Impedance Scaling

In this scaling, the magnitude of the impedance is increased or decreased. To scale in magnitude, z(s) (the impedance) is multiplied by a constant factor K_m. It scales magnitude of elements in order to maintain a desirable physical property.

fig: Impedance of Z(s)

An impedance |Z(jw)| is said to be magnitude scaled by a factor of ‘K_m’ if it’s multiplied by a real positive constant ‘K_m’.

If K_m > 1, it’s called Scaled Up.

If K_m < 1, it’s called Scaled Down.

Values of Transfer function remains same even after impedance scaling.

Let R(_old), L(_old), C(_old) be the old values of Resister, Inductor and Capacitor. To achieve impedance change, the magnitude of every element in a network should be changed.

Then,

Z_R (_new) = K_m * Z_{R (old)} …………………….  1

L(_new) = K_m * X_L ………………………….. 2

C(_new) = Xc * K_m      ……………………. ...3

For Resistor,

|Z_R| = R(_old)

After Scaling by factor of ‘K_m’, we have

K_m * |Z_R| = K_m * R_(old)

From eq 1

R(_new) = K_m * R_(old)

For Inductor,

|Z_L| = w*L_(old)

After Scaling by factor of ‘K_m’, we have

K_m * X_L = K_m * s * L_(old)

K_m * X_L = (K_m* L_(old)) * s = = L(_new) *S

From eq 2

i.e., L(new) = K_m* L_(old)

For Capacitor,

\[
X_C \times K_m = \frac{1}{s \, C_{\text{old}}} \quad \Rightarrow \quad
K_m = \frac{1}{s \left(\frac{C_{\text{old}}}{K_m}\right)} = \frac{1}{s \, C_{\text{new}}}
\]

From eq 3

\[
\text{i.e.,} \quad C_{\text{new}} = \frac{C_{\text{old}}}{K_m}
\]

Then, the new Values of Resister, Inductor and Capacitors are

R(_new) = K_m * R_(old)

L(_new) = K_m * L_(old)

\[
\text{i.e.,} \quad C_{\text{new}} = \frac{C_{\text{old}}}{K_m}
\]

If all the impedance in the Network is scaled be ‘K_m’ then,

·      All the network function that are dimensionless such as voltage ratio (V2/V1 ), current ratio (I2/I1)  remains unchanged.

_·       All the network function that has dimension of impedance (Z_in = Vo/Vi) i.e., Transfer Function (T(s)) is multiplied by K_m.

·         All the network function that has dimension of admittance is divided by K_m i.e., Capacitance