9.14:
Scaling
Magnitude and frequency scaling simplifies filter design by calibrating filter responses and circuit parameters to fit within usable ranges.
Magnitude scaling involves amplifying all impedances in a network by a factor, maintaining consistent frequency responses.
The scaled impedance and frequency values are then expressed using the scaling factor.
This process does not alter the circuit's resonance frequency, quality factor, bandwidth, or transfer functions.
Frequency scaling shifts the frequency response of a network either up or down the frequency axis while leaving the impedance unchanged. It is accomplished by multiplying the frequency by a scaling factor.
Frequency scaling influences the impedances of frequency-dependent reactive elements, but the resistor remains unaffected.
The scaling factor alters the resonant frequency and bandwidth, but the quality factor remains the same.
A general expression for impedance and frequency scaling accommodates simultaneous magnitude and frequency scaling.
For cases when magnitude scaling is not applied, the magnitude scaling factor is set to one. Similarly, without frequency scaling, the frequency scaling factor is set to one.
9.14:
Scaling
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that scaling techniques can later adjust these values to more practical levels.
Scaling a circuit can be done in two ways: magnitude (or impedance) scaling and frequency scaling. These methods adjust the circuit's components to work within practical ranges. Magnitude scaling changes the size of the components without affecting how the circuit responds to different frequencies. On the other hand, frequency scaling moves the circuit's response to higher or lower frequencies on the spectrum.
Magnitude Scaling:
Magnitude scaling involves adjusting the sizes of the circuit components (such as resistors, inductors, and capacitors) by a certain factor, but without changing the way the circuit responds to different frequencies. The impedances of the circuit are in terms of resistors (R), inductors (L), and capacitors (C) in a circuit. When magnitude scaling Km is applied, these components are transformed as follows:
Frequency Scaling:
Frequency scaling shifts the frequency response of a circuit along the frequency axis, either up or down, without altering the impedance levels. This is achieved by multiplying the frequency by a scaling factor, denoted by Kf. The new values of the inductance and capacitance are determined by:
If a circuit is scaled for both the parameters- magnitude and frequency at the same time, then:
If magnitude and frequency scaling factors are equal, neither magnitude nor frequency scaling occurs.