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Ch. 26 - DC Circuits
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Giancoli Douglas 5th editionCh. 26 - DC Circuits문제 95
25장, 문제 95

The circuit shown in Fig. 26–89 is a primitive 4-bit digital-to-analog converter (DAC). In this circuit, to represent each digit (2n) of a binary number, a “1” has the nᵗʰ switch closed whereas zero (“0”) has the switch open. For example, 0010 is represented by closing switch n = 1, while all other switches are open. Show that the voltage V across the 1.0 - Ω resistor for the binary numbers 0001, 0010, 0100, and 1010 (which represent 1, 2, 4, 10) follows the pattern that you expect for a 4-bit DAC.


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Step 1: Understand the circuit configuration. The circuit is a 4-bit digital-to-analog converter (DAC) where each switch corresponds to a binary digit (bit). When a switch is closed, it connects a voltage source to the circuit, contributing a current proportional to the bit's weight (2ⁿ). The total current through the 1.0-Ω resistor determines the output voltage V, as per Ohm's Law (V = IR).
Step 2: Analyze the contribution of each bit. Each bit corresponds to a specific current contribution based on its weight. For example, the least significant bit (LSB, n=0) contributes a current of I₀ = V₀/2⁰, the next bit (n=1) contributes I₁ = V₀/2¹, and so on. The total current is the sum of the contributions from all closed switches.
Step 3: Calculate the output voltage for each binary number. For 0001, only the LSB (n=0) is closed, so the current is I = V₀/2⁰. For 0010, only the n=1 switch is closed, so the current is I = V₀/2¹. For 0100, only the n=2 switch is closed, so the current is I = V₀/2². For 1010, the n=1 and n=3 switches are closed, so the current is I = V₀/2¹ + V₀/2³.
Step 4: Use Ohm's Law to find the voltage across the 1.0-Ω resistor. Since V = IR and the resistor has a resistance of 1.0 Ω, the voltage V is numerically equal to the total current I. For each binary number, substitute the calculated current into V = I to find the corresponding voltage.
Step 5: Verify the pattern. Observe that the output voltage V is proportional to the decimal value of the binary number. For example, 0001 (decimal 1) produces a voltage proportional to 1, 0010 (decimal 2) produces a voltage proportional to 2, and so on. This confirms the expected behavior of a 4-bit DAC, where the output voltage is directly proportional to the binary input value.

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Digital-to-Analog Conversion

Digital-to-Analog Conversion (DAC) is the process of converting digital signals, typically binary numbers, into analog voltages. In a 4-bit DAC, each bit represents a power of two, allowing for 16 distinct voltage levels. The configuration of switches in the circuit determines which bits are active, thus influencing the output voltage across a load, such as a resistor.

Binary Number Representation

Binary number representation uses two symbols, 0 and 1, to express values. Each digit in a binary number corresponds to a power of two, where the rightmost digit represents 2^0, the next 2^1, and so on. In the context of the DAC, the closed switches represent '1's, contributing to the total voltage output, while open switches represent '0's, contributing nothing.

Voltage Division in Circuits

Voltage division is a fundamental principle in electrical circuits where the total voltage across a series of resistors is divided among them based on their resistance values. In the DAC circuit, the output voltage across the 1.0-Ω resistor is determined by the configuration of the closed switches, which effectively creates a voltage divider that reflects the binary input as an analog voltage.
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