When working with resonant circuits, we can use components with more convenient values (ie: 1 ohm, 1 H or 1F) and transform them by scaling.
Two types of scaling
- Magnitude/Impedance Scaling
- Frequency Scaling
Magnitude Scaling
With magnitude scaling (sometimes called impedance scaling) all of the impedance values are increased by a certain factor while the frequency response remains unchanged. We begin by recalling the following impedance definitions for a resistor, inductor and capacitor: $$ \mathbb{Z}_R = R \;, \;\;\; \mathbb{Z}_L = j\omega L \;, \;\;\; \mathbb{Z}_C = \frac{1}{j\omega C} \quad,(Eqns\;1) $$
In order to apply magnitude scaling, we multiply each of the above impedance values (equations #1) by a factor (Km) and hold the frequency constant: $$ (Equations \;2) $$ $$ \mathbb{Z}'_R = K_m\mathbb{Z}_R = K_mR $$ $$ \mathbb{Z}'_L = K_m\mathbb{Z}_L = j\omega K_m L $$ $$ \mathbb{Z}'_C = K_m\mathbb{Z}_C = \frac{K_m}{j\omega C} = \frac{1}{j\omega \frac{C}{K_m}} $$
If we compare equations 1 and 2 we get the following changes in element values: $$ R' = K_mR \;,\;\;\;L' = K_mL \;,\;\;\; C' = \frac{C}{K_m} \quad,(Eqns\;3)$$
Resonant Frequency:
We now consider the resulting resonant frequency (after magnitude scaling is performed): $$ \omega_o' = \frac{1}{\sqrt{L'C'}} $$ substituting equations #3 into the above gives us: $$ \;\;\; = \frac{1}{\sqrt{K_mL(\frac{C}{K_m})}} $$ $$ \omega_o' = \frac{1}{\sqrt{LC}} = \omega_o $$ With magnitude scaling, we see that the resonant frequency has not changed. Additionally, magnitude scaling does not affect transfer functions of the following forms (being that they are dimensionless quantities): $$ \frac{\mathbb{V}_o}{\mathbb{V}_i} \;\;\; or \;\;\; \frac{\mathbb{I}_o}{\mathbb{I}_i} $$
Frequency Scaling
With frequency scaling, the frequency response is shifted up or down the frequency axis while the impedance values are left unchanged. In this situation, the frequencies are multiplied by a factor (Kf). Once again, we recall the following impedance definitions for a resistor, inductor and capacitor:: $$ \mathbb{Z}_R = R \;, \;\;\; \mathbb{Z}_L = j\omega L \;, \;\;\; \mathbb{Z}_C = \frac{1}{j\omega C} \quad,(Eqns\;4) $$ Keeping in mind that the impedance values of the inductor and capacitor are frequency dependent, we multiply their respective frequencies by the frequency factor (Kf) and get the following impedance values for the inductor and capacitor: $$ \mathbb{Z}_L = j(\omega K_f)L' = j\omega L $$ $$ \mathbb{Z}_C = \frac{1}{j(\omega K_f)C'} = \frac{1}{j\omega C} $$ Notice that in order to scale the frequency AND have the resulting impedances be the same as the pre-scaled impedances, we need different values for the inductor and capacitor. We get these new element values by comparing the above equations to equations 4. Doing so gives us the following new element values:
Frequency-scaled circuit element values: $$ R' = R \;,\;\;\; L' = \frac{L}{K_f} \;,\;\;\; C' = \frac{C}{K_f} \quad,(Eqns\;5) $$ ...where: $$ \omega' = K_f\omega $$
Resonant Frequency
Consider the expression for the resonant frequency of a series or parallel resonant circuit: $$ \omega_o' = \frac{1}{\sqrt{LC}} $$ ...For our frequency scaled circuit we have: $$ \omega_o' = \frac{1}{\sqrt{L'C'}} $$ substituting eqns #5 into the above expression gives us: $$ \;\;\; = \frac{1}{\sqrt{(\frac{L}{K_f})(\frac{C}{K_f})}} $$ $$ \;\;\; = \frac{1}{\sqrt{\frac{LC}{(K_f)^2}}} $$
$$ \omega_o' = \frac{K_f}{\sqrt{LC}} = K_f \omega_o $$ Furthermore, the bandwidth and quality factor of a frequency scaled circuit are expressed as: $$ B' = K_fB $$ $$ Q' = Q $$
Magnitude and Frequency Scaling
To scale both magnitude and frequency we have:
$$ (Eqns\;6) $$ $$ R' = K_mR \;,\;\;\; L' = \Big( \frac{K_m}{K_f} \Big)L \;,\;\;\; C' = \Big( \frac{1}{K_mK_f} \Big) C \;,\;\;\; $$ $$ \omega' = K_f\omega $$
In the next page we will look at a scaling example problem.
Continue on to scaling example problem #1...