---
title: Capacitor Networks
module: Capacitance
moduleNumber: 4
lessonNumber: 2
order: 402
summary: |
  Wire several capacitors together and the source sees one equivalent capacitance — but
  which? The answer comes not from how the symbols are drawn but from which conductors
  share a node: parallel branches hold a common voltage and add, $C_{\rm eq}=\sum_iC_i$,
  while series branches share a common charge and add reciprocally. We derive both rules
  from charge conservation on the floating internal node, then extend the node-charge
  method to unequal, precharged, and stray-coupled branches and carry a worked reduction
  through to the charge and voltage on every element.
topics: [Capacitance]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 24 — Capacitance and Dielectrics; §24-3"
---

## Terminal capacitance and conductor nodes

A capacitor network is defined by conductors and their connections, not by the
visual spacing of component symbols. Every region joined by ideal wire is one
equipotential conductor node. A two-terminal equivalent capacitance is the ratio of
charge supplied at one chosen terminal to the potential difference imposed between
that terminal and the other chosen terminal, after every internal conductor charge
constraint has been satisfied. The result depends on the terminal pair and on whether
any internal conductor is grounded, connected to a source, or left floating.

A branch capacitance $C_{ij}$ joining nodes $i$ and $j$ contributes the charge
assigned to node $i$,

$$
Q_{i\leftarrow j}=C_{ij}(V_i-V_j).
$$

The opposite charge is assigned to node $j$. Summing this expression over every
capacitor incident on a node gives its net free charge. A terminal connected to a
source has a prescribed potential or supplied charge; a floating internal conductor
has a prescribed net charge, commonly zero. This node-charge statement is the
capacitive analogue of a conservation equation. It prevents the error of applying a
series or parallel rule before the conductor connections have been identified.

## Parallel combinations: common voltage, additive charge

Capacitors are in parallel when each branch connects to the same two conductor nodes.
Their potential difference is therefore common. If the terminal voltage is
$\Delta V$, source charge is the sum of charges supplied to the positive node:

$$
Q_{\rm term}=C_1\Delta V+C_2\Delta V+\cdots.
$$

Dividing by the shared terminal voltage gives

$$
C_{\rm eq}=C_1+C_2+\cdots.
$$

This derivation explains why parallel capacitance increases: more than one field
region accepts charge at the same potential difference. Unequal branch values pose no
special difficulty. A very small parallel branch contributes very little terminal
charge, whereas a large branch dominates the equivalent. The rule fails if a symbol
that appears to share a node is actually separated by a switch, a measurement input,
or a crossing wire with no junction.

$$
% caption: Parallel branches connect the same two conductor nodes. The common terminal voltage appears across every branch while the high-terminal charge is the branch-charge sum, so $C_{\rm eq}=\sum_i C_i$.
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$$

## Series combinations: neutral floating conductors

In a series branch, an internal conductor joins one plate of one capacitor to one
plate of the next. If that conductor is disconnected from every source, begins neutral,
and has no stray connection to another object, its net charge remains zero. The
capacitor plate on one side then carries charge $-Q$ when the plate on the
other side carries $+Q$. The terminal charge magnitude is the same on each series
capacitor, but their voltage drops need not be equal.

For two series capacitors,

$$
\Delta V=\frac{Q}{C_1}+\frac{Q}{C_2},
\qquad
C_{\rm eq}=\frac{C_1C_2}{C_1+C_2}.
$$

For more branches, reciprocal capacitances add. The voltage division follows
$V_1=Q/C_1$ and $V_2=Q/C_2$, so $V_1/V_2=C_2/C_1$. The smaller capacitance
takes the larger potential difference. The equivalent is smaller than either branch,
because the same terminal charge must produce the sum of the drops through both field
regions. In the limit that one capacitance becomes extremely large, its voltage drop
vanishes and the equivalent approaches the smaller capacitance. If one capacitance
tends to zero, the branch equivalent tends to zero.

$$
% caption: Series capacitors joined by a neutral floating conductor. The internal node has no source path and no net free charge, so adjacent plates carry equal and opposite charge; equal-charge branches divide the terminal voltage unequally.
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The neutral-node assumption is a condition, not a component property. If the internal
conductor has net charge $Q_f$, its potential must satisfy a charge equation. For
two capacitors joining the internal node to terminal potentials $V_a$ and $V_b$,

$$
C_1(V_f-V_a)+C_2(V_f-V_b)=Q_f.
$$

An attached probe, a shield, or a nearby enclosure adds another capacitance term to
this equation. A source connected temporarily to the internal conductor can leave it
with nonzero charge after the source is removed. In either case, treating the branch
as an ordinary neutral series pair gives an incorrect terminal capacitance and
incorrect internal voltage division. The node equation remains valid and provides the
direct route to the solution.

