How to clone a PowerShell PSCustomObject variable to disconnect it from another variable?

Question

I'm looking for a better solution to the following problem. This code:

$a = "{value:'1'}" | ConvertFrom-Json
$b = $null
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$b = $a
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$a.value = '2'
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$b.value = '3'
Write-Output ("a: {0} - b: {1}" -f $a, $b)

Returns this:

a: @{value=1} - b: 
a: @{value=1} - b: @{value=1}
a: @{value=2} - b: @{value=2}
a: @{value=3} - b: @{value=3}

What I need is this: after setting $b to $a I should be able to change $a while $b remains unchanged.

The only solution I could come up with is do a convert to and from JSON, like this:

$a = "{value:'1'}" | ConvertFrom-Json
$b = $null
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$b = $a | ConvertTo-Json -depth 100 | ConvertFrom-Json
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$a.value = '2'
Write-Output ("a: {0} - b: {1}" -f $a, $b)
$b.value = '3'
Write-Output ("a: {0} - b: {1}" -f $a, $b)

That returns my desired result:

a: @{value=1} - b: 
a: @{value=1} - b: @{value=1}
a: @{value=2} - b: @{value=1}
a: @{value=2} - b: @{value=3}

I'm looking for a better solution because in my real-world example the variables are large JSON files that are loaded from disk, updated and saved via a PUT to an API.

Update: PSObject.Copy() does not solve my real-world problem because that JSON has nested objects (as Ansgar Wiechers predicted in his answer).

See this example for the difference in behaviour between ConvertTo/From-Json and PSObject.Copy():

$a = "{value:'1',nested:{foo:'original value'}}" | ConvertFrom-Json
$b = $a | ConvertTo-Json -depth 100 | ConvertFrom-Json
$a.nested.foo = 'changing $a does not change $b'
$b | ConvertTo-Json -Depth 1
$c = $a.PSObject.Copy()
$a.nested.foo = 'changing $a changes $c!'
$c | ConvertTo-Json -Depth 1

Result:

{
    "value":  "1",
    "nested":  {
                   "foo":  "original value"
               }
}
{
    "value":  "1",
    "nested":  {
                   "foo":  "changing $a changes $c!"
               }
}

Update 2: the code in the other question that my question is marked as a duplicate of does indeed work in my case (see below), but I'll stick to my ConvertTo/From-Json because it's a lot simpler...

$a = "{value:'1',nested:{foo:'original value'}}" | ConvertFrom-Json

$ms = New-Object System.IO.MemoryStream
$bf = New-Object System.Runtime.Serialization.Formatters.Binary.BinaryFormatter
$bf.Serialize($ms, $a)
$ms.Position = 0
$b = $bf.Deserialize($ms)
$ms.Close()

$a.nested.foo = 'changing $a does not change $b'
$b | ConvertTo-Json -Depth 1

Result:

{
    "value":  "1",
    "nested":  {
                   "foo":  "original value"
               }
}