Resonance and the Transfer Function

Lets now consider putting a sinusoidal source in our series RCL circuit and consider the voltage across one of the circuit elements. The resistor for example in figure 2.7

 
Figure 2.7:  Driven series RCL circuit.

Applying Ohm's law across the resistor gives (cf. a voltage divider)

where is known as the transfer function in the frequency domain. We have changed independent variables from to for convenience.

We define

contains all the information needed to characterize the circuit. In exponential form

where

and

has a maximum (resonance) given by . Or is the resonant frequency.

Example: Consider the series LCR circuit (figure 2.8) driven by a voltage phasor .

 
Figure 2.8:  Driven series LCR circuit.

1. At an angular frequency such that and , write the current phasor in terms of and R.

   v(t) is given by v(t)=Zi(t), where . At and

   

   Therefore

   

2. At the instant when is exactly real, calculate the three phasors representing the voltage developed across the R, C, and L circuit elements.

   v(t) is real at t=0. Thus

   

   And

   

   Also

   

3. Algebraically and with a sketch on the complex plane, show that the complex voltage sum around the closed loop is zero.

   The three voltage phasors are

   

   

   

   Around the closed loop . If this expression is zero at t=0 it will be zero for all time. Therefore .

 
Figure 2.9: Complex voltage sum around the closed loop of the driven LCR circuit.

Example: Sketch simplified versions of the circuit shown in figure 2.10 that would be valid at:

 
Figure 2.10:  Example LCR circuit.

1. ;

   .

    
   Figure 2.14: Example circuit for .
   

2. very low frequencies but not ;

   When is small C and L are in parallel and

   

   2L and 100R in parallel gives

   

    
   Figure 2.12: Example circuit for very low frequencies but not .
   

3. very high frequencies but not ;

   large (note: ).

    
   Figure 2.13: Example circuit for very high frequencies but not .
   

4. .

   .

    
   Figure: Example circuit for .
   

Example: For the circuit shown in figure 2.15 plot as a function of frequency over the range rad/s to rad/s.

 
Figure 2.15:  Example circuit with components in parallel.

The equivalent impedance for the three components in parallel is

Plugging in the numerical values gives

A table of values and its plot follows.

 
Table 2.1: Numerical values for example circuit.

 
Figure 2.16: Plot of for example circuit.