}}{R_{\rm V}}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | The load current | + | The piecewise-linear model is |
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | I_{\rm L} | + | u_{\rm Z} \approx |
| - | = | + | |
| - | \frac{U_{\rm Z}}{R_{\rm L}}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | The Z-diode | + | <panel type=" |
| + | * Z-diodes are useful for voltage limitation and voltage stabilization. | ||
| + | * Z-diodes have a huge variety of breakdown voltages: $U_{\rm Z} \approx 1.0 ~\rm V... 400 ~ V $ \\ Z-diodes allow to get "knee voltages" | ||
| + | * Z-diodes are still conventional diodes in the forward direction. | ||
| + | </ | ||
| + | |||
| + | \\ | ||
| + | |||
| + | A typical application | ||
| + | |||
| + | <panel type=" | ||
| + | Use this simulation to observe how a Z-diode limits the output voltage. | ||
| + | |||
| + | Things to try: | ||
| + | |||
| + | * change the input voltage, | ||
| + | * change the load resistance, | ||
| + | * observe when the Z-diode current becomes too small for stabilization. | ||
| + | |||
| + | < | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | |||
| + | </ | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ | + | {\color{blue }{I_{\rm V}}} &= \frac{U_{\rm I}-U_{\rm Z}}{R_{\rm V}} && |
| - | I_{\rm Z} | + | \\ |
| - | = | + | {\color{green}{I_{\rm L}}} &= \frac{U_{\rm Z}}{R_{\rm L}} && |
| - | I_{\rm V}-I_{\rm L} | + | \\ |
| - | } | + | {\color{red}{I_{\rm Z}}} & |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | and must stay in the allowed | + | The Z-diode can stabilize the voltage only if \({\color{red}{I_{\rm Z}}}\) remains inside |
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | I_{\rm Z,min} | + | I_{\rm Z,min} \leq {\color{red}{I_{\rm Z}}} \leq I_{\rm Z,max}. |
| - | \leq | + | |
| - | I_{\rm Z} | + | |
| - | \leq | + | |
| - | I_{\rm Z,max}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 216: | Line 260: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | P_{\rm Z} | + | P_{\rm Z} = U_{\rm Z}\cdot {\color{red}{I_{\rm Z}}} \leq P_{\rm tot}. |
| - | = | + | |
| - | U_{\rm Z}I_{\rm Z} | + | |
| - | \leq | + | |
| - | P_{\rm tot}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | <panel type=" | + | A Z-diode stabilizer is simple, but not efficient for large load currents. |
| - | \[ | + | |
| - | \begin{align*} | + | |
| - | {\color{blue}{I_{\rm V}}} | + | |
| - | &= | + | |
| - | \frac{U_{\rm E}-U_{\rm Z}}{R_{\rm V}} | + | |
| - | && | + | |
| - | \\ | + | |
| - | {\color{green}{I_{\rm L}}} | + | |
| - | &= | + | |
| - | \frac{U_{\rm Z}}{R_{\rm L}} | + | |
| - | && | + | |
| - | \\ | + | |
| - | {\color{red}{I_{\rm Z}}} | + | |
| - | &= | + | |
| - | {\color{blue}{I_{\rm V}}} | + | |
| - | - | + | |
| - | {\color{green}{I_{\rm L}}} | + | |
| - | && | + | |
| - | \end{align*} | + | |
| - | \] | + | |
| - | + | ||
| - | The Z-diode can stabilize the voltage only if \({\color{red}{I_{\rm Z}}}\) remains inside the allowed range. | + | |
| - | </ | + | |
| - | + | ||
| - | <callout type=" | + | |
| - | A Z-diode stabilizer is simple, but not efficient for large load currents. | + | |
| It is useful for voltage limitation, small reference voltages, and robust simple circuits. | It is useful for voltage limitation, small reference voltages, and robust simple circuits. | ||
| - | </ | ||
| - | <panel type=" | ||
| - | Use this simulation to observe how a Z-diode limits the output voltage. | ||
| - | |||
| - | Things to try: | ||
| - | |||
| - | * change the input voltage, | ||
| - | * change the load resistance, | ||
| - | * observe when the Z-diode current becomes too small for stabilization. | ||
| - | |||
| - | {{url> | ||
| - | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Line 279: | Line 281: | ||
| If a relay coil, solenoid, or small motor is switched off, the current tries to continue flowing. Without a safe current path, the voltage can become very large. | If a relay coil, solenoid, or small motor is switched off, the current tries to continue flowing. Without a safe current path, the voltage can become very large. | ||
| - | |||
| - | <WRAP group> | ||
| - | <WRAP column half> | ||
| - | <panel type=" | ||
| - | < | ||
| - | {{drawio> | ||
| - | </ | ||
| - | </ | ||
| - | |||
| - | <WRAP column half> | ||
| - | <panel type=" | ||
| - | < | ||
| - | {{drawio> | ||
| - | </ | ||
| - | </ | ||
| - | </ | ||
