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.]

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:
where Z = √{(R*I = Vcos*(*ωt - θ*))*/Z*^{2}+ (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}= Ve^{j0}/(Ze^{jθ}) = (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:
- Ie
^{j(ωt-θ)}= Ve^{j(ωt)}/(Ze^{j(ωt+θ)}) = (V/Z)e^{j(ωt-θ)} - I∠-θ = V∠0/Z∠θ = V∠-θ

*readily*with*any number*of RLCs in series-parallel combinations.- Graphs arise from the solution of a differential equation:

*Introduction*. This page.*ParSer I for series-parallel resistors*. Each circuit comprises*resistors*only, and we wish to calculate the*resistance*(Ω ohms) between two*nodes*of a*port*(and to calculate the current (A amperes) into the port when a cell (V Volt) is attached to the port).*Interlude on RLC-circuits*. An RLC-circuit comprises*resistors*,*capacitors*and*inductors*where the node-pair at a port is a*cosinusoidal*voltage (*V*cos(*ωt*)) with given frequency*f*equal to*ω*/(2π). Their measures (value,unit, symbol) are resistance (*R*, ohm, Ω), inductance (*L*, henry, H), capacitance (*C*, farad, F). An RLC-calculator, with*given topology*, is introduced in preparation for ParSer II which allows*arbitrary topology*(restricted to series-parallel topology). [A side track reviews one's understanding of the extension of Ohm's law to resistors, inductors and capacitors.]*ParSer II for series-parallel RLC-circuits*. Each circuit comprises RCLs, and we wish to calculate the*impedance*(Ω ohms) between two*nodes*of a*port*(and to calculate the steady-state current (I amperes) when a voltage (*V cos ωt*) is applied across the port).*Harry Broeders' applet*. We see ParSer II and Broeders' applet working in tandem, and inviting a reader to combine the two into a single applet.