## Worked reduction and terminal check

> **Worked example (mixed series-parallel reduction).** A neutral series branch of
> $C_1=120\ \mathrm{pF}$ and $C_2=330\ \mathrm{pF}$ sits in parallel with
> $C_3=47.0\ \mathrm{pF}$. The series branch is
>
> $$
> C_{12}=\frac{(120)(330)}{120+330}\ \mathrm{pF}=88.0\ \mathrm{pF},
> $$
>
> so the terminal equivalent is $C_{\rm eq}=88.0+47.0=135\ \mathrm{pF}$. At an applied
> $15.0\ \mathrm V$, the total terminal charge is $Q_{\rm term}=C_{\rm eq}\Delta V
> =2.03\ \mathrm{nC}$. The parallel $47.0\ \mathrm{pF}$ branch carries
> $0.705\ \mathrm{nC}$; the series branch carries the remaining $1.32\ \mathrm{nC}$ on
> each capacitor, dropping $11.0\ \mathrm V$ across $C_1$ and $4.00\ \mathrm V$ across
> $C_2$. Their sum returns the imposed $15.0\ \mathrm V$, and the smaller capacitance
> takes the larger drop — two independent checks on the reduction.

An AC bridge can test this terminal equivalent without measuring the floating-node
potential. Balance the unknown network $C_x$ against a traceable standard $C_s$
and noninductive ratio arms. In the ideal loss-free balance,

$$
\frac{C_x}{C_s}=\frac{R_2}{R_1}.
$$

With a $100\ \mathrm{pF}$ standard, the worked network should balance at
$R_2/R_1=1.35$. A discrepancy can arise from a wiring error, fixture capacitance,
an unrecognized stray path from the floating conductor, or a lossy branch at the
test frequency. Repeating the balance with a changed shield connection tests whether
the measured object is the intended two-terminal network.

The terminal definition must remain unchanged between calculation and bridge test.
Moving one measurement lead from a floating node to a shielded enclosure creates a
new capacitance branch and changes the very network being verified. Record every
connection, including guards and unused terminals, with the reported equivalent.

$$
% caption: Worked mixed network. A neutral series pair ($C_1$ over $C_2$) forms one branch in parallel with $C_3$; terminal charge divides between branches while the internal series-node charge stays constrained, and an AC bridge compares the two-terminal capacitance with a standard at null.
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$$

## Network reduction, bridge balance, and measurement loading

Systematic reduction starts by assigning a potential to every conductor node. Choose
one terminal as the reference, assign known source-terminal potentials, and write one
charge equation for each floating conductor. If node $k$ is connected by
capacitances $C_{kj}$ to nodes $j$, its equation is

$$
\sum_j C_{kj}(V_k-V_j)=Q_k^{\rm free}.
$$

The sum is over actual capacitive connections, including deliberate components,
fixture capacitance to a shield, a probe input, and any known connection to the
reference enclosure. For a neutral floating node the right side is zero. After the
unknown node potentials are found, terminal charge is the sum of branch charges
leaving the driven terminal; division by the imposed terminal voltage gives the
two-terminal equivalent capacitance. This procedure works for reducible series and
parallel networks and for layouts in which no shortcut is valid.

For several floating nodes, the equations form a symmetric capacitance matrix. Each
diagonal coefficient is the sum of capacitances incident on one floating node; each
off-diagonal coefficient is the negative capacitance directly joining the two nodes.
The known terminal potentials appear on the right-hand side after their terms are
moved. A singular matrix usually signals an internal conductor with no capacitive
reference path or a missing constraint, not a numerical inconvenience. Adding a tiny
stray capacitance to an enclosure may make the matrix solvable, but it also changes
the physical network and must be reported rather than treated as a harmless
regularization.

$$
% caption: Node-potential formulation. Each floating conductor gets one charge-balance equation from every capacitance incident on it; terminal charge is recovered only after the internal potentials are solved, so no series or parallel shortcut is assumed in advance.
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An AC bridge measures terminal capacitance by comparing impedances at null. In a
simple ratio bridge, the unknown network and a calibrated standard occupy matching
arms, while noninductive ratio resistors establish the balance. For ideal loss-free
capacitors, the null condition reduces to $C_x/C_s=R_2/R_1$. The detector sees no
voltage at balance, which reduces sensitivity to detector gain, but it does not make
the result independent of source frequency, standard accuracy, lead routing, or
network loading. A bridge measurement is meaningful only after its high, low, and
guard terminals have been mapped to the same conductor nodes used in the calculation.

Source and meter connections can alter a small network. A source with series
resistance $R_s$ does not change the final ideal dc charge ratio, but it sets the
settling scale for a two-terminal network, approximately $\tau=R_sC_{\rm eq}$ when
one dominant capacitance is charged through that resistance. A bridge or digitizer
input can add capacitance to a floating node; its input resistance provides a leakage
path that changes long-time charge retention. In the time domain, readings must be
taken after the relevant node voltages have settled. A reading captured within a few
time constants can be a transient redistribution result rather than the requested
terminal capacitance.

