A Tale of Two Topologies

The questions below are due on Friday April 17, 2020; 10:00:00 AM.

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Question Focus

In the previous section you worked on questions dealing with compensators in a mathematical domain using Matlab tools like bode and margin to help reason about systems based on their transfer functions. These exercises shift the focus to exploring different circuit implementations (also known as topologies) of those compensators. However, still feel free to use matlab or other computational tools to aid in answering the questions.

Compensator Topologies

We will focus on two different topologies shown below.

We will refer to this topology as Topology 1 or the series topology for this question

We will refer to this topology as Topology 2 or the parallel topology for this question

For both topologies, if we lump together R_{lead}, R_{in}, and C_{lead} into a single element with impedance Z_{eq}, what is the transfer function from V_{in} to V_{out} (or equivalently 2.5 - V_{in} and V_{out} - 2.5) in terms of R_f and Z_{eq}? Use Z_eq for Z_{eq} and R_f for R_f.

\frac{V_{out}}{V_{in}}

Topology 1

Let's focus on topology 1 reproduced below:

What is the equivalent impedance, $Z_{1}$ of compensator formed by $R_{lead}, R_{in},$ and $C_{lead}$ in terms of $R_{lead}, R_{in},s,$ and $C_{lead}$? Use R_lead for $R_{lead}$, C_lead for $C_{lead}$, R_in for $R_{in}$.

Z_{1}

One useful way to write the system transfer function from V_{in} to V_{out} is in the form

K(s) = \frac{A}{B} * \frac{s + C}{s+D}

since it becomes much more obvious what the value of the pole and the zero are when it is written like this.

What are the values of \frac{A}{B}, C, and D for the entire system transfer function in terms of R_{lead}, R_{in}, C_{lead}, R_{f}, and s. Use R_lead for R_{lead}, C_lead for C_{lead}, R_in for R_{in}, R_f for R_{f}.

\frac{A}{B}

Often times we are interested in what happens at extremely high frequencies and extremely low frequencies (DC).

What is transfer function of whole network including R_f as the frequency approaches zero? This is known as the DC gain of the system. Assume R_{lead} = 100 Ohms, R_{in} = 10000 Ohms, C_{lead} = 10 uF, and R_{f} = 220000 Ohms.

DC Gain:

What is transfer function of whole network including R_f as the frequency approaches infinity? Assume R_{lead} = 100 Ohms, R_{in} = 10000 Ohms, C_{lead} = 10 uF, and R_{f} = 220000 Ohms.

High Frequency Gain:

What values of R_{lead}, R_{in}, and C_{lead} would you use to create the following transfer function

K(s) = \frac{s+25}{s+1000}

Assume R_f = 220000 Ohms.

R_{lead} (Ohms)

R_{in} (Ohms)

C_{lead} (nF)

Topology 2

Let's focus on topology 2 reproduced below:

What is the equivalent impedance, $Z_{2}$ of compensator formed by $R_{lead}, R_{in},$ and $C_{lead}$ in terms of $R_{lead}, R_{in},s,$ and $C_{lead}$? Use R_lead for $R_{lead}$, C_lead for $C_{lead}$, R_in for $R_{in}$.

Z_{2}

One useful way to write the system transfer function from V_{in} to V_{out} is in the form

K(s) = \frac{A}{B} * \frac{s + C}{s+D}

since it becomes much more obvious what the value of the pole and the zero are when it is written like this.

\frac{A}{B}

Often times we are interested in what happens at extremely high frequencies and extremely low frequencies (DC).

DC Gain:

What is transfer function of whole network including R_f as the frequency approaches infinity? Assume R_{lead} = 100 Ohms, R_{in} = 10000 Ohms, C_{lead} = 10 uF, and R_{f} = 220000 Ohms.

High Frequency Gain:

What values of R_{lead}, R_{in}, and C_{lead} would you use to create the following transfer function

K(s) = \frac{s+25}{s+1000}

Assume R_f = 220000 Ohms.

R_{lead} (Ohms)

R_{in} (Ohms)

C_{lead} (nF)

Put it all together

For a given compensator transfer function, can either topology be used to create it? (With different component values)

The difference between the two topologies may seem small at first, but lets see what happens as we change the value of R_{lead} relative to R_{in}. Assume R_{in} = 10000 Ohms, C_{lead} = 10 uF, and R_{f} = 220000 Ohms.

For R_{lead} = 100 Ohms

Topology 1 High Frequency Gain:

Topology 1 DC Gain:

Topology 2 High Frequency Gain:

Topology 2 DC Gain:

For R_{lead} = 1000 Ohms

Topology 1 High Frequency Gain:

Topology 1 DC Gain:

Topology 2 High Frequency Gain:

Topology 2 DC Gain:

For R_{lead} = 10000 Ohms

Topology 1 High Frequency Gain:

Topology 1 DC Gain:

Topology 2 High Frequency Gain:

Topology 2 DC Gain: