EE142 Lecture7

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Amin ArbabianJan M. Rabaey

Lecture 7: Two-Ports (2)

EE142 – Fall 2010

Sept. 16th, 2010

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Announcements

HW3 Posted, due Thursday September 23rd, 330pm EE142 drop box.

Office hours

Two-Port notes by Professor Niknejad as well as Chapter 7 of the book are posted online

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Review

Two-Port Networks– Abstraction

– Transformations

– Connecting two-ports

– Transformer as a two-port

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Connecting Two-Ports

Notice that a series connection of two two-ports implies the same current flows through both two-ports whereas the voltage across the two-ports is the sum of the individual voltages.

On the other hand, a shunt connection of two two-ports implies the same voltage is applied across both two-ports whereas the current into the two-ports is the sum of the individual currents.

These simple observations allow us to simply sum two-port parameters for various shunt/series interconnections of two-ports.

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Y-parameters from Hybrid-pi Model

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Y-parameters from Hybrid-pi Model (2)

Note that the hybrid- model is unilateral if Yμ = sCμ = 0. Therefore it’s unilateral at DC.– Stability more of a concern at higher frequencies

A good amplifier has a high ratio y21/y12 because we expect the forward transconductance to dominate the behavior.

How else could we quickly derive the Y-parameters?

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Selection of Two Ports

Practical issues– Measurements (Y-Params vs Z-Params)

– Difficulty of presenting an AC open circuit

– Short circuit, perhaps up to 100MHz

Circuit topology– Feedback

– Connections: Cascade, Series, Shunt,…

– Physical interpretation

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Feedback

Ideal feedback – Separate feedforward and feedback networks

– Easy to analyze

Transformation to the ideal form– Real networks are bilateral and introduce loading

– Use knowledge of two-ports

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Feedback (2)

Since the overall two-port parameters of the amplifier in closed loop is simply the sum of the amplifier and feedback network two-port parameters, we can simply move the non-idealities of the feedback network (loading and feed-forward) into the main amplifier and likewise move the intrinsic feedback of the amplifier to the feedback network.

Now we can use ideal feedback analysis.

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Example (Shunt-Shunt Feedback)

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Example (continued)

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Voltage Gain

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Input/Output Impedance

The input admittance is easily calculated from the voltage gain

Output impedance

Unilateral case:

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External Voltage Gain

The gain from the voltage source to the output can be derived by a simple voltage divider equation

Low Frequencies:

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Feedback Representation

If we unilaterize the two-port by arbitrarily setting Y12 = 0, we have an “open” loop forward gain of:

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Feedback Factor

Using the last equation also allows us to identify the feedback factor

If we include the loading by the source YS, the input admittance of the amplifier is given by

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Effect of Feedback on input/output Admittance

The last equation can be re-written as

Since YS + y11 is the input admittance of a unilateral amplifier, we can interpret the action of the feedback as raising the input admittance by a factor of 1 + T.

Likewise, the same analysis yields

It’s interesting to note that the same equations are valid for series feedback using Z parameters, in which case the action of the feedback is to boost the input and output impedance.

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Two Port Stability

When the source provides energy the ratio of V/I is negative

If the real part of input impedance is negative then the active network is sourcing power to our tank– Where is this power coming from?

Stability concerns if this compensates the losses in the tank…

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Quantitative Treatment of Stability

The two-port network is unstable if it supports non-zero currents/voltages with passive terminations

But we have terminations on both sides

The only way to have a non-trial solution is for the determinant of the matrix to be zero at a particular frequency

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Stability and Loop Gain

Taking the determinant of the matrix we have

Where we have identified the loop gain T. We can clearly see that instability implies that T = -1, which is exactly what we learned in feedback system analysis.

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Stability and Impedances

Starting from the determinant we have:

Or:

This can be expressed as:

Rewriting from the output yields

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Impedance Conditions

This means that the terminated network is unstable if the net admittances are zero at input/output . This implies that:

Can also show that:

This agrees with our initial (intuitive) observations.

What if the termination values are not known/fixed?

yields

For a specific termination

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Stability Conditions

Inherent stability

For a unilateral amplifier we have:

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Stability Factor

In general, it can be shown that a two-port is absolutely stable iff

The stability factor k is given by

The stability of a unilateral amplifier with y12 = 0 is infinite k = ∞ which implies absolute stability since as long as

(y11) > 0 and (y22) > 0

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More on Stability Factor

A amplifier with absolute stability or unconditional stability (k > 1) means that the two-port is stable for all passive terminations at either the load or the source.

If k < 1, then the system can be conditionally stable, or stable for a range of source/load impedances. This range of impedance is very easily calculated using scattering parameters. It’s also possible for a system to be completely unstable.

Unconditional stability is very conservative if the source and load impedance is well specified and well controlled.

But in certain situations the load or source impedance may vary greatly. For instance the input impedance of an antenna can vary if the antenna is disconnected, bent, shorted, or broken.

An unstable two-port can be stabilized by adding sufficient loss at the input or output to overcome the negative conductance.

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Power Gain

We can define power gain in many different ways. The power gain Gp is defined as follows

We note that this power gain is a function of the load admittance YL and the two-port parameters Yij.