This project is based on a project by Brian Winkel. https://www.simiode.org/resources/518

4-23-S-circuit.png

The goal of this scenario is to determine what the circuit shown above does. There is a voltage source on the left of the circuit. We will measure the gain of the circuit by finding the ratio of the amplitude of the voltage across the resistor labeled $R_{load}$ to the amplitude of the voltage source. The voltage across $R_{load}$ is $R_{load}$ times the current through the resistor. So, we will focus on the current.

Label the currents:

  • current across $R_1$: $i_1$
  • current across $C_1$: $i_2$
  • current across $C_2$ and $R_{load}$: $i_3$
  • Use Kirchhoff's Voltage Law around Loop I to write one differential equation.
  • Use Kirchhoff's Voltage Law around Loop II to write a second differential equation.
  • Use Kirchhoff's Current Law at the node labeled 2 to find a relationship between the 3 currents.

Enter these equations into a Matlab live scripts file.

Here are the basic steps:

  • Enter the 3 equations.
  • Have Matlab take the Laplace transform of each.
  • The Laplace transform of $i_3$ is $I_3$ - have Matlab find $I_3$
  • Find the inverse Laplace transform of $I_3$
  • The voltage across $R_3$ is $R_3 \cdot i_3$
  • Finally compute the ratio: $\frac{\text{ output voltage amplitude}}{\text{ input voltage amplitude}}$, this is called the gain

When you find $i_3$ is should look something like: $$A\cos(\omega t) + B \sin(\omega t) +e^{- \text{something}}\cdot \text{a big mess}$$ The last term is transient: it goes to 0 as $t \rightarrow \infty$. One can use trig identities to show that the amplitude of this current is $\sqrt{A^2+B^2}$. Read the values of $A$ and $B$ from the Matlab output and have Matlab compute the amplitude of $i_3$. (Keep lots of decimal places.)

Everyone in class has been assigned a different value of $\omega$.

Come to class with your value of $\omega$ and your computed gain.

The Matlab file has the values of the resistors and capacitors entered. The voltage source is $v(t)=\sin(\omega t)$; this is also entered in the Matlab file.

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