Networks with more than one floating node can have more than one settling mode. A
single product $R_sC_{\rm eq}$ is then only a terminal-scale estimate, not proof
that every internal potential has reached its final value. A high-resistance probe or
leakage path can create a slow mode that is invisible in a rapid bridge balance but
appears in a dc hold measurement. Inspect the internal-node prediction as well as the
terminal signal when timing matters. A practical protocol applies a step, records the
terminal response over a range of delays, and repeats after changing the source
resistance. If the apparent capacitance changes with delay, source impedance, or
detector range, the measurement has not isolated a static two-terminal equivalent.
This diagnostic remains within the network model: it tests which conductive and
capacitive paths are active during the stated observation interval.

The reported two-terminal quantity is obtained from the source-terminal charge:

$$
C_{\rm eq}=\frac{Q_a}{V_a-V_b},
\qquad
Q_a=\sum_j C_{aj}(V_a-V_j).
$$

| recorded item | node-level meaning | consequence of omission |
| --- | --- | --- |
| high and low terminals | define $V_a-V_b$ and the charge counted in $Q_a$ | a different network can be measured |
| guard or shield | adds or removes a capacitance to a reference node | floating-node potentials change |
| probe input | supplies a finite capacitance and resistance | loading and a slow leakage mode appear |
| observation delay | selects a point on the network transient | an unsettled response is reported as a static value |

The calculation and the bridge record must use the same terminal map. An omitted node
creates an unmodeled branch in the reported equivalent capacitance.

> **Worked example (floating-node loading).** A driven terminal at $V_a=10.0\ \mathrm V$
> connects to a floating node through $C_1=100\ \mathrm{pF}$; the node connects to the
> reference terminal through $C_2=220\ \mathrm{pF}$ and has a stray
> $C_g=22.0\ \mathrm{pF}$ to the shield, also at reference potential. The neutral-node
> equation (capacitances in picofarads, voltage in volts) is
>
> $$
> 100(V_f-10.0)+(220+22.0)V_f=0,
> $$
>
> so $V_f=2.92\ \mathrm V$. The source-terminal charge is
> $Q_a=100(10.0-2.92)=708\ \mathrm{pC}$, giving $C_{\rm eq}=Q_a/V_a=70.8\ \mathrm{pF}$.
> Omitting the shield capacitance would predict only $68.8\ \mathrm{pF}$ — the small
> stray shifts the terminal result by nearly three percent. Adding a meter of
> $15.0\ \mathrm{pF}$ input capacitance from the node to reference replaces $C_g$ by
> $37.0\ \mathrm{pF}$ and raises the equivalent to $72.0\ \mathrm{pF}$: a loading effect,
> not a change in the components. With source resistance $R_s=10.0\ \mathrm{k\Omega}$ the
> settling scale is $R_sC_{\rm eq}\approx0.71\ \mathrm{\mu s}$, so a static reading needs
> a few microseconds absent any slower leakage or instrument filter.

$$
% caption: Worked loading diagnostic for a floating node. A stray capacitance $C_g$ from the internal node to shield changes the solved node potential and terminal equivalent; a meter input capacitance changes it again, while source resistance sets how long the reading must settle.
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$$

The diagnostic is checked by deliberate perturbation. First measure with the meter
input disconnected or replaced by a specified low-capacitance probe; then repeat with
an added known capacitor from the floating node to shield. The change in equivalent
capacitance should follow the node equation. Next vary source resistance and confirm
that the approach to the final reading scales with the predicted time constant. A
bridge balance that agrees only after a long wait but disagrees immediately after a
switch event is behaving consistently with the network; a balance that changes when
the guard is touched identifies an unmodeled terminal path. These tests distinguish
component tolerance from loading, stray capacitance, and incomplete settling before
more elaborate network models are introduced.

## Distributed capacitance, parasitics, and transient network tests

An ideal capacitor network treats each drawn node as an equipotential conductor and
each branch capacitance as a localized element. That approximation has a scale limit.
A cable has capacitance distributed continuously between its conductors; a probe has
input capacitance to its reference lead; adjacent traces have mutual capacitance; and
every floating metal part has some capacitance to the enclosure. At sufficiently low
frequency and small physical size, these effects can be collected into a few lumped
branches. At higher frequency, voltage varies along the conductors and a cable must be
treated as a transmission structure rather than one node plus one capacitance.

