Solving for θ givesState the GoalSp

Solving for θ gives

State the Goal
Specify values of resistors R1 and R2, the potentiometer resistance RP, and the ampl ifier gain b
that will cause the meter voltage, vG, to be related to the angle θ by Eq. 4.12-1.
Generate a Plan
Analyze the circuit shown in Figure 4.12-2 to determine the relationship between v, and θ.
Select values of R1 R2, and Rp. Use these values to simplify the relationship between Vi and
θ. If possible, calculate the value of b that will cause the meter voltage, vQ, to be related to
the angle θ by Eq. 4.12-1. If this isn’t possible, adjust the values of R1, R2, and Rp and try
again.
Act on the Plan
The circuit has been redrawn in Figure 4.12-3. A single node equation will provide the
relationship between between v, and θ:

Solving for vi gives

This equation is quite complicated. Let’s put some restrictions on R 1,R 2, and Rp that will make
it possible to simplify this equation. First, let R1 = R2= R. Second, require that both R and Rp be
much smaller than 2 MΩ (for example, R < 2 0 kΩ). Then,

That is, the first term in the denominator of the left side of Eq. 4.12-4 is negligible compared to
the second term. Equation 4.12-4 can be simplified to

fIGURE 4.12-3 The redrawn circuit showing the mode vi.