7.4 Power Supply Design & Voltage Regulators

7.4 Power Supply Design & Voltage Regulators

Quick Answer: Power supply filters smooth pulsating DC from rectifiers into steady DC. Capacitor filter ripple is $V_{r(p-p)} = \frac{I_{DC}}{f_{ripple} C}$. Zener regulators hold voltage $V_Z$ constant across load changes. Fixed 3-terminal regulators (78xx positive, 79xx negative) deliver fixed voltages. Adjustable regulators like LM317 produce $V_{out} = 1.25\text{V} \left(1 + \frac{R_2}{R_1}\right) + I_{adj} R_2$. Switched-Mode Power Supplies (SMPS)—Buck (step-down: $V_{out} = D V_{in}$), Boost (step-up: $V_{out} = \frac{V_{in}}{1-D}$), and Buck-Boost ($V_{out} = -\frac{D}{1-D} V_{in}$)—achieve $85-95\%$ efficiency compared to $40-60\%$ for linear regulators.

Power Supply Filtering & Smoothing Circuits

The output of a rectifier is pulsating DC containing a desired DC component $V_{dc}$ and an undesirable AC ripple voltage component $V_r$.

1. Filter Topologies & Ripple Expressions

• Capacitor Input Filter (C-Filter): A large electrolytic capacitor placed in parallel across the load charges to peak voltage $V_m$ during diode conduction and discharges into $R_L$ during non-conduction.
  • Peak-to-Peak Ripple Voltage: $$V_{r(p-p)} = \frac{I_{DC}}{f_{ripple} C} = \frac{V_{DC}}{f_{ripple} R_L C}$$
  • DC Output Voltage under Load: $$V_{DC} = V_m - \frac{V_{r(p-p)}}{2} = V_m - \frac{I_{DC}}{2 f_{ripple} C}$$
  • RMS Ripple Voltage & Ripple Factor: $$V_{r(rms)} = \frac{V_{r(p-p)}}{2\sqrt{3}} \implies \gamma = \frac{V_{r(rms)}}{V_{DC}} = \frac{1}{2\sqrt{3} f_{ripple} R_L C}$$
• LC Filter (Choke Input Filter): Series inductor $L$ blocks high-frequency AC ripple currents, while shunt capacitor $C$ bypasses remaining AC ripple to ground. Ripple factor is independent of load current: $$\gamma = \frac{\sqrt{2}}{3 \omega^2 L C} \quad (\text{for full-wave rectifier at } 2\omega)$$
• $\pi$-Filter (CRC / CLC Filter): Combines an input capacitor, series choke or resistor, and output capacitor, yielding extremely low ripple factor at the expense of higher component cost and size.

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Linear Voltage Regulators: Zener & Integrated Circuits

Voltage regulation ensures the output DC voltage remains stable despite changes in AC line voltage (Line Regulation) or load current (Load Regulation).

• Line Regulation (%): $\frac{\Delta V_{out}}{\Delta V_{in}} \times 100\%$
• Load Regulation (%): $\frac{V_{NL} - V_{FL}}{V_{FL}} \times 100\%$

1. Zener Diode Shunt Regulator

A Zener diode connected in reverse bias across the load regulates output voltage at $V_Z$.

• Series current-limiting resistor $R_S$: $$R_S = \frac{V_{in} - V_Z}{I_S} = \frac{V_{in} - V_Z}{I_Z + I_L}$$
• Operational Bounds:
  • To maintain regulation at maximum load ($I_{L(max)}$), Zener current must not fall below $I_{Z(min)}$ (typically $5\text{ mA}$): $$R_{S(max)} = \frac{V_{in(min)} - V_Z}{I_{Z(min)} + I_{L(max)}}$$
  • At zero load ($I_L = 0$), Zener current must not exceed maximum power dissipation $I_{Z(max)} = \frac{P_{Z(max)}}{V_Z}$: $$R_{S(min)} = \frac{V_{in(max)} - V_Z}{I_{Z(max)}}$$

2. Fixed IC Regulators (78xx & 79xx Series)

Standard 3-terminal linear voltage regulators feature internal thermal shutdown, current limiting, and safe-area protection.

