Series-parallel     Parser I     Interlude on RLC-circuits     Parser II     Harry Broeders' applet

## Series/parallel circuits

This topic follows a track to three peaks, with optional spur tracks. It begins with a gentle climb to ParSer I followed by a steeper climb to ParSer II; then there is a saddle to visit Broeders' applet. A "Whither now?" is a guide to the peaks track.

The purpose of this chapter begins with calculations of the resistance measured across a node-pair of a small network of resistors. Examples:-

• Find the resistance across two resistors in series: 3 Ω and 4Ω. [Answer. 7 Ω]
• Find the resistance across two resistors in parallel: 2 Ω and 5 Ω. [Answer. Apply 1 V across the node-pair. The current through the resistors will be 1/2 A and 1/5 A, totalling 7/10 A. Hence the resistance of the pair-in-parallel is 1/(7/10) Ω, that is 10/7 Ω (that is, the reciprocal of the sum of the reciprocals of the resistances).]
• Now assume that a network between two nodes comprises (a) in series with (b). [Answer. The resistance between the nodes is 7 Ω + 10/7 Ω, that is 59/7 Ω.]
Our purpose is to tackle an arbitrarily larger network with a series/parallel structure by replacing a sequence of numerical calculations with a single numerical expression to be evaluated within JavaScript. The centrepiece of the chapter is the calculator ParSer with two versions: ParSer I for resistors, and ParSer II for resistors, inductors and capacitors (RLCs). I chose the name, ParSer, because it has two connotations: one is the series-parallel structure of the circuits; and the other is the parsing of the strings. [Note that a general circuit does not necessarily have the series-parallel structure (cf delta and wye networks).]

ParSer I and ParSer II addresses different audiences:-

• ParSer I for High-school students. You will have applied Ohm's law (I = E/R)) to a few resistors in series or in parallel. You will see in ParSer I how to deal readily with any number of resistors in series-parallel combinations.
• ParSer II for University students. You will have understood how the transition from resistors to RLCs transforms Ohm's law to the forms for the steady-state solution of an RLC-circuit driven by a voltage Vcos(ωt) across a node-pair input. Consider, for example, various views of a series RLC circuit:-
• Graphs arise from the solution of a differential equation:
I = Vcos(ωt - θ))/Z
where Z = √{(R2 + (Lω - (Cω)-1)2} and tan θ = (Lω - (Cω)-1)/R.
• The sides and angles of a right-angled triangle are expressed in terms of polar coordinates:
I∠-θ = V∠0/Z∠θ = V∠-θ
• The polar forms are transformed into complex cartesian form:
Ie-jθ = Vej0/(Ze) = (V/Z)e-jθ
• Phasors arise from re-introducing ω:
I∠(ωt-θ) = V∠(ωt)/Z∠θ = (V/Z)∠(ωt-θ)
• Phasors and the complex cartesian form are combined into:
Iej(ωt-θ) = Vej(ωt)/(Zej(ωt+θ)) = (V/Z)ej(ωt-θ)
You will see,in ParSer II how to deal readily with any number of RLCs in series-parallel combinations.
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