| When the switch is opened, the freewheeling diode becomes forward-biased. The inductor current circulates through the diode and the coil. | When the switch is opened, the freewheeling diode becomes forward-biased. The inductor current circulates through the diode and the coil. | ||
| Line 316: | Line 302: | ||
| \] | \] | ||
| - | With a freewheeling diode, the switch voltage is limited to a safe value. | + | With a freewheeling diode, the switch voltage is limited to a safe value. |
| The disadvantage is that the current decays more slowly, so a relay or solenoid may release more slowly. | The disadvantage is that the current decays more slowly, so a relay or solenoid may release more slowly. | ||
| Line 325: | Line 311: | ||
| <panel type=" | <panel type=" | ||
| - | Use this simulation to observe the overvoltage when switching an inductive load, and how a diode limits it. | + | Use this simulation to observe the overvoltage when switching an inductive load like a motor, and how a diode limits it. |
| Things to try: | Things to try: | ||
| Line 332: | Line 318: | ||
| * compare the circuit with and without the protection diode, | * compare the circuit with and without the protection diode, | ||
| * observe the voltage across the switch. | * observe the voltage across the switch. | ||
| - | |||
| - | {{url> | ||
| - | </ | ||
| - | |||
| - | ==== Clamp diodes for sensitive inputs ==== | ||
| - | |||
| - | Microcontroller and sensor inputs tolerate only a limited voltage range. | ||
| - | Clamp diodes can conduct disturbances away from the sensitive input. | ||
| < | < | ||
| - | <panel type=" | + | {{url>https://www.falstad.com/ |
| - | < | + | </ |
| - | {{drawio>block12_input_clamp_diodes.svg}} | + | |
| </ | </ | ||
| - | </ | ||
| - | For a \(5~{\rm V}\) input, the input node is often clamped approximately to | ||
| - | \[ | + | ~~PAGEBREAK~~ ~~CLEARFIX~~ |
| - | \begin{align*} | + | |
| - | -0.7~{\rm V} | + | |
| - | \lesssim | + | |
| - | u_{\rm in} | + | |
| - | \lesssim | + | |
| - | 5.7~{\rm V}. | + | |
| - | \end{align*} | + | |
| - | \] | + | |
| - | The resistor \(R_{\rm V}\) limits the clamp current: | + | ==== Half-wave rectifier |
| - | \[ | + | A rectifier converts an AC voltage into a unidirectional voltage. |
| - | \begin{align*} | + | |
| - | I_{\rm clamp} | + | |
| - | \approx | + | |
| - | \frac{U_{\rm disturb}-U_{\rm clamp}}{R_{\rm V}}. | + | |
| - | \end{align*} | + | |
| - | \] | + | |
| - | <callout | + | <panel type="info" |
| - | Clamp diodes are not a substitute for proper EMC design. | + | Use this simulation to observe how one half-wave is removed by a diode. |
| - | For external connectors, use suitable protection components and check the datasheets. | + | |
| - | </ | + | |
| - | <panel type=" | + | Things to try: |
| - | A sensor cable near a motor cable can pick up short disturbance pulses. | + | |
| - | Clamp diodes can prevent the input voltage from exceeding the allowed range, while the series resistor limits the injected current. | + | |
| - | </ | + | |
| - | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | * reverse the diode direction, |
| - | + | * change the load resistance, | |
| - | ==== Half-wave rectifier M1 ==== | + | * change capacitor, |
| - | + | * compare input and output | |
| - | A rectifier converts an AC voltage into a unidirectional | + | |
| < | < | ||
| - | <panel type=" | + | {{url>https://www.falstad.com/ |
| - | < | + | </ |
| - | {{drawio>block12_half_wave_rectifier_m1.svg}} | + | |
| </ | </ | ||
| - | </ | ||
| Assumptions for the basic formulas: | Assumptions for the basic formulas: | ||
| Line 404: | Line 357: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ | + | \boxed{ U_{\rm di} = \frac{\sqrt{2}}{\pi}U_\sim } |
| - | U_{\rm di} | + | |
| - | = | + | |
| - | \frac{\sqrt{2}}{\pi}U_\sim | + | |
| - | } | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 420: | Line 369: | ||
| \] | \] | ||
| - | The ripple | + | The output voltage can be split into an average DC value and an AC ripple |
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | w_U | + | u_{\rm out}(t) = U_{\rm di} + u_\sigma(t). |
| - | = | + | |
| - | \frac{U_\sigma}{U_{\rm di}} | + | |
| - | \approx | + | |
| - | 1.21. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| - | < | + | Here |
| - | The half-wave rectifier is simple, but it uses only one half-wave. | + | |
| - | Therefore the ripple is large and the transformer is used poorly. | + | |
| - | </ | + | |
| - | <panel type=" | + | * \(U_{\rm di}\) is the average value, i.e. the DC component, |
| - | Use this simulation to observe how one half-wave is removed by a diode. | + | * \(u_\sigma(t)\) is the time-dependent ripple component, |