Cable capacitance is often the dominant unintended branch in picofarad networks. A
one-metre coaxial cable can contribute about one hundred picofarads from its centre
conductor to shield, comparable with or larger than the device under test. A nominally
high-resistance voltage probe can still load a node through its input capacitance.
A floating node has that input as a branch to the probe reference conductor, and the
branch must appear explicitly in the node-charge equation. Disconnecting the probe changes the
network; moving its ground clip can change which enclosure or return path receives
the displaced charge.

$$
% caption: Cable and probe capacitance as network branches. The cable centre conductor, shield, and probe input add capacitances from the measured node to the return path; for small intended capacitors these branches can dominate the terminal equivalent and belong in the same node model as the deliberate components.
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$$

Layout coupling is not limited to cables. Two traces that run in parallel over a
return plane form a distributed three-conductor geometry. Their mutual capacitance
couples a fast voltage change on one trace into the other, creating a displacement
current even when no intentional component joins them. Reducing parallel run length,
increasing spacing, adding a grounded shield trace, or using a defined return plane
changes that coupling. These are changes to the capacitance matrix of the layout.
They cannot be described accurately by saying that one trace merely "picks up noise"
without identifying the source node, victim node, and return conductor.

The lumped-network model applies when physical dimensions are short enough that
propagation delay is negligible compared with the transition or sinusoidal period of
interest, and when a small set of measured parasitic capacitors reproduces the
observed response. It fails when a cable has significant phase variation along its
length, when lead inductance creates resonance with capacitance, or when multiple
unmeasured paths shift results after the fixture is rearranged. At that point,
measuring cable parameters per unit length or using a transmission-line model is more
defensible than adding an arbitrary capacitor to force a fit.

$$
% caption: Layout coupling between adjacent conductors. A changing voltage on the aggressor trace drives displacement current through mutual capacitance into the victim trace and its return path; spacing, parallel length, shielding, and return-plane geometry set this parasitic branch, not an abstract noise label.
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$$

Transient testing exposes parasitic branches by comparing an expected step response
with a controlled perturbation. A source resistor $R_s$ driving a nominal terminal
capacitance $C_N$ predicts a one-pole rise with time constant $R_sC_N$ when the
network has one dominant mode. Adding cable, probe, and fixture capacitances at the
same terminal changes the effective value to approximately their parallel sum. A
multi-node network can instead produce several exponential modes: a prompt change
near the driven terminal followed by slower redistribution through resistive source,
meter, or leakage paths. Fitting only the first part of a waveform can therefore
return a capacitance different from a bridge measured after settling.

> **Worked example (parasitic loading of a step response).** Drive a nominal
> $100\ \mathrm{pF}$ network through $R_s=1.00\ \mathrm{M\Omega}$, giving an ideal time
> constant $R_sC=100\ \mathrm{\mu s}$. A cable contributes $50.0\ \mathrm{pF}$ and a probe
> adds $15.0\ \mathrm{pF}$ at the same terminal, so the observed capacitance is
> $165\ \mathrm{pF}$ and the dominant time constant becomes $165\ \mathrm{\mu s}$. After
> one time constant the ideal step reaches about $63\%$ of its final value and after five
> the remaining difference is under one percent, so a trace sampled at
> $100\ \mathrm{\mu s}$ would read the loaded network as an incompletely charged ideal one
> unless the cable and probe branches were included.

$$
% caption: Transient step test for parasitic capacitance. The nominal network rises with its predicted time constant, while cable and probe capacitance raise the observed time constant and slow the trace; comparing both curves after controlled additions separates incomplete settling from a real change in the intended capacitor.
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\node[black] at (3.6,1.35) {with cable and probe};
\end{tikzpicture}
$$

Frequency-domain measurements provide a complementary limit check. At low frequency,
leakage resistance and instrument input resistance can carry appreciable current and
distort the apparent capacitance. At intermediate frequency, a bridge can isolate the
capacitive quadrature current if its guard and reference are controlled. At high
frequency, cable inductance, connector geometry, and propagation delay introduce
phase shifts that a single parallel capacitance cannot represent. Sweep frequency
while preserving the same terminal configuration, then compare magnitude and phase
with the lumped model. A constant capacitance with growing phase error indicates a
missing series inductance or distributed path; a capacitance that changes when a
probe is moved indicates loading. The reported equivalent must consequently state
frequency, source impedance, observation delay, and terminal map rather than appear
as a context-free component label.

A controlled calibration addition makes the interpretation quantitative. Connect a
known small capacitor at the same terminal pair and repeat both the bridge and step
measurements. In a valid lumped model, the bridge equivalent increases by the known
addition and the dominant settling time increases by $R_s\Delta C$, within source
and timing uncertainty. If either change is smaller than expected, part of the added
branch is being shielded, bypassed, or placed at a different node than assumed. If
the change is larger, a probe or cable return has been moved with the added part.
Repeat with the cable disconnected, then with its shield connected at the documented
reference node. This sequence separates a genuine device capacitance from fixture
coupling without requiring a speculative correction after the measurement.

For rapid transitions, compare cable propagation delay with the desired rise-time
resolution. A delay that is negligible for a one-millisecond bridge excitation can
be significant for a nanosecond step test. The same physical cable may therefore be
adequately represented by a lumped capacitance in one experiment and require a
distributed model in another. State the time scale at which the equivalent network is
claimed to apply; without it, a disagreement between frequency and step measurements
cannot be diagnosed as either a component effect or a model-domain error.

