# Is there a high-frequency μ transform?

-June 10, 2014

In the high-frequency (hf) region of transistor operation, the β transform can be used to refer gyrated circuit elements of the base (gate) to the emitter (source). Does the dual μ transform have a similar effect in the hf region whereby emitter (source) or collector (drain) elements are gyrated?

The β Transform

A basic circuit theorem that has dual parts to it (as do multiple circuit theorems) is the reduction theorem. It has current and voltage forms. The current form of the theorem is familiar because of its frequent use in circuit analysis and is the β transform, illustrated below.

It is the transform used to move equivalent resistances between base and emitter branches of the BJT input loop.

The μ Transform

The β transform has a less familiar dual, the μ transform. It is applied to electron tube and JFET circuits because of their relatively low plate or drain resistance. With the need to include ro in the BJT model, it is derived from the Early voltage, VA, as shown below.

where

The introduction of ro leads to the quasistatic (frequency-independent, or “low-frequency ac”) μ, or

where rm = 1/gm­ = re/α0 of the BJT; rm­ is the BJT transresistance and re is the incremental emitter resistance (» 26 mV/|IE|) of the BJT T model, and is omitted in the above diagram. For a typical BJT such as the PN3904 for which typical β0 = 150, VA≈100 V. Then for |VCE| << VA,

For electron tubes and JFETs, μ is considerably lower, in the range of 20 to a few hundred.

The μ transform is presented in the diagrams below. N1 represents the transistor input (base or gate) circuit and N2 is the output (collector or drain) circuit. Between them is the dependent voltage source, μ·v. The μ transform, like the β transform, refers resistance between N1 and N2 as shown in the lower two block diagrams. Referring N2 resistance to the N1 circuit reduces it by μ0 + 1. Resistance in N1 looks μ0 + 1 times larger from N2. For a BJT with μ0 = 2999, an emitter resistance of 1 kΩ will be referred to (or “be viewed from”) the collector terminal as 3 MΩ - a large resistance that can be (and usually is) ignored.

The BJT T model can be modified by converting it to a hybrid-π model (using the β transform) and then thevenizing it, as shown below. The dependent voltage source in series with ro is μ·vbe. Then β·ib is expressed in vbe as

The dependent voltage source corresponds to the source shown above in the general μ-transform circuit. The polarity of the voltage source, as shown in the lower diagram below, is consistent with the polarity of voltage developed in the Norton circuit (upper right) given the polarity of the dependent current source.

In the quasistatic (no reactances) hybrid-π model, β = β0 and μ = μ0. Then ib is the quasistatic current through rπ and vbe is the quasistatic voltage across it.