| + | * \(U_\sigma\) | ||
| - | Things to try: | + | \[ |
| + | \begin{align*} | ||
| + | U_\sigma = \sqrt{ \frac{1}{T} \int_0^T u_\sigma^2(t)\, | ||
| + | \end{align*} | ||
| + | \] | ||
| - | * reverse | + | The ripple factor for the ideal circuit is |
| - | * change the load resistance, | + | |
| - | * compare input and output voltage. | + | |
| - | {{url> | + | \[ |
| - | </ | + | \begin{align*} |
| + | w_U = \frac{U_\sigma}{U_{\rm di}} \approx 1.21. | ||
| + | \end{align*} | ||
| + | \] | ||
| - | ==== Center-tap rectifier | + | < |
| + | The half-wave rectifier | ||
| + | Therefore the ripple is large and the transformer is used poorly. \\ | ||
| + | Damping capacitors must be relatively large. | ||
| + | </ | ||
| - | A full-wave rectifier uses both half-waves. | ||
| - | <WRAP group> | + | ==== Bridge |
| - | <WRAP column half> | + | |
| - | <panel type=" | + | |
| - | < | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | </ | + | |
| - | <WRAP column | + | A full-wave rectifier uses both half-waves. |
| - | <panel type=" | + | |
| - | < | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | </ | + | |
| - | </ | + | |
| - | For the center-tap rectifier M2: | + | <panel type=" |
| + | Use this simulation to compare half-wave and full-wave rectification. | ||
| - | \[ | + | Things to try: |
| - | \begin{align*} | + | |
| - | U_{\rm di} | + | |
| - | = | + | |
| - | \frac{2\sqrt{2}}{\pi}U_{1{\rm N}} | + | |
| - | = | + | |
| - | \frac{\sqrt{2}}{\pi}U_{\rm S}. | + | |
| - | \end{align*} | + | |
| - | \] | + | |
| - | Here \(U_{1{\rm N}}\) is the RMS voltage of one half of the secondary winding | + | * observe which two diodes conduct in each half-wave, |
| + | * compare input and output | ||
| + | * add or remove smoothing if available in the simulation. | ||
| + | |||
| + | {{url> | ||
| + | </ | ||
| For the bridge rectifier B2: | For the bridge rectifier B2: | ||
| Line 488: | Line 425: | ||
| \begin{align*} | \begin{align*} | ||
| \boxed{ | \boxed{ | ||
| - | U_{\rm di} | + | U_{\rm di} = \frac{2\sqrt{2}}{\pi}U_\sim |
| - | = | + | |
| - | \frac{2\sqrt{2}}{\pi}U_\sim | + | |
| } | } | ||
| \end{align*} | \end{align*} | ||
| Line 512: | Line 447: | ||
| <panel type=" | <panel type=" | ||
| - | In a bridge rectifier, two diodes conduct at the same time. | + | In a bridge rectifier, two diodes conduct at the same time. |
| Therefore, for silicon diodes, the output voltage is roughly reduced by | Therefore, for silicon diodes, the output voltage is roughly reduced by | ||
| Line 527: | Line 462: | ||
| ^ Circuit ^ Uses half-waves ^ Ideal average voltage \(U_{\rm di}\) ^ Ripple frequency ^ | ^ Circuit ^ Uses half-waves ^ Ideal average voltage \(U_{\rm di}\) ^ Ripple frequency ^ | ||
| - | | M1 half-wave | one half-wave | \(\frac{\sqrt{2}}{\pi}U_\sim\) | \(f\) | | + | | M1 half-wave |
| - | | M2 center-tap | both half-waves | \(\frac{2\sqrt{2}}{\pi}U_{1{\rm N}}\) | \(2f\) | + | | B2 bridge |
| - | | B2 bridge | both half-waves | \(\frac{2\sqrt{2}}{\pi}U_\sim\) | \(2f\) | | + | </ |
| - | <panel type=" | ||
| - | Use this simulation to compare half-wave and full-wave rectification. | ||
| - | |||
| - | Things to try: | ||
| - | |||
| - | * observe which two diodes conduct in each half-wave, | ||
| - | * compare input and output voltage, | ||
| - | * add or remove smoothing if available in the simulation. | ||
| - | |||
| - | {{url> | ||
| - | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Line 547: | Line 471: | ||
| ==== Capacitor smoothing ==== | ==== Capacitor smoothing ==== | ||
| - | A rectifier output is not constant. A smoothing capacitor stores charge near the voltage maximum and supplies the load between maxima. | + | A rectifier output is not constant. |
| + | A smoothing capacitor stores charge near the voltage maximum and supplies the load between maxima. | ||
| + | < | ||
| < | < | ||
| - | <panel type=" | + | {{url>https://www.falstad.com/ |
| - | < | + | </ |
| - | {{drawio>block12_bridge_rectifier_with_capacitor.svg}} | + | |
| </ | </ | ||
| - | </ | ||
| For a bridge rectifier with a sufficiently large smoothing capacitor, the DC voltage is approximately | For a bridge rectifier with a sufficiently large smoothing capacitor, the DC voltage is approximately | ||
| Line 560: | Line 484: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{\rm di} | + | U_{\rm di} \approx \sqrt{2}U_\sim \end{align*} |
| - | \approx | + | |
| - | \sqrt{2}U_\sim | + | |
| - | \end{align*} | + | |
| \] | \] | ||
