Hybrid Mu-follower Output Impedance

Where to start?

Well, I often get the question “what is the output impedance of the gyrator circuit?”. My response has always been “it gets close to 1/gfs as a good approximation”. Recently, I was asked the question again, but this time I decided to crack on the formulae, which is a daunting task for someone who has ditched calculus after obtaining Ms in Engineering degree more than 20 years ago.

For simplification let’s start with a simple hybrid mu-follower stage (a.k.a. Gyrator load) like the following:

When the output is taken from the source this single-ended push-pull (SEPP) circuit is called mu-follower. For simplicity I omitted the constant-voltage version and instead a constant current (CCS) mode is shown above. Either way the principle is the same.

The triode can be replaced with its equivalent anode resistance (Ra or Rp). When we look at the impedance from the output, the input is shorted so we can connect the equivalent Ra to ground like is shown below:

The following diagram represent the equivalent model of the MOSFET with transconductance gm (or gfs) and internal resistance of Ro (or Rds). The impedance at the output can be found by placing an ideal voltage source (Vx) and measure its current looking at the circuit (Ix) when the input is held to ground (Vi=0):

We can rearrange the previous circuit to make it simpler to our eyes before we embark on the maths and group some of the Rs and 1/gm resistances under R’:

I spent one lunch break to crack on the formulae, which requires a bit of work:

So what does this formula tells us?

• The output impedance of the valve (Rp) has an impact on the output impedance. However, this is reduced by the bootstrapping effect of the FET.
• The higher the transconductance (gm or gfs) the better. Why? Higher gm will reduce the output impedance by bootstrapping more Rs.
• Higher gm or gfs means more current. This has a penalty on both the power dissipated across the FET as well as the voltage drop across Rs and therefore the increased need for more volts at HT.
• When Rs>>Rp and Rs*Gm>>1 then the output impedance approaches to 1/gm or 1/gfs. This means you need a low anode resistance triode (Rp), large anode current and Rs to approach to this value.  Most of the jFET used in this topology will approach to 20mS, whereas other MOSFETs can do 200-250mS at same current levels (e.g. 20-30mA). This means that the output impedance can get as low as 4-5Ω. In reality, you will see a contribution of Rp regardless as is unlikely you will achieve Rs>>Rp without sacrificing too many volts dropped across the gyrator due to the size of Rs.

The proof is in the pudding

Let’s use LTSpice to show this formula in action. A simple example is illustrated below with a BF862 jFET. An ideal valve is represented with a 1KΩ Rp and idle current of 3mA. The transconductance (gm or gfs) of the BF862 is shown on the graph to the right. At 3mA is about 20.9mS in the LTSpice model.

I used an Rs of 100Ω. If we plug in all the values in the formula:

Which correlates well with the simulation below:

Doggy bag

It’s good to have the formula to understand well how all components play on the output impedance. In particular we need to remember this:

• A good FET is needed with good gfs at the idle anode current. Typically the jFETs used (2SK170, BF862, etc.) have 10 times less gfs than a MOSFET. However, if we are looking to implement a CCS load with mu output, the depletion devices are the ones which work. The enhancement devices (ie. MOSFETs) can only be used in the gyrator topology where is a constant voltage CCS load.
• Higher gm or gfs means more current. This has a penalty on both the power dissipated across the FET as well as the voltage drop across Rs and therefore the increased need for more volts at HT. It’s always a question of compromise, as there is no free lunch. I typically use 470Ω to 1K5Ω as Rs (or R-MU as I call it in my circuit design).  With an BSH111BK MOSFET at 25-30mA and a 470Ω Rs, the output impedance is reduced by a factor of 120. So if you have a 4P1L with 1K6Ω anode resistance, the output impedance of the gyrator stage is as low as 18Ω.
• The higher Rs also the better for the valve. The bootstrapping effect of the FET will increase the reflected load into the valve and therefore improve the linearity of the stage. This is a good thing!

I ran out of time today, so hope this is useful to folks out there!  Enjoy the weekend!