## Charged switches, redistribution, and circuit-energy bookkeeping

Switching networks require an initial state and a final connection state. A capacitor
voltage cannot change discontinuously through an ordinary finite-resistance path, so
each capacitor carries its pre-switch voltage into the instant after the switch
changes. Conductors newly joined by a switch subsequently share one potential, but
the redistribution needed to reach that common potential is governed by charge
conservation and the paths provided by the real circuit. Drawing two capacitor symbols
beside a switch is not enough: the polarity of each initial voltage and the exact
terminals joined by the switch determine whether charge differences add or cancel.

Consider two initially isolated capacitors whose negative terminals are joined and
whose positive terminals are then connected together. Let their signed initial
voltages relative to the common negative node be $V_{1i}$ and $V_{2i}$. The total
free charge on the joined positive conductor is conserved after the source is removed:

$$
Q_{+}=C_1V_{1i}+C_2V_{2i}=(C_1+C_2)V_f.
$$

The final common voltage is consequently

$$
V_f=\frac{C_1V_{1i}+C_2V_{2i}}{C_1+C_2}.
$$

The formula uses signed voltages. A capacitor initially connected with opposite
polarity has a negative $V_i$ in this convention, so it can lower the final voltage
or produce a zero final value. The equation does not state that charge is conserved
on each original positive plate; charge can move through the switch. It states that
the net free charge on the newly joined conductor is conserved when no external path
exists.

$$
% caption: State change when initially charged capacitors are connected. Before closure each capacitor holds its own signed voltage; after closure the joined positive plates form one conductor at a common final voltage, set by conservation of total free charge on that conductor rather than by an average of voltages.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (0.55,2.0)--(0.55,1.0); \draw[thick] (0.8,2.0)--(0.8,1.0);
\draw[thick] (2.05,2.0)--(2.05,1.0); \draw[thick] (2.3,2.0)--(2.3,1.0);
\draw[black] (0.675,2.0)--(1.2,2.0);
\draw[black] (1.6,2.0)--(2.175,2.0);
\draw[black] (1.2,2.0)--(1.55,2.32);
\filldraw[black] (1.2,2.0) circle (1.3pt);
\filldraw[black] (1.6,2.0) circle (1.3pt);
\draw[black] (0.675,1.0)--(2.175,1.0);
\node[black,above] at (1.35,2.34) {open switch};
\node[black,below] at (1.4,1.0) {initial state};
\draw[->,thick] (2.75,1.5)--(3.4,1.5);
\draw[thick] (3.85,2.0)--(3.85,1.0); \draw[thick] (4.1,2.0)--(4.1,1.0);
\draw[thick] (5.35,2.0)--(5.35,1.0); \draw[thick] (5.6,2.0)--(5.6,1.0);
\draw[acc,thick] (3.975,2.0)--(5.475,2.0);
\draw[black] (3.975,1.0)--(5.475,1.0);
\node[black,below] at (4.725,1.0) {joined state};
\end{tikzpicture}
$$

The field energy usually decreases during this redistribution even though charge is
conserved. Initial and final values are

$$
U_i=\frac12C_1V_{1i}^2+\frac12C_2V_{2i}^2,
\qquad
U_f=\frac12(C_1+C_2)V_f^2.
$$

Subtracting after inserting the charge-conservation result gives

$$
U_i-U_f=\frac12\frac{C_1C_2}{C_1+C_2}(V_{1i}-V_{2i})^2\ge0.
$$

Equality holds only when the two initial voltages already match. The nonnegative
difference is not destroyed by a correct circuit model. It becomes thermal energy in
switch and lead resistance, dielectric loss, and in very fast transitions a small
amount of electromagnetic radiation or ringing energy that is later dissipated. An
ideal zero-resistance switch predicts an impulsive current and hides the physical
path of this energy. It remains adequate for the final-state constraint but omits the
transient mechanism.

$$
% caption: Energy bookkeeping for capacitor redistribution. Charge conservation fixes the final common voltage, while the drop from initial to final field energy is nonnegative and goes into the resistive, lossy, and radiative parts of the real switching path; an ideal switch hides the mechanism, not the energy difference.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.2,0) node[right,black] {state};
\draw[->,black] (0,0)--(0,3.0) node[above,black] {stored energy};
\draw[black,fill=black!14] (0.9,0) rectangle (2.0,2.45);
\draw[black,fill=black!16] (3.1,0) rectangle (4.2,1.6);
\draw[black,dashed] (2.0,2.45)--(4.2,2.45);
\draw[->,acc,thick] (2.25,1.55)--(2.9,1.55);
\node[acc,above] at (2.57,1.6) {loss};
\node[black,below] at (1.45,0) {before closure};
\node[black,below] at (3.65,0) {after settling};
\end{tikzpicture}
$$