| Line 572: | Line 493: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{\rm di} | + | U_{\rm di} \approx \sqrt{2}U_\sim - 2U_{\rm TO} - \frac{\Delta U}{2}. |
| - | \approx | + | |
| - | \sqrt{2}U_\sim | + | |
| - | - | + | |
| - | 2U_{\rm TO} | + | |
| - | - | + | |
| - | \frac{\Delta U}{2}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 588: | Line 503: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ | + | \boxed{ C \approx \frac{I_{\rm d}}{f_\sigma\Delta U} } |
| - | C | + | |
| - | \approx | + | |
| - | \frac{I_{\rm d}}{f_\sigma\Delta U} | + | |
| - | } | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 607: | Line 518: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | C | + | C \approx k\frac{I_{\rm d}}{f_\sigma U_\sigma}. |
| - | \approx | + | |
| - | k\frac{I_{\rm d}}{f_\sigma U_\sigma}. | + | |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 624: | Line 533: | ||
| <callout type=" | <callout type=" | ||
| - | A larger capacitor reduces ripple, but it also creates short high charging-current pulses through the diodes and transformer. | + | A larger capacitor reduces ripple, but it also creates short high charging-current pulses through the diodes and transformer. |
| For power supplies, check diode peak current, transformer rating, capacitor ripple current, and inrush current. | For power supplies, check diode peak current, transformer rating, capacitor ripple current, and inrush current. | ||
| </ | </ | ||
| - | ==== Application overview | + | ===== Exercises |
| + | |||
| + | # | ||
| + | # | ||
| + | |||
| + | The following simulation includes multiple diodes and several lamps. | ||
| + | A lamp lights brightly when a voltage of approximately | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm lamp}\geq 5~{\rm V} | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | drops across it. | ||
| + | |||
| + | Close the switch in the simulation. | ||
| + | |||
| + | * Which lamps light up brightly? | ||
| + | * Which lamps remain dark? | ||
| + | * Explain the result using the idea of diode bypass paths. | ||
| + | |||
| + | {{url> | ||
| + | |||
| + | |||
| + | 1. Determine which lamps light up brightly. | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | Number the lamps from left to right: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | L_1, | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | After closing the switch, check the voltage across each lamp in the simulation. | ||
| + | |||
| + | A lamp is assumed to light brightly if | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm lamp}\geq 5~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The simulation shows that the outer lamps have a sufficiently large voltage across them, while the inner lamps are bypassed by conducting diodes. | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | The lamps | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | L_1 | ||
| + | \quad \text{and} \quad | ||
| + | L_5 | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | light up brightly. | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | 2. Determine which lamps remain dark. | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | The inner lamps are connected in parts of the circuit that are bypassed by forward-biased diodes. | ||
| + | |||
| + | A forward-biased diode has only a small voltage drop. Therefore, a lamp in parallel with such a diode path receives only a small voltage. | ||
| + | |||
| + | If | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm lamp}< | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | the lamp does not light brightly. | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | The lamps | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | L_2, | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | remain dark or almost dark. | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | # | ||
| + | |||
| + | The following simulation includes two diodes and two resistors. | ||
| + | |||
| + | Assume a simple constant-voltage diode model: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm F}=0.6~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The source voltage is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_0=4.0~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The resistors are | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | R_1=200~\Omega, | ||
| + | \qquad | ||
| + | R_2=100~\Omega. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Calculate the currents through | ||
| + | |||
| + | * \(D_1\), | ||
