9.6: Summary
- Page ID
- 98507
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A series connection is any connection in which the current through one component must be identical to the current flowing through any other component in that connection. This remains true for AC circuits employing resistors, capacitors and inductors. The equivalent impedance of a set of resistors, inductors and capacitors placed in series is the vector sum of their resistance and reactance values. Inductive and capacitive reactance have opposing signs and will partially cancel each other. While resistance remains constant across frequency, reactance does not. Capacitive reactance decreases with frequency while inductive reactance increases. Therefore, the precise impedance value will vary with frequency. If the impedance angle is positive, the circuit is said to be inductive. If the impedance angle is negative, the circuit is said to be capacitive.
Once inductance and capacitance values have been turned into their corresponding reactances at the frequency of interest, Ohm's law can be used to find the voltage across any component. The voltage will equal the product of the resistance or reactance and the current flowing through it. This is a vector computation. The current through a resistor will be in phase with the voltage across it. In contrast, the inductor voltage will lead the current and the resistor voltage by 90 degrees. Also, the capacitor voltage will lag the current and the resistor voltage by 90 degrees. From these observations it is apparent that the inductor and capacitor voltages in a series connection must be 180 degrees out of phase.
Kirchhoff's voltage law (KVL) states that the sum of voltage rises around any series loop must equal the sum of voltage drops around that loop. This remains true for AC analysis, however, it is imperative that a vector summation is performed and not just a simple summation of the magnitudes. If the phase angles are ignored, it is possible for the voltage magnitudes to sum to considerably more than the source voltage. The voltage divider rule (VDR) remains true for AC analysis, but again, it is a vector computation. Thus, a rendering using just magnitudes will not produce a correct result. For example, a divider between a resistor and inductive reactance of equal value will not yield the source voltage splitting in half across each component.
For highest accuracy, the coil resistance, or \(R_{coil}\), of inductors should not be ignored. It is treated as a small resistance in series with and integral to the inductor. Coil resistance is computed from the inductor's \(Q\), or quality factor. \(Q\) is defined as the ratio of \(X_L\) to \(R_{coil}\).
Review Questions
1. How is the equivalent impedance for a string of series connected resistors, capacitors and inductors computed?
2. Will the impedance of an RLC circuit be the same at all frequencies? Why/why not?
3. How are series connected AC voltage sources combined?
4. Does Kirchhoff's voltage law change for AC circuit analysis?
5. Is it possible for the voltage magnitudes of an AC RLC circuit to sum to more than the source voltage? Why/why not?
6. How is a phasor impedance plot for a series network related to its phasor voltage plot?