The switching transient can be derived when the two positive nodes are connected
through a resistance $R$. Let $v_1(t)$ and $v_2(t)$ be capacitor voltages relative
to the common negative node. Current through the resistor is $(v_1-v_2)/R$. Charge
balance on the two capacitors gives a decaying voltage difference,

$$
v_1-v_2=(V_{1i}-V_{2i})e^{-t/\tau},
\qquad
\tau=R\frac{C_1C_2}{C_1+C_2}.
$$

The capacitance in this time constant is the series combination of the two
capacitors, even though the final connected state has their capacitances in parallel.
This distinction follows from the differential mode: one capacitor loses charge while
the other gains the same amount. The weighted average
$(C_1v_1+C_2v_2)/(C_1+C_2)$ remains equal to $V_f$ throughout the isolated
redistribution, while the voltage difference decays. A switch with contact bounce,
lead inductance, or a nonlinear resistance can add ringing or multiple time scales;
the final charge-conservation result remains the check on any transient solution.

> **Worked example (charge redistribution on closing a switch).** Capacitor
> $C_1=100\ \mathrm{pF}$ at $12.0\ \mathrm V$ and $C_2=300\ \mathrm{pF}$ at
> $4.00\ \mathrm V$ have like-polarity terminals connected. Conservation of the joined
> free charge gives the final common voltage
>
> $$
> V_f=\frac{(100)(12.0)+(300)(4.00)}{100+300}=6.00\ \mathrm V.
> $$
>
> Field energy falls from $U_i=9.60\ \mathrm{nJ}$ to $U_f=7.20\ \mathrm{nJ}$, so the
> switching path dissipates $2.40\ \mathrm{nJ}$ — the value the squared-difference
> formula also gives. With connection resistance $1.00\ \mathrm{k\Omega}$ the
> differential-mode time constant is $\tau=RC_1C_2/(C_1+C_2)=75.0\ \mathrm{ns}$ and the
> initial current is $(12.0-4.00)/1000=8.00\ \mathrm{mA}$; after five time constants the
> voltage difference is under one percent of its initial value. A meter across either
> capacitor during the transient adds a branch capacitance and can shift both the final
> voltage and the observed time constant.

$$
% caption: Redistribution transient for unequal initial capacitor voltages. The two capacitor voltages approach the charge-conserving common value while their difference decays with $R$ times the series capacitance; the current starts at the voltage difference over switching resistance, then falls to zero.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.3,0) node[right,black] {time};
\draw[->,black] (0,0)--(0,3.0) node[above,black] {capacitor voltage};
\draw[black,dashed] (0,1.5)--(5.0,1.5);
\draw[acc,thick] plot[domain=0:5.0,samples=140] ({\x},{1.5+1.0*exp(-1.05*\x)});
\draw[black,thick] plot[domain=0:5.0,samples=140] ({\x},{1.5-0.75*exp(-1.05*\x)});
\draw[black,dashed] (0.95,0)--(0.95,1.32);
\node[black,below] at (0.95,0) {one time constant};
\node[acc,right] at (2.2,1.98) {$C_1$ voltage};
\node[black,right] at (2.2,1.02) {$C_2$ voltage};
\node[black,right] at (4.0,1.75) {common value};
\end{tikzpicture}
$$

Switching calculations should state which conductors are isolated before and after
closure. Connecting only the positive plates while negative plates remain separate is
not the same network as joining both terminal pairs. Connecting opposite polarities
changes the signed initial charges and can create a larger transient current. A source
left connected supplies or absorbs charge, invalidating the isolated total-charge
constraint used above. Before using a final-voltage formula, draw the post-switch
nodes, list which nodes have external charge paths, and preserve the signed initial
capacitor voltages. That network-state record makes charge conservation and energy
bookkeeping testable rather than mnemonic rules.

A practical transient verification uses both voltage and current records. Measure the
two capacitor voltages with probes whose input capacitances are included in the model,
and place a known small resistance in the switching path if current must be inferred.
The signed current should initially equal the voltage difference divided by the total
series resistance and should integrate to the charge transferred from one capacitor
node to the other. The final measured voltages must agree within uncertainty and must
equal the charge-weighted common voltage predicted from the documented pre-switch
state. A current trace that reverses unexpectedly, a final voltage outside the
weighted range for like-polarity capacitors, or a nonzero late current indicates an
unaccounted source, leakage path, contact state, or probe branch.

Opposite-polarity tests are particularly diagnostic because they exercise sign
conventions. If equal capacitances begin at equal and opposite voltages and matching
terminals are joined with the stated polarity, the final common voltage is zero while
the initial field energies are dissipated in the redistribution path. If the measured
final voltage is not near zero, the likely causes are unequal actual capacitances,
unmatched initial voltages, an offset reference, or an external capacitive path. This
test should be performed at a voltage and switching speed consistent with the rating
of the components and the current capability of the switch. It verifies the node
charge model without requiring a separate energy measurement.

