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After a DC source is connected, capacitor voltage approaches the source voltage exponentially through a resistor.
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Formula
UC and U0 in V; R in Ω; C in F; t and τ in s
| Symbol | Meaning | Unit |
|---|---|---|
| Capacitor voltage at time t | V | |
| Constant ideal source voltage | V | |
| Total series resistance | Ω | |
| Capacitance | F | |
| RC time constant | s |
U_C(t)=U_0(1-e^{-t/(RC)}),\qquad \tau=RCThis response assumes a linear RC circuit, constant DC source, initially uncharged capacitor and ideal components; source and load resistance belong in R. After one time constant, voltage reaches about 63.2% of its final value. For discharge, UC=Ui e^(−t/RC). The charging equation approaches its final value asymptotically.
This response assumes a linear RC circuit, constant DC source, initially uncharged capacitor and ideal components; source and load resistance belong in R. After one time constant, voltage reaches about 63.2% of its final value. For discharge, UC=Ui e^(−t/RC). The charging equation approaches its final value asymptotically.
After a DC source is connected, capacitor voltage approaches the source voltage exponentially through a resistor.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Time response of an RC charging circuit” and check its unit and magnitude.
With R=10 kΩ and C=100 μF, τ=1 s.
After one second, a capacitor charged from 5 V reaches about 3.16 V.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: UC and U0 in V; R in Ω; C in F; t and τ in s.
Formula cluster
After a DC source is connected, capacitor voltage approaches the source voltage exponentially through a resistor.
With R=10 kΩ and C=100 μF, τ=1 s. After one second, a capacitor charged from 5 V reaches about 3.16 V.
Model timing, filters and capacitor charge transients in simple circuits.
Here is how to work through a typical Time response of an RC charging circuit (UC(t)=U₀(1−e⁻ᵗ/ᴿᶜ); τ=RC) task step by step:
Task
Work through the numerical example for “Time response of an RC charging circuit” and check its unit and magnitude.
Solution path
With R=10 kΩ and C=100 μF, τ=1 s.
After one second, a capacitor charged from 5 V reaches about 3.16 V.
τ is the RC network’s time constant. At t=τ, an ideal initially uncharged capacitor reaches about 63.2% of its final step value; during discharge about 36.8% of initial voltage remains. Larger R or C makes the process slower. The product has units Ω·F=s. This first-order response assumes a linear resistor and capacitor, constant source and defined initial voltage. Include source resistance and load in effective R. The exponential never reaches its final value exactly; practical settling is specified to a tolerance. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
If a capacitor initially charged to Ui discharges through R, UC(t)=Ui e−ᵗ/ᴿᶜ. Unlike charging, the exponential decays toward zero rather than a source voltage. Current sign depends on reference direction because discharge current reverses relative to charging. This first-order response assumes a linear resistor and capacitor, constant source and defined initial voltage. Include source resistance and load in effective R. The exponential never reaches its final value exactly; practical settling is specified to a tolerance. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Mathematically, UC(t) approaches its final value asymptotically and never reaches it exactly in finite time. After five time constants, less than one percent of the original error remains. Practical circuits often treat that as charged, but accuracy depends on tolerance and leakage. This first-order response assumes a linear resistor and capacitor, constant source and defined initial voltage. Include source resistance and load in effective R. The exponential never reaches its final value exactly; practical settling is specified to a tolerance. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: UC and U0 in V; R in Ω; C in F; t and τ in s. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example With R=10 kΩ and C=100 μF, τ=1 s. After one second, a capacitor charged from 5 V reaches about 3.16 V. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. This first-order response assumes a linear resistor and capacitor, constant source and defined initial voltage. Include source resistance and load in effective R. The exponential never reaches its final value exactly; practical settling is specified to a tolerance. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: This response assumes a linear RC circuit, constant DC source, initially uncharged capacitor and ideal components; source and. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. This first-order response assumes a linear resistor and capacitor, constant source and defined initial voltage. Include source resistance and load in effective R. The exponential never reaches its final value exactly; practical settling is specified to a tolerance. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
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How do you rearrange Time response of an RC charging circuit for Base equation for the target quantity?
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Which common mistake happens with Time response of an RC charging circuit?
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