plug v_x = v_h((1/(s C_4 R_4)) + 1) into v_x (s R_3 C_3 + s R_3 C_4 +1) - v_i (s R_3 C_3) - v_h (s R_3 C_4), then expand this new equation, then simplify the equation in a form where (v_h / v_i) = ___

Let's break this problem into manageable steps and systematically solve for \( \frac{v_h}{v_i} \). 

### Step 1: Plug in \( v_x = v_h \left( \frac{1}{s C_4 R_4} + 1 \right) \) into the given equation.

We are given the equation:

\[
v_x \left( s R_3 C_3 + s R_3 C_4 + 1 \right) - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

Now, substitute \( v_x = v_h \left( \frac{1}{s C_4 R_4} + 1 \right) \) into the above equation:

\[
v_h \left( \frac{1}{s C_4 R_4} + 1 \right) \left( s R_3 C_3 + s R_3 C_4 + 1 \right) - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

### Step 2: Expand the equation.

Expand the first term on the left-hand side:

\[
v_h \left[ \left( \frac{1}{s C_4 R_4} + 1 \right) \left( s R_3 C_3 + s R_3 C_4 + 1 \right) \right] - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

Distribute \( \left( \frac{1}{s C_4 R_4} + 1 \right) \) over \( \left( s R_3 C_3 + s R_3 C_4 + 1 \right) \):

\[
v_h \left[ \frac{1}{s C_4 R_4} \left( s R_3 C_3 + s R_3 C_4 + 1 \right) + \left( s R_3 C_3 + s R_3 C_4 + 1 \right) \right] - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

Simplify each part:

\[
v_h \left[ \frac{s R_3 C_3}{s C_4 R_4} + \frac{s R_3 C_4}{s C_4 R_4} + \frac{1}{s C_4 R_4} + s R_3 C_3 + s R_3 C_4 + 1 \right] - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

Cancel out the common terms where possible:

\[
v_h \left[ \frac{R_3 C_3}{C_4 R_4} + \frac{R_3 C_4}{C_4 R_4} + \frac{1}{s C_4 R_4} + s R_3 C_3 + s R_3 C_4 + 1 \right] - v_i \left( s R_3 C_3 \right) - v_h \left( s R_3 C_4 \right) = 0
\]

### Step 3: Simplify and collect like terms.

First, combine terms that include \( v_h \):

\[
v_h \left[ \frac{R_3 (C_3 + C_4)}{C_4 R_4} + \frac{1}{s C_4 R_4} + s R_3 (C_3 + C_4) + 1 \right] = v_i \left( s R_3 C_3 \right)
\]

### Step 4: Isolate \( \frac{v_h}{v_i} \).

Now, divide both sides by \( v_i \) and simplify:

\[
\frac{v_h}{v_i} = \frac{s R_3 C_3}{\frac{R_3 (C_3 + C_4)}{C_4 R_4} + \frac{1}{s C_4 R_4} + s R_3 (C_3 + C_4) + 1}
\]

This is the simplified form for \( \frac{v_h}{v_i} \).

### Conclusion:

The final simplified equation for \( \frac{v_h}{v_i} \) is:

\[
\frac{v_h}{v_i} = \frac{s R_3 C_3}{\frac{R_3 (C_3 + C_4)}{C_4 R_4} + \frac{1}{s C_4 R_4} + s R_3 (C_3 + C_4) + 1}
\]

Would you like further clarification or details on any of the steps?

### Related Questions:
1. What are the physical meanings of \( C_3 \), \( C_4 \), \( R_3 \), and \( R_4 \) in this circuit?
2. How would this equation change if \( C_4 \) is much smaller than \( C_3 \)?
3. Can we analyze the high-frequency and low-frequency behavior of this system using the derived equation?
4. How would the solution change if \( v_h = v_i \)?
5. What does the term \( \frac{v_h}{v_i} \) represent in the context of this circuit?

### Tip:
When simplifying complex equations, always aim to collect like terms and isolate the variable of interest early to avoid unnecessary complexity later.

Learn how to simplify a complex RC circuit equation by solving for v_h / v_i. This problem involves applying algebraic manipulations and Laplace transforms, and is suitable for electrical engineering or physics students at the undergraduate level.

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