## Tolerance analysis, calibration, and model limits

An equivalent capacitance is a derived quantity, so its uncertainty follows the
network constraint rather than a single component tolerance printed on a package. For
parallel branches, independent absolute standard uncertainties combine as

$$
u^2(C_{\rm parallel})=u^2(C_1)+u^2(C_2)+\cdots.
$$

For two series capacitors,
$C_s=C_1C_2/(C_1+C_2)$, logarithmic differentiation gives

$$
\frac{\d C_s}{C_s}=
\frac{C_2}{C_1+C_2}\frac{\d C_1}{C_1}+
\frac{C_1}{C_1+C_2}\frac{\d C_2}{C_2}.
$$

The larger capacitance receives the larger weight multiplying the fractional
uncertainty of the other branch. This is consistent with the voltage division: the
smaller series capacitance controls more of the equivalent behavior. Independent
random tolerance contributions can be combined in quadrature after these sensitivity
coefficients are applied. Correlated effects, such as temperature drift shared by
capacitors from the same assembly or a common calibration scale, require covariance
terms or a worst-case bound; treating them as independent can understate uncertainty.

Tolerance is also state-dependent in a floating network. A stray capacitance to a
shield changes the sensitivity of terminal capacitance to an internal branch. A probe
that is present during calibration but absent during use changes the node matrix
itself and can dominate uncertainty beyond individual measurement digits. List
intended terminal potentials, floating-node charge assumptions, enclosure connection,
cable routing, and unused terminals before propagating component values. An
uncertainty calculation cannot repair a terminal definition that differs between the
model and the measurement.

A bridge null is traceable only when its standard, ratio, frequency, amplitude, and
terminal configuration are recorded. At balance, an ideal comparison gives
$C_x/C_s=R_2/R_1$, but the practical ratio is corrected by standard uncertainty,
ratio-arm calibration, detector resolution, and fixture effects. Calibrate the
instrument with standards that bracket the expected unknown value at the same
frequency and voltage range. Retain the reference certificate and its stated
conditions rather than treating a nominal standard label as an exact value.

The terminal definition is part of that calibration. A two-terminal measurement
includes every branch between the high and low leads. A guarded three-terminal
measurement excludes current supplied to the guard path and can give a different
number for the same physical assembly. Open correction estimates the fixture branch
remaining when no specimen is attached; short correction estimates residual series
impedance. Both corrections are valid only for the cable placement, shield routing,
and range in which they were obtained. Reconnecting a floating node to a shield for
convenience can move charge through a new branch and invalidate a prior calibration.

Residual analysis tests whether the calibrated network model is adequate. Measure
terminal capacitance across several frequencies, source resistances, added reference
capacitors, and probe conditions. Subtract the prediction from each reading and plot
the residual against the control variable. Random residual scatter at the stated
repeatability supports the model over that range. A constant offset suggests a fixed
parallel fixture capacitance; a change with cable length suggests distributed
capacitance; a delay-dependent residual points to incomplete settling; and a curved
frequency trend indicates that a single ideal capacitance cannot describe the
measurement band. Residual shape distinguishes these mechanisms more strongly than
one favorable bridge null.

Use an acceptance test that is independent of the data used to adjust the model. For
example, calibrate bridge scale and fixture correction with two standards, fit the
network using several source resistances, then predict the terminal capacitance after
adding a known capacitor at a documented node. The prediction must include the
uncertainty of the added standard, the connection repeatability, and the sensitivity
of every floating-node equation to that branch. Agreement within a stated coverage
interval supports both the calibration and terminal map. Failure localizes the next
diagnostic: a uniform offset points toward the fixture correction, a mismatch only
with the added branch points toward node assignment, and a mismatch that grows with
frequency points beyond the lumped-model range.

Reporting should distinguish a standard uncertainty from a tolerance interval. A
component tolerance may be a manufacturer limit over temperature and aging, whereas a
bridge repeatability estimate describes short-term scatter under one configuration.
Combine quantities only after assigning their meaning and probability model. State the
coverage factor used for an expanded uncertainty and identify contributions retained
as bounds because a distribution is not justified. This prevents a precise bridge
balance from being presented as an accurate network equivalent when terminal paths or
model limits dominate the result.

Distributed-model limits set a boundary on every tolerance claim. A cable can be
represented by a lumped capacitance only when its propagation delay and inductive
effects are negligible over the measurement frequency or step rise time. Beyond that
range, the capacitance matrix of a small network is replaced by a distributed line
with position-dependent voltage and current. No amount of calibration at one low
frequency validates a lumped equivalent at a fast edge. State the frequency band,
source impedance, observation delay, and allowed residual when reporting an
equivalent capacitance.

