8.5: Summary
- Page ID
- 98514
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter we have examined parallel circuits using either a single voltage source, or one or more AC current sources, along with two or more resistors, capacitors and inductors. The defining characteristic of a parallel configuration is that all components are connected to just two nodes and that all elements in this configuration see the same voltage. If multiple AC current sources are present, they may be combined into a single equivalent current source via vector addition.
Unlike the purely resistive case, the equivalent impedance of a group of parallel RLC components will not always be smaller than the smallest resistance or reactance in the group due to cancellations between the inductance and capacitance. As voltage is identical for all components, then the currents through the capacitors must be 180 degrees out of phase with the currents through the inductors. Thus it is quite possible that the current through one of the reactance branches could be larger than the current supplied by the source. In general, the effective impedance is found by summing the individual conductances and susceptances to find the total admittance of the group, and then taking the reciprocal of this value. The product-sum rule may be used so long as the angles are taken into account (i.e., a vector process).
Kirchhoff's current law (KCL) states that the sum of currents entering a node must equal the sum of currents leaving that node. This remains true for AC circuit analysis, however, it must be remembered to always use a vector summation. A simple summation of current magnitudes will not achieve proper results.
Individual branch currents in a circuit driven by a voltage source may be determined by using Ohm's law: simply divide the source voltage by the individual resistance and reactance values. These branch currents must sum to the total current delivered by the voltage source thanks to KCL. If the parallel network is driven by current sources, the individual branch currents can be found by first determining the effective parallel impedance and then using Ohm's law to find the system voltage. Once the voltage is known, Ohm's law is used again on each component to find the associated branch current. Alternately, the current divider rule may be used repeatedly on successive pairings of components.
Review Questions
1. How is the equivalent impedance for a group of parallel connected resistors, inductors and capacitors computed?
2. How is the equivalent value for parallel connected AC current sources computed?
3. Is the product-sum rule still applicable for AC analysis? Can it be used for reactance and/or complex impedance?
4. Define Kirchhoff's current law for AC circuit analysis.
5. Is it possible for a branch current to be larger than the source current in an AC parallel circuit? Explain why/why not.