Solving sinusoidal networks
Assemble the full AC workflow: freeze physical references and frequency, convert sources to AoN peak phasors, build frequency-specific Z/Y models, derive series/parallel reductions from Kirchhoff, solve with KCL/KVL, return to the real waveform and audit the result by units, reconstruction, limiting cases and model validity. Includes checked RC series, RC parallel and RLC examples.
- Freeze topology, voltage/current references, frequency and the AoN cosine/peak convention before complex arithmetic.
- Separate DC and distinct AC frequencies into independent LTI subproblems before applying phasors.
- Build R/C/L impedance and admittance models at the declared angular frequency.
- Derive series impedance and parallel admittance reduction from KVL/KCL instead of using topology-blind formulas.
- Solve and verify a series RC network by complex KVL.
- Solve and verify a parallel RC network by nodal admittance and complex KCL.
- Interpret the sign of net RLC reactance and the corresponding voltage/current phase relation.
- Return the solved phasor to a real time-domain waveform and audit units, Kirchhoff residuals, limiting cases, convention consistency and model validity.
Chapters 01–04 built the pieces: why phasors work, how complex amplitudes are manipulated, how R/C/L impedance is derived, and why KCL/KVL retain their algebraic form. The final task is to assemble those pieces into a repeatable network-solving method.
The goal is not merely to obtain one complex number. A good solution must make its references, frequency, convention, model and checks visible enough that another engineer can reconstruct and audit the reasoning.
05.1Freeze the physical problem before touching complex arithmetic
Before converting anything to a phasor, write down five facts: the topology, all voltage/current reference directions, the sinusoidal frequency, the waveform convention, and which part of the response is requested.
AC-01 uses cosine-reference, peak-amplitude phasors unless stated otherwise. Initial-condition transients are outside this steady-state calculation.
Figure 5.1 — A defensible AC solution in eight checks
The one question is: in what order should a sinusoidal problem be transformed without losing references or physical meaning?
Complex arithmetic sits in the middle of the chain, not at the beginning. Physical references and frequency are fixed before it; units, reconstruction and model limits are checked after it.
05.2Decompose frequencies before solving the network
One phasor problem represents one angular frequency. A DC offset and two AC frequencies therefore become separate linear subproblems.
For an LTI circuit: solve the DC problem, solve the ω1 phasor problem, solve the ω2 phasor problem, and superpose the resulting real time-domain responses. Never add phasors belonging to different frequencies.
If nonlinear elements generate harmonics or mixing products, the one-frequency impedance workflow is no longer sufficient; conservation still holds, but a multi-frequency nonlinear method is required.
05.3Build the frequency-specific element model from the laws already derived
At a declared nonzero ω, the derived ideal-element relations may now be used as a compact working summary:
Use impedance naturally with common-current/series structures and admittance naturally with common-voltage/parallel or nodal structures. This is a choice of algebraic coordinates, not a change in physics.
05.4Series and parallel reductions are consequences of Kirchhoff
If elements truly carry the same branch current, KVL gives
If elements truly share the same terminal voltage, KCL gives
The words “same current” and “same voltage” are the topology test. Do not combine components merely because they look visually adjacent.
05.5Worked example A — series RC by KVL
A 1 kΩ resistor and a 100 nF ideal capacitor are in series with
The source phasor and angular frequency are
The element impedances are
KVL gives one common current:
Therefore
The check is vectorial, not a magnitude sum:
In the time domain, the current is i(t)=5.320 cos(2π·1000t+87.858°) mA.
05.6Worked example B — parallel RC by nodal admittance
A current source is=4∠20° mA feeds a parallel 2 kΩ resistor and 50 nF capacitor at 2 kHz. Let v be the node voltage relative to the return node.
KCL gives
The branch currents are
Their complex sum reconstructs the source:
This example shows why admittance is the natural coordinate for nodal equations: parallel branch currents become direct products Yv̲.
05.7Worked example C — mixed RLC and the meaning of the net reactance
For a series network at 5 kHz with R=330 Ω, L=10 mH, C=220 nF, and vs=8∠−15° V:
The positive net imaginary part means the network is inductive at this frequency: current lags the source voltage by 27.183°. That conclusion comes from the sign of the net reactance, not from counting how many capacitors or inductors are present.
Near a series resonance the ideal reactive terms can cancel, but real component losses, parasitics and self-resonances determine how far the ideal model can be trusted.
05.8Return to the real waveform only after the complex solution is complete
The frequency used in reconstruction must be the same frequency used to build every impedance in that phasor problem. Preserve the physical unit and reference direction of the solved quantity.
If a computed phasor is reported with a negative magnitude, normalize it before returning to time:
A phase outside a preferred display interval may also be shifted by integer multiples of 360° without changing the waveform.
05.9Audit the answer before trusting it
A calculator returning a complex number is not a validation. The strongest routine check is to substitute the solved phasors back into the original KCL/KVL equations and verify the residual is zero within rounding.
05.10Mastery exercises
- A source is 12 cos(2π·500t−20°) V across a series 470 Ω resistor and 1 µF capacitor. Find the current phasor and time-domain current.
- For the circuit in exercise 1, compute both element voltage phasors and verify KVL by complex addition.
- A 3∠40° mA source drives a parallel 1 kΩ resistor and 100 nF capacitor at 1 kHz. Write the nodal equation before evaluating it.
- Explain why adding |Z_R|+|Z_C| cannot replace complex series addition.
- A source contains DC, 1 kHz and 2 kHz components. Describe the correct LTI solution sequence.
- State four checks that should be performed before accepting a numerical AC answer.
- At a chosen frequency, a series RLC equivalent impedance has a negative imaginary part. Is the net behavior capacitive or inductive, and what does that imply for current phase relative to voltage?
- Explain why a real component’s datasheet self-resonant frequency can invalidate a mathematically correct ideal-element calculation.
Solutions
1. ω=3141.59 rad/s and ZC=−j318.31 Ω. Thus Zeq=470−j318.31=567.65∠−34.11° Ω, so i=21.140∠14.11° mA. Therefore i(t)=21.140 cos(2π·500t+14.11°) mA.
2. vR=9.936∠14.11° V and vC=6.729∠−75.89° V. Their complex sum returns approximately 12∠−20° V.
3. With currents referenced from the node into the branches: 3∠40° mA=(1/1 kΩ+jω·100 nF)v̲.
4. Magnitudes discard the −90° phase of the capacitor impedance; KVL requires complex voltage addition.
5. Solve the DC network, the 1 kHz phasor network and the 2 kHz phasor network separately, then superpose the real responses.
6. Examples: dimensional units, KCL/KVL reconstruction, phase/sign consistency, limiting cases, peak/RMS convention and model validity range.
7. Negative net reactance is capacitive; for the receiver convention the current leads the applied voltage by the magnitude of the impedance phase.
8. The ideal lumped model omits parasitic L/C and loss; around self-resonance those omitted terms can dominate the physical impedance.
AC-01 is now conceptually complete. AC-02 can build frequency response, transfer functions and Bode reasoning by repeating this single-frequency solution over frequency rather than introducing a disconnected new formalism.