How to change the radius markers and number of concentric circles in a polar plot?

Question

I'm using PolarPlot to plot a function in [0, 1] in a log scale plot, capped at -40 dB, as in the following example:

logscale[x_, minDb_] =
  If[ x < 10^(minDb/ 10), minDb, 10 Log10[x] ]/(-minDb) + 1 ;

PolarPlot[
 logscale[Abs[Cos[t]], -40],
 {t, 0, Pi},
 PolarAxesOrigin -> {0, 1 },
 PolarGridLines -> Automatic,
 PolarAxes -> True,
 PlotStyle -> Thick,
 PolarTicks -> {Table[{N[Pi/6 i],
     ToString[30 i] <> "\[Degree]"}, {i, -5, 6}], Automatic}
 ]

I'd like to know how to make it so that there are three concentric circles marking the radial depth in the interior (instead of 4), and would like the radial tick numbers (from the outside) to be 0, -10, -20, -30, and -40 at the origin, instead of the automatic values 1, 0.8, ...

Answer 1

How about this?

PolarPlot[
  logscale[Abs[Cos[t]], -40],
  {t, 0, Pi},
  PolarAxesOrigin -> {0, 1},
  PolarGridLines -> {Automatic, {1/4, 2/4, 3/4}},
  PolarAxes -> True,
  PlotStyle -> Thick,
  PolarTicks -> {
    Table[{N[Pi/6 i], ToString[30 i] <> "\[Degree]"}, {i, -5, 6}],
    {{1, 0}, {3/4, -10}, {2/4, -20}, {1/4, -30}, {0, -40}}
  }
]