Extra Element Theorem, Understanding Gain and Phase Margin, and Bode Plots

 

The Extra Element Theorem helps to find/determine what gain is due to a single element. We want to use this theorem to study extra capacitors, and calculate the gain and the frequency dependence. 

We care comparing the gain with or without the element using the following equation: 

Here, H∞ it the open circuit gain, Z(s) is the capacitor impedance, Z(n) is the impedance when the output is 0, which puts the test source at the position of its drive current. Z(d) is the impedance looking into the rest of the circuit. Z(n) = Vt/It and we want to choose Thevenin current such that the output = 0. The source attachment can look very similar to the following:

There is a second equation to help determine the H(s), the short circuit impedence, and it reverses several terms, as follows: 

We now want to calculate the gain of a system with negative feedback. The gain is the following with feedback: 

We're examining where on the bode plot that the 1 + βA = 0. βA = -1. You can't have βA = -1,, as this corresponds without instability. 

Here, Wgc is the gain crossing frequency, and Wpc is the phase crossing frequency. You want Wgc < Wpc since you can't have  βA = -1 which is before Φg reaches -π. 

The gain margin is how far the gain is below 0, which is the gain margin. How far the phase is above -π is called the phase margin.

It's good to combine these plots together for reference.

The next section discusses on Bode plots. It follows logarithmic units.

Suppose X is a complex frequency with the equation S = σ + jw. Every step on the Bode Plot's horizontal axis corresponds to ascension on the vertical axis. This is the reversed when g = 1/jw, with the opposite slope.

Bode plots have polynomial factors and they are define by 4 factors:

(1 + s/wi), (1/(1 + s/wi)), (1 + wi/s), and (1/(1/ + wi/s)).

These are the following diagrams for the Bode plots:

1 + s/wi

1/(1 + s/wi)

Here again is a list of what the phase looks like.

Now to the homework.

I'm going to do problem 1 which is as follows:

Given the following circuit, calculate a capacitor's effects on frequency response, both directly, and by using the extra element theorem.

Here is the circuit: 

and here is the initial solution: 

Now this calculation will be represented by the extra element theorem, as follows: 

For the effect of the Capacitor CS on the Loop Gain TD, the EET states that 

TD(s) = TD0 (1 + Z(s)/ZN)/(1 + Z(s)/ZD) since removing CS in series with Rs requires shorting CS. 

Consider this diagram:

And here is the solution:

Now, all that needs to happen is the substitution of the Loop Gain with the Extra Element Theorem.

The 2 alternative forms that I'm supposed to write the Loop Gain in are the open and closed circuit. 

The next question that I'm going to cover asks to draw the pole at Wc. 

Here, G = 1 / (1 + s/wc).

The next plot that I want to draw is 1 + wc/s and it goes like the  following:

The next question is (1/(1 + wc/s))

1/i = -i.

And the other equation is (1 + s/w1) (1 + w2/s)

Now I want to discuss the quiz. 

We inject the circuit element on whatever we take out.