Block 12 — Diode Applications

Learning objectives

90-minute plan

• Warm-up (10 min):
  • What happens if an LED is connected directly to \(24~{\rm V}\)?
  • Recall from Block 11: diode polarity, forward voltage, reverse blocking.
  • Recall from switching transients: inductor current cannot jump.

• Core concepts (55 min):
  • LED operation with a series resistor.
  • Z-diode voltage limitation and stabilization.
  • Freewheeling diode for inductive loads.
  • Clamp diodes for sensitive inputs.
  • Diode rectifiers: M1, B2.
  • Capacitor smoothing and ripple.

• Practice (20 min):
  • Calculate an LED series resistor.
  • Check Z-diode current limits.
  • Estimate switching overvoltages in an inductive load.
  • Calculate average rectifier voltages and smoothing capacitors.

• Wrap-up (5 min):
  • Which diode application belongs to which engineering problem?
  • Preview: bipolar transistors as controlled switches and amplifiers.

Conceptual overview

Core content

Operating point with a series resistor

A diode must usually be operated with a current-limiting element.

For the circuit in figure 1 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

\[ \begin{align*} I_{\rm D} \approx \frac{U_{\rm E}-U_{\rm TO}}{R}. \end{align*} \]

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 \(U_{\rm E}\), an LED forward voltage \(U_{\rm F}\), and a desired LED current \(I_{\rm F}\), the series resistor is

\[ \begin{align*} \boxed{ R_{\rm V} = \frac{U_{\rm E}-U_{\rm F}}{I_{\rm F}} } \end{align*} \]

For circuit design it is important the check the real resistor power with the absolute maximum ratings of the resistors

\[ \begin{align*} P_R = (U_{\rm E}-U_{\rm F})I_{\rm F} = R_{\rm V}I_{\rm F}^2 \quad \overset{!}{<} \quad P_{R, \rm max} \end{align*} \]

The LED power is approximately

\[ \begin{align*} P_{\rm LED} = U_{\rm F}I_{\rm F}. \end{align*} \]

Z-Diodes

If the reverse voltage of a diode becomes too large, the diode enters breakdown. In this region, the reverse current rises strongly.

Two physical effects can cause breakdown:

• avalanche breakdown: charge carriers gain enough energy to free additional charge carriers by collisions.
• Zener breakdown: in strongly doped pn junctions, charge carriers can cross the barrier by a quantum-mechanical effect.

For ordinary diodes, breakdown is usually unwanted and can destroy the diode if the current is not limited.
Z-diodes are designed to operate safely in this reverse-breakdown region at a defined voltage \(U_{\rm Z}\).

The current must still be limited by the surrounding circuit.
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

\[ \begin{align*} u_{\rm Z} \approx U_{\rm Z}+r_{\rm Z} \cdot i_{\rm Z}. \end{align*} \]

A typical application is a Z-diode stabilizer

\[ \begin{align*} {\color{blue }{I_{\rm V}}} &= \frac{U_{\rm E}-U_{\rm Z}}{R_{\rm V}} &&\text{current supplied through the series resistor}, \\ {\color{green}{I_{\rm L}}} &= \frac{U_{\rm Z}}{R_{\rm L}} &&\text{useful load current}, \\ {\color{red}{I_{\rm Z}}} &= {\color{blue}{I_{\rm V}}} - {\color{green}{I_{\rm L}}} &&\text{remaining current through the Z-diode}. \end{align*} \]

The Z-diode can stabilize the voltage only if \({\color{red}{I_{\rm Z}}}\) remains inside the allowed range:

\[ \begin{align*} I_{\rm Z,min} \leq {\color{red}{I_{\rm Z}}} \leq I_{\rm Z,max}. \end{align*} \]

The power limit is

\[ \begin{align*} P_{\rm Z} = U_{\rm Z}\cdot {\color{red}{I_{\rm Z}}} \leq P_{\rm tot}. \end{align*} \]

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.

Freewheeling diode for inductive loads

Inductors resist a sudden change of current:

\[ \begin{align*} u_L=L\frac{{\rm d}i_L}{{\rm d}t}. \end{align*} \]

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.

