Resolving Total Impedance (Zt) from Nodal Analysis Equations

Published on March 13, 2026

Algebra

Derive the expression for total impedance Zt = V0/Ix by algebraically eliminating the intermediate node voltage Vx from two coupled equations representing nodal analysis in an AC circuit.

Question

Resolve Zt

Answer

This problem requires resolving the total impedance

using the two given equations, which appear to be derived from nodal analysis (Kirchhoff’s Current Law) in an AC circuit containing impedances (likely an operational amplifier circuit).

The given equations are:

We need to manipulate these equations to find an expression for

in terms of the circuit components (

First, let’s simplify the impedance terms. The term

represents a parallel combination of a resistor

and a capacitor

, whose impedance is

Let

.
Let

The equations become:

Our goal is to eliminate

and express

in terms of

Step 1: Rearrange Equation (1) to isolate terms.

Step 2: Rearrange Equation (2) to isolate terms.

Group terms involving

on one side and

on the other:

Step 3: Substitute from Step 2 into the expression for derived from Step 1 (or simply equate the two expressions for if we used a different path, but here we substitute the expression for derived from (2) into the terms in (1)).

It is usually easier to substitute

into the equation for

(Equation 1).

Substitute

from Step 2 into Equation (1):

Substitute the expression for

from Step 2:

Step 4: Solve for .

Factor out

Now, invert this expression to find

Step 5: Simplify the denominator.

Let

be the denominator:

Combine terms inside the parentheses:

Substitute these back into the first term of

Now substitute back into

To combine these, we need a common denominator

Finally,

Step 6: Substitute back the definitions of and .

Recall:

The resulting expression for

is generally complex, involving all component values and the complex frequency

. The final simplified algebraic expression in terms of

and

is:

and

were infinite (or open circuits, i.e.,

and

), the expression would simplify. Given the notation

, we must use the derived algebraic expression involving

and

where

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