Current Switching DAC The circuit in given Figure is a four-bit DAC. The DAC output is the voltage vO and the input is the binary code represented by bits b₁, b₂, b₃, and b₄. The input bits are either 0 (low) or 1 (high), and each controls one of the four switches in the figure.

When bits are low, their switches are in the left position, directing the 2R leg currents to ground. When bits are high, their switches move to the right position, directing the 2R leg currents to the OP AMP's inverting input. The 2R leg currents do not change when switching from left to right because the inverting input is a virtual ground ð Þ vN = vP = 0 . The purpose of this problem is to show that this constantcurrent switching produces the following input-output relationship.

(a) Since the inverting input is a virtual ground, show that the currents in the 2R legs are i₁ = V_REF/2R, i₂ = V_REF/4R, i₃ = V_REF/8R, and i₄ = V_REF/16R, regardless of switch positions.

(b) Show that the sum of currents at the inverting input is

b₁ i₁ + b₂ i₂ + b₃ i₃ + b₄ i₄ + i_F = 0 where bits b_k(k = 1, 2, 3, 4) are either 0 or 1.

(c) Use the results in parts (a) and (b) to show that the OP AMP output voltage is