| + | * \(R_1\), | ||
| + | * \(R_2\). | ||
| + | |||
| + | {{url> | ||
| + | |||
| + | 1. Calculate the current through \(R_1\). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | The current through \(R_1\) passes through one forward-biased diode. | ||
| + | |||
| + | Therefore the voltage across \(R_1\) is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{R1} | ||
| + | = | ||
| + | U_0-U_{\rm F}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Insert the values: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{R1} | ||
| + | &= | ||
| + | 4.0~{\rm V}-0.6~{\rm V} | ||
| + | \\ | ||
| + | &= | ||
| + | 3.4~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Now apply Ohm's law: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1} | ||
| + | = | ||
| + | \frac{U_{R1}}{R_1} | ||
| + | = | ||
| + | \frac{3.4~{\rm V}}{200~\Omega} | ||
| + | = | ||
| + | 17~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1}=17~{\rm mA} | ||
| + | \end{align*} | ||
| + | \] | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | 2. Calculate the current through \(R_2\). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | The current through \(R_2\) passes through two forward-biased diodes. | ||
| + | |||
| + | Therefore the voltage across \(R_2\) is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{R2} | ||
| + | = | ||
| + | U_0-2U_{\rm F}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Insert the values: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{R2} | ||
| + | &= | ||
| + | 4.0~{\rm V}-2\cdot 0.6~{\rm V} | ||
| + | \\ | ||
| + | &= | ||
| + | 2.8~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Now apply Ohm's law: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R2} | ||
| + | = | ||
| + | \frac{U_{R2}}{R_2} | ||
| + | = | ||
| + | \frac{2.8~{\rm V}}{100~\Omega} | ||
| + | = | ||
| + | 28~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R2}=28~{\rm mA} | ||
| + | \end{align*} | ||
| + | \] | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | 3. Calculate the current through \(D_1\). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | The diode \(D_1\) supplies both current paths. | ||
| + | |||
| + | Therefore, by Kirchhoff' | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1} | ||
| + | = | ||
| + | I_{R1}+I_{R2}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Insert the values: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1} | ||
| + | &= | ||
| + | 17~{\rm mA}+28~{\rm mA} | ||
| + | \\ | ||
| + | &= | ||
| + | 45~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}=45~{\rm mA} | ||
| + | \end{align*} | ||
| + | \] | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | # | ||
| + | |||
| + | The following simulation includes two diodes and a switch. | ||
| + | |||
| + | Assume a simple constant-voltage diode model: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm F}=0.7~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The source voltage is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_0=5.0~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The resistor is | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | R_1=1.0~{\rm k}\Omega. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Calculate the currents through | ||
| + | |||
| + | * \(R_1\), | ||
| + | * \(D_1\), | ||
| + | * \(D_2\), | ||
| + | |||
| + | depending on the switch state \(S\). | ||
| + | |||
| + | {{url> | ||
| + | |||
| + | 1. Calculate the currents for open switch \(S\). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | With the switch open, only \(D_1\) is connected to the resistor path. | ||
| + | |||
| + | The conducting diode clamps the node voltage to approximately | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm node}\approx U_{\rm F}=0.7~{\rm V}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | The resistor current is therefore | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1} | ||
| + | &= | ||
| + | \frac{U_0-U_{\rm F}}{R_1} | ||
| + | \\ | ||
| + | &= | ||
| + | \frac{5.0~{\rm V}-0.7~{\rm V}}{1.0~{\rm k}\Omega} | ||
| + | \\ | ||
| + | &= | ||
| + | 4.3~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Since only \(D_1\) conducts, | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}=I_{R1}, | ||
| + | \qquad | ||
| + | I_{D2}=0. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | For open switch: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1}& | ||
| + | \\ | ||
| + | I_{D1}& | ||
| + | \\ | ||
| + | I_{D2}& | ||
| + | \end{align*} | ||
