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# 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.    Next: Four-Terminal Networks Up: Alternating Current Circuits Previous: Combined Impedances

Doug Gingrich
Tue Jul 13 16:55:15 EDT 1999