Wheatstone bridge schematic

For Wheatstone bridge with known resistances R1, R3, R4 adjusted for im = 0:

	R2 = R1*R4/R3
For unbalanced bridges, given v, im and fixed resistors,
get unknown i1, i2, i3, i4, R2 values
using equations [1], [2], [3], [4], [5] by Kirchhoff's circuit laws:
	i1 = im + i2          [1]
	im = i4 - i3          [2]
	im*Rm = i3*R3 - i1*R1 [3]
	im*Rm = i2*R2 - i4*R4 [4]
Instead of ideal EMF source v,
nanoVNA provides stimulus Vf with source impedance Rs ~ Rm: 
	Vf + (i1 + i3)*Rs + i1*R1 + i2*R2 = 0 [5]
click for im vs R2 derivation If R1 = R3 = R4 = Rm, then: 
	im = i3 - i1                [6] // from [3]
	2*im = i3 - i2              [7] // [6], [1]
	2*im = i4 - i1              [8] // [6], [2]
	i3 = i2 + 2*im              [9] // [7]
	i4 = 2*im + i1 = 3*im + i2 [10] // [8], [1]
Eliminate i1, i3 using [1], [9]: 
	Vf + (im + i2 + i2 + 2*im)*Rs + (im + i2)*R1 + i2*R2 = 0 // [5]
	Vf + 3*im*Rm + 2*i2*Rm + im*Rm + i2*Rm + i2*R2 = 0       // Rs ~ Rm = R1
	i2 = - (Vf + im*3*Rm + im*Rm)/(3*Rm + Rm)
	   = -im - Vf/(4*Rm)                      [11]

	i2 = im*Rm/R2 + i4*Rm/R2                        // [4]
	   = im*Rm/R2 + (3*im + i2)*Rm/R2               // [10]
	   = 4*im*Rm/R2 + i2*Rm/R2
Multiply by R2/Rm and rearrange:
	4*im = (-1 + R2/Rm)*i2
	     = (-1 + R2/Rm)*(-im - Vf/(4*Rm))           // [11]
	     = (1 - R2/Rm)*(im + Vf/(4*im))
	     = im + Vf/(4*im) - im*R2/Rm -Vf*R2/(4*im*Rm)
	3*im = Vf/(4*im) - im*R2/Rm -Vf*R2/(4*im*Rm)
Multiple by 4*im*Rm and rearrange: 
	12*Rm*im*im = Vf*Rm - 4*R2*im*im - Vf*R2
	im*im*(12*Rm + 4*R2) = Vf*(Rm - R2)
	im = sqrt(Vf*(Rm - R2)/(12*Rm + 4*R2))

Unbalanced bridge response

observations

  1. Bridge response from 0 to 50 Ohms matches 50 to 150 Ohms response.
  2. Upper 50% of R2 range provokes less than 10% response change.
  3. Impedances farther from 50 Ohms provoke larger im signals, improving signal-to-noise, but with less sensitivity to impedance differences.
  4. Polarity change confounds response to AC stimulus of nonlinear 50 Ohm loads.
gnuplot commands for bridge response plot 
set xrange [0:1000]
set yrange [-4:2.5]
set xtics 50
set ytics 0.5
set grid xtics ytics
set ylabel "arbitrarily scaled 50 Ohm response"
set xlabel "unknown R2 in Ohms"
set title "unbalanced 50 Ohm bridge"
p (x<50)?8*sqrt((50-x)/(600 + 4*x)) : -8*sqrt((x-50)/(600 + 4*x)) title "meter current i_m"