> **Worked example (network uncertainty budget).** A $100\pm1\ \mathrm{pF}$ capacitor in
> series with $300\pm3\ \mathrm{pF}$ gives a $75.0\ \mathrm{pF}$ series result, placed in
> parallel with $47.0\pm0.47\ \mathrm{pF}$. The series sensitivity weights are $0.75$ and
> $0.25$, giving a series standard uncertainty of $0.593\ \mathrm{pF}$. Combining with the
> parallel branch gives $C_{\rm eq}=122.0\ \mathrm{pF}$ with component-only uncertainty
> $0.757\ \mathrm{pF}$. Adding a bridge-standard contribution of $0.24\ \mathrm{pF}$,
> ratio and repeatability contributions of $0.12\ \mathrm{pF}$ each, and a guarded-fixture
> correction of $0.30\ \mathrm{pF}$ combines in quadrature to about $0.87\ \mathrm{pF}$,
> or $0.71\%$, provided the contributions are independent and residual tests show no
> further model discrepancy. A residual trend exceeding that budget is a model limit, not
> a reason to round the uncertainty downward.

## Reporting a capacitance-network result

A capacitance-network result is incomplete unless it names the terminals that define
the reported quantity. Label the driven terminal, reference terminal, every internal
conductor, shield, guard, and enclosure connection. A notation such as $C_{AB}$
means the charge supplied at terminal A divided by $V_A-V_B$, with every other
conductor state specified separately. It does not mean that a component marked A is
intrinsically a capacitor of that value. Moving the reference lead from B to a guard
or enclosure can change the node-charge equations and therefore change the measured
equivalent.

Document the state before measurement. For a static bridge result, record whether
internal conductors were neutral, grounded, precharged, or connected through a
resistive path. For a switched test, record all initial capacitor voltages with
polarity, switch position before and after the event, source connection, observation
delay, and the point at which the reported value was sampled. A floating node is not
defined merely by having no visible wire in a schematic; probe capacitance, shield
coupling, or a prior source connection can give it a nonzero free charge or a path to
reference.

Instrument connections are part of the terminal definition. Report bridge frequency,
test amplitude, high and low leads, guard connection, cable type and length, open and
short corrections, standard identifier, and balance criterion. For a time-domain
result, report source resistance, probe model or input capacitance, sample rate,
analog bandwidth, and wait time expressed relative to the predicted network modes.
These details permit another measurement system to reproduce the same electrical
network rather than merely repeat the same nominal component values.

Use at least one independent check path. A node-potential calculation using measured
branch values can predict the terminal equivalent. A bridge null can compare that
equivalent with a traceable standard. A controlled step test can check whether the
same terminal model predicts the observed settling response. Agreement among these
paths supports the stated node map. A disagreement should be reported with its
residual pattern and tested connection changes; replacing it with a rounded average
removes the evidence needed to locate a stray branch or an incorrect initial-state
assumption.

Include a diagram identifier or photograph reference for the physical wiring. Mark
which switch contacts were open or closed, the orientation of polarized components if
relevant, and whether unused conductors were left open, grounded, or guarded. These
details make a reported network state recoverable after the fixture has been changed.

A concise worked report might read: “Between terminal A and terminal B, with the
guard connected to the documented enclosure and the internal node initially neutral,
the network capacitance was $C_{AB}=(122.0\pm1.7)\ \mathrm{pF}$, expanded
uncertainty with coverage factor two. Measurement used a guarded bridge at
1.00 kHz and 1.00 V rms, a 100 pF traceable standard, open and short corrections,
and the stated cable routing. A node-charge calculation predicted 122.0 pF; a
separate added-capacitor test agreed within the uncertainty budget.” This format
states the measurand, conditions, uncertainty meaning, and independent validation
without claiming a context-free capacitance.

The final report should retain the raw readings or a traceable data reference alongside
the reduced capacitance. That record allows a later reviewer to recalculate the
equivalent after a corrected standard value, revised cable correction, or new
understanding of an internal-node path. It also distinguishes a network specification
from a measurement result: the former lists intended component values and topology;
the latter states the terminal-defined behavior observed under documented conditions.

Frequency belongs in the reported result whenever dielectric loss, lead inductance,
or a distributed cable branch is appreciable. A bridge may report an equivalent
parallel capacitance and loss at one frequency, whereas a transient test identifies a
settling response over a band of frequencies. Those results can both be valid while
referring to different terminal models. State the fitting model, frequency range, and
whether the quoted capacitance is a series or parallel equivalent. A single number
without that convention can conceal a measurable loss conductance or a resonance
outside the calibration band.

Uncertainty terms also need their correlation stated. Repeating a bridge balance
reduces random reading scatter, but it does not average away a common standard error,
a fixture correction, or the same unmodeled cable branch used on every repeat. Group
the uncertainty budget into random repeatability, calibrated standards, geometry or
fixture terms, and model discrepancy. A frequency sweep, a cable substitution, and a
guard-state reversal test different terms in that budget. Preserve those raw
comparisons with the result. They show whether the quoted uncertainty describes a
stable two-terminal capacitance or only a repeatable instrument reading under one
particular wiring arrangement.