When the switch is opened, the freewheeling diode becomes forward-biased. The inductor current circulates through the diode and the coil.

The magnetic energy stored in the inductance is

\[ \begin{align*} W_L = \frac{1}{2}LI_0^2. \end{align*} \]

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.

Half-wave rectifier (M1)

A rectifier converts an AC voltage into a unidirectional voltage.

Assumptions for the basic formulas:

• sinusoidal input voltage,
• RMS value \(U_\sim\),
• ohmic load,
• ideal diode.

For a half-wave rectifier:

\[ \begin{align*} \boxed{ U_{\rm di} = \frac{\sqrt{2}}{\pi}U_\sim } \end{align*} \]

The ripple frequency is

\[ \begin{align*} f_\sigma=f. \end{align*} \]

The output voltage can be split into an average DC value and an AC ripple component:

\[ \begin{align*} u_{\rm out}(t) = U_{\rm di} + u_\sigma(t). \end{align*} \]

Here

• \(U_{\rm di}\) is the average value, i.e. the DC component,
• \(u_\sigma(t)\) is the time-dependent ripple component,
• \(U_\sigma\) is the RMS value of this ripple component.

\[ \begin{align*} U_\sigma = \sqrt{ \frac{1}{T} \int_0^T u_\sigma^2(t)\,{\rm d}t }. \end{align*} \]

The ripple factor for the ideal circuit is

\[ \begin{align*} w_U = \frac{U_\sigma}{U_{\rm di}} \approx 1.21. \end{align*} \]

Bridge rectifier B2

A full-wave rectifier uses both half-waves.

For the bridge rectifier B2:

\[ \begin{align*} \boxed{ U_{\rm di} = \frac{2\sqrt{2}}{\pi}U_\sim } \end{align*} \]

The ripple frequency is

\[ \begin{align*} f_\sigma=2f. \end{align*} \]

The ideal ripple factor is

\[ \begin{align*} w_U\approx 0.48. \end{align*} \]

Capacitor smoothing

A rectifier output is not constant.
A smoothing capacitor stores charge near the voltage maximum and supplies the load between maxima.

For a bridge rectifier with a sufficiently large smoothing capacitor, the DC voltage is approximately

\[ \begin{align*} U_{\rm di} \approx \sqrt{2}U_\sim \end{align*} \]

for ideal diodes and small ripple.

With real silicon diodes in a bridge rectifier:

\[ \begin{align*} U_{\rm di} \approx \sqrt{2}U_\sim - 2U_{\rm TO} - \frac{\Delta U}{2}. \end{align*} \]

Here \(\Delta U\) is the approximate peak-to-peak ripple voltage.

A simple estimate for the smoothing capacitor is

\[ \begin{align*} \boxed{ C \approx \frac{I_{\rm d}}{f_\sigma\Delta U} } \end{align*} \]

with

• \(I_{\rm d}\): load current,
• \(f_\sigma\): ripple frequency,
• \(\Delta U\): allowed peak-to-peak ripple voltage.

Exercises

Common pitfalls

• Connecting LEDs without current limitation: The LED current can become destructive.
• Forgetting resistor power: In \(24~{\rm V}\) control cabinets, LED resistors can dissipate noticeable heat.
• Using a Z-diode without load-current check: The Z-current must remain between \(I_{\rm Z,min}\) and \(I_{\rm Z,max}\).
• Using clamp diodes without a series resistor: The clamp current must be limited.
• Thinking the freewheeling diode removes energy instantly: It gives the current a safe path, but turn-off may become slower.
• Ignoring diode drops in bridge rectifiers: Two diodes conduct at the same time.
• Confusing RMS and peak values: A \(12~{\rm V}\) RMS sine has a peak value of about \(17~{\rm V}\).
• Assuming a smoothing capacitor creates perfect DC: The output still has ripple and charging-current pulses.
• Using capacitor formulas without checking ratings: Check voltage rating, ripple current, polarity, and inrush current.

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