• 78xx Series: Positive fixed voltage regulators (e.g., 7805 = $+5\text{V}$, 7812 = $+12\text{V}$, 7824 = $+24\text{V}$).
• 79xx Series: Negative fixed voltage regulators (e.g., 7905 = $-5\text{V}$, 7912 = $-12\text{V}$).
• Pinouts (TO-220 Package):
  • 78xx: Pin 1 = Input, Pin 2 = Ground, Pin 3 = Output.
  • 79xx: Pin 1 = Ground, Pin 2 = Input, Pin 3 = Output.

3. Adjustable IC Regulator (LM317)

The LM317 is a 3-terminal positive adjustable regulator maintaining a precise reference voltage $V_{ref} = 1.25\text{ V}$ between its Output and Adjust pins.

• Output Voltage Formula: $$V_{out} = V_{ref} \left(1 + \frac{R_2}{R_1}\right) + I_{adj} R_2 = 1.25\text{V} \left(1 + \frac{R_2}{R_1}\right) + I_{adj} R_2$$
• Since adjustment pin current $I_{adj}$ is small ($\approx 50\ \mu\text{A}$), it is often neglected: $$V_{out} \approx 1.25\text{V} \left(1 + \frac{R_2}{R_1}\right)$$
• Recommended setting: Choose $R_1 \approx 240\ \Omega$ to ensure minimum load current ($\approx 5\text{ mA}$).

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Switched-Mode Power Supplies (SMPS) & DC-DC Converters

Switched-Mode Power Supplies switch a power transistor (MOSFET) fully ON and fully OFF at high frequencies ($50\text{ kHz} - 1\text{ MHz}$), controlling output voltage by varying the Pulse-Width Modulation (PWM) duty cycle $D = \frac{t_{on}}{T}$.

Basic Non-Isolated Converter Topologies

Linear vs. Switched-Mode Power Supply Comparison

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Step-by-Step Worked Examples

Example 1: LM317 Output Voltage Calculation

Problem: An LM317 regulator circuit uses $R_1 = 240\ \Omega$ and $R_2 = 2.4\text{ k}\Omega$ ($2400\ \Omega$). The adjust pin current is $I_{adj} = 50\ \mu\text{A}$. Calculate the regulated output voltage $V_{out}$.

Solution:

1. Identify Formula: $$V_{out} = 1.25 \left(1 + \frac{R_2}{R_1}\right) + I_{adj} R_2$$
2. Calculate Main Term: $$1.25 \left(1 + \frac{2400}{240}\right) = 1.25 (1 + 10) = 1.25 \times 11 = 13.75\text{ V}$$
3. Calculate $I_{adj}$ Offset Term: $$I_{adj} R_2 = (50 \times 10^{-6}\text{ A}) \times (2400\ \Omega) = 0.12\text{ V}$$
4. Total Output Voltage: $$V_{out} = 13.75\text{ V} + 0.12\text{ V} = 13.87\text{ V}$$

Example 2: Buck Converter Duty Cycle & Output Voltage

Problem: A Buck DC-DC converter operates with an input voltage $V_{in} = 24\text{ V}$ and a switching frequency of $100\text{ kHz}$. If the switch is ON for $t_{on} = 3\ \mu\text{s}$ during each period, calculate: (a) Switching period $T$, (b) Duty cycle $D$, and (c) Regulated output voltage $V_{out}$.

Solution:

1. Calculate Switching Period $T$: $$T = \frac{1}{f_s} = \frac{1}{100 \times 10^3\text{ Hz}} = 10\ \mu\text{s}$$
2. Calculate Duty Cycle $D$: $$D = \frac{t_{on}}{T} = \frac{3\ \mu\text{s}}{10\ \mu\text{s}} = 0.30 \quad (30\%)$$
3. Calculate Output Voltage $V_{out}$: $$V_{out} = D \times V_{in} = 0.30 \times 24\text{ V} = 7.2\text{ V}$$