| + | \] | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | 2. Calculate the currents for closed switch \(S\). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | With the switch closed, \(D_1\) and \(D_2\) are connected in parallel. | ||
| + | |||
| + | The resistor current is still determined by the source voltage, the forward diode voltage, and \(R_1\): | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1} | ||
| + | &= | ||
| + | \frac{U_0-U_{\rm F}}{R_1} | ||
| + | \\ | ||
| + | &= | ||
| + | \frac{5.0~{\rm V}-0.7~{\rm V}}{1.0~{\rm k}\Omega} | ||
| + | \\ | ||
| + | &= | ||
| + | 4.3~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | Kirchhoff' | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}+I_{D2}=I_{R1}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | With the ideal constant-voltage diode model, the individual currents through two parallel diodes are not uniquely determined. | ||
| + | |||
| + | If both real diodes are approximately identical, the current splits approximately equally: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1} | ||
| + | \approx | ||
| + | I_{D2} | ||
| + | \approx | ||
| + | \frac{4.3~{\rm mA}}{2} | ||
| + | = | ||
| + | 2.15~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | For closed switch: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{R1}=4.3~{\rm mA} | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | and | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}+I_{D2}=4.3~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | For approximately identical real diodes: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1} | ||
| + | \approx | ||
| + | I_{D2} | ||
| + | \approx | ||
| + | 2.15~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | 3. Explain why the current sharing is not unique in the simple model. | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column rightalign> | ||
| + | # | ||
| + | <WRAP leftalign> | ||
| + | The constant-voltage diode model assumes that each conducting diode has exactly the same voltage drop: | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | U_{\rm D}=U_{\rm F}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | For two parallel diodes, this condition is true for many possible current distributions. | ||
| + | |||
| + | Therefore, the model only determines the sum | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}+I_{D2}, | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | not the individual diode currents. | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | |||
| + | <WRAP half column> | ||
| + | # | ||
| + | The constant-voltage diode model determines only | ||
| + | |||
| + | \[ | ||
| + | \begin{align*} | ||
| + | I_{D1}+I_{D2}=4.3~{\rm mA}. | ||
| + | \end{align*} | ||
| + | \] | ||
| + | |||
| + | It does not uniquely determine \(I_{D1}\) and \(I_{D2}\) separately. | ||
| + | |||
| + | <callout type=" | ||
| + | Parallel diodes are sensitive to small differences in real diode characteristics. | ||
| + | Current sharing should not be assumed to be perfect without checking the design. | ||
| + | </ | ||
| + | # | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | # | ||
| - | < | ||
| - | ^ Problem ^ Diode application ^ Main design question ^ | ||
| - | | Status indication | LED with resistor | Which current and resistor value? | | ||
| - | | Small reference voltage | Z-diode stabilizer | Is \(I_{\rm Z}\) inside the allowed range? | | ||
| - | | Relay or solenoid switch-off | freewheeling diode | Where can the inductor current flow? | | ||
| - | | Sensor input disturbance | clamp diodes | Is the clamp current limited? | | ||
| - | | AC to DC conversion | rectifier | M1, M2, or B2? | | ||
| - | | DC supply with lower ripple | smoothing capacitor | Which ripple voltage is acceptable? | | ||
| - | ===== Exercises ===== | ||
| # | # | ||
| Line 649: | Line 1100: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{\rm | + | U_{\rm |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 675: | Line 1126: | ||
| R_{\rm V} | R_{\rm V} | ||
| &= | &= | ||
| - | \frac{U_{\rm | + | \frac{U_{\rm |
| \\ | \\ | ||
| &= | &= | ||
| Line 699: | Line 1150: | ||
| P_R | P_R | ||
| &= | &= | ||
| - | (U_{\rm | + | (U_{\rm |
| \\ | \\ | ||
| &= | &= | ||
| Line 730: | Line 1181: | ||
| \[ | \[ | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{\rm | + | U_{\rm |
| \end{align*} | \end{align*} | ||
| \] | \] | ||
| Line 763: | Line 1214: | ||
| I_{\rm V} | I_{\rm V} | ||
| &= | &= | ||
| - | \frac{U_{\rm | + | \frac{U_{\rm |
| \\ | \\ | ||
| &= | &= | ||