Ladder Input Impedance

After you have analyzed the Ladder Circuit, someone request an equation for the input impedance. If you work for a company with ISO recognized methodology, this can not happen. But in case you don’t, K9 Analysis can provide a quick solution. The general circuit schematic is shown below:

There are many ways to find the input impedance of a circuit. K9 tries to keep things simple. A simple approach is to use Ohm’s law. The input impedance Zin is equal to Vin divided by the Current following in Z1, I(Z1).

Zin = Va / I(Z1)

There are a few problems here:

• I(Z1) is not in the SFG
• I(Z1) is not an input
• Mason’s Gain Formula only applies for input to output transfer functions.

These problems are easy to solve.

We can add I(Z1) to the SFG by writing a defining equation. Ohm’s law can be used as follows:

I(Z1) = (Va – V1) / Z1

The above equation in SFG format is:

I(Z1) = 1/Z1 * Va – 1/Z1 * V1

The SFG with the I(Z1) node is shown below.

Whenever you add a node to a SFG, you are adding an equation. You need to verify that the new set of equations is valid. If the new SFG node has only incoming branches, it can not create loops and the SFG remains valid.

The last problem is that Mason’s Gain Formula does not apply to Va / I(Z1). We can however use Mason’s Gain Formula to find the inverse of Zin.

1/Zin = I(Z1) / Va.

In this formula Va is an input, and I(Z1) is an output. Problems solved. Let’s get the answer.

Analyze the SFG

The I(Z1) node did not add any loops to the SFG. From the Ladder analysis

L1 = ZP2/Z2 * ZP1/Z2

L2 = ZP3/Z3 * ZP2/Z3

L3 = ZPout/Z4 * ZP3/Z4

L1 and L3 are non-touching loops.

The determinant is:

D = 1 – (L1 + L2 + L3) + (L1 * L3)

There are two path from Va to I(Z1). The direct path does not touch any loops, the associated delta for this path is the same as D.

D1 = 1 – (L1 + L2 + L3) + (L1 * L3)

The second path via V1 touches L1. The associated delta for this path is

D2 = 1 – ( L2 + L3)

The transfer function from Va to I(Z1) is

Va/I(Z1) = (1/Z1 * D1 + ( ZP1/Z1)*(-1/Z1)*D2)/D

Since D1 is equal to D, the equation can be simplified.

1/Zin = Va/I(Z1) = 1/Z1 - ( ZP1/Z1)*(1/Z1)*D2)/D

Check the Answer

If Z6 is shorted the input impedance should be equal to Z1. Shorting Z6 will force ZP1 to be equal to zero. This makes

Va / I(Z1) = 1 / Z1

and

Zin = Z1

Conclusion

This example illustrated how to add a current variable to the SFG. A Current variable can be used to derive an input impedance equation. The input voltage source stimulates the input and the current variable measures the input current.

We now have the equation for the input impedance of a ladder circuit. You can extend the analysis to bigger ladders. Note that only the ∆ terms change.

Summary

If a variable is missing from the SFG, you can add a node and the equation that defines the node.

Mason’s Gain Formula only applies for Input to Output transfer functions (gains). Try to arrange the equation as a circuit gain.

In VSA the Input must be an ideal voltage source.

You can also add test inputs. This is illustrated in the Ladder Output Impedance example.

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