# 8.4: Summary

- Page ID
- 25289

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Resonance can be described as a preferred mode of vibration, or a frequency at which a system operates particularly well. Resonant systems can be used to filter out or select specific frequencies across the spectrum. Obvious uses include tuning circuits, oscillators, filters and the like. In electrical systems there are two basic forms; series RLC resonance and parallel RLC resonance. Series resonance tends to be the less complicated of the two.

For series resonance, the resonant frequency, \(f_0\), is defined as the frequency at which the magnitude of \(X_L\) equals the magnitude of \(X_C\). In this instance, the reactances cancel, leaving the series impedance as \(R\). This creates a U-shaped curve for the impedance as it varies across the frequency spectrum. At low frequencies, the capacitive reactance dominates and the series impedance is high in magnitude and capacitive. At frequencies above the resonant frequency, the inductive reactance dominates and the series impedance is again high in magnitude but it is inductive. If this circuit is driven with a constant voltage source, the current will be maximum at resonance and tail off at lower and higher frequencies. The sharpness of the current curve across frequency is a function of the system \(Q\), or quality factor. A high \(Q\) circuit is one with a very sharp and narrow curve. The “shoulders” of the curve are defined as the frequencies at which the power has dropped to one half of the value at resonance. This corresponds to 0.707 times the current at resonance. The lower frequency is \(f_1\) and the upper frequency is \(f_2\). The difference between the two is called the bandwidth, \(BW\). The ratio of resonant frequency to the bandwidth yields the circuit \(Q\). Circuit \(Q\) can also be found by dividing the magnitude of reactance at resonance to the total circuit resistance. In high \(Q\) circuits it is possible for the voltage across the inductor or capacitor to be many times higher than the source voltage, higher in fact, by a factor of \(Q\).

Parallel resonance is similar to series resonance but in some ways is like its mirror image. In a parallel resonant circuit the inductor will dominate at low frequencies and produce a small net impedance. At high frequencies, the capacitor will dominate and also produce a small net impedance. At resonance, the two effectively will cancel and yield a large impedance. In other words, the impedance versus frequency curve will appear like an upside down U, producing maximum impedance at resonance, and the opposite of the series impedance curve. If this system is driven by a constant current source, the resulting voltage will echo the shape of the impedance curve, producing maximum voltage at resonance. The upper and lower frequencies, along with the bandwidth and system \(Q\), are defined in the same manner as they are in the series case (with one exception regarding finding \(Q\) via resistance and reactance).

There is one important caveat regarding parallel resonant circuits. Practical inductors contain a non-trivial series coil resistance. This can play a dominant role in the system response. Analysis is generally handled by performing a series to parallel transform which creates a parallel resistance out of the inductor's series resistance. As a result, system \(Q\) can be found as the ratio of effective parallel resistance to maximum reactive magnitude, the opposite of the series case. For high \(Q\) systems, generally taken as 10 or higher, the resonant frequency can use the same equation as the series case. For low \(Q\) systems, the series to parallel transform creates a shift in resonant frequency, making it somewhat lower than the value obtained from the basic series equation. Also, the inductor and capacitor currents will be approximately \(Q\) times higher than the source current, made possible because they are 180 degrees out of phase with each other and effectively cancel.

In both series and parallel systems, for high \(Q\), \(f_1\) and \(f_2\) are assumed to lie equidistant from \(f_0\), splitting \(BW\) in half on either side. This is just an approximation and errors will grow as the \(Q\) decreases. More accurately, the two frequencies lie where the ratio of \(f_1/f_0\) is the same as the ratio of \(f_0/f_2\).

## Review Questions

1. Describe the concept of resonance. How is resonance defined in a series RLC network?

2. Sketch the impedance versus frequency plot for series resonance.

3. Sketch the impedance versus frequency plot for parallel resonance.

4. Define the terms resonant frequency, bandwidth and \(Q\).

5. How does inductor \(Q\) impact system \(Q\) in resonant circuits?