Transient Analysis of RC and RL Circuits

Summary

This chapter analyzes the time-domain behavior of first-order circuits containing resistors with either capacitors (RC) or inductors (RL). Students will learn how these circuits respond to sudden changes like switching events, developing intuition for time constants and exponential responses. The chapter covers charging and discharging behavior, initial and final conditions, and the concepts of natural and forced response. Understanding transient analysis is crucial for analyzing real-world circuits where signals change over time.

Concepts Covered

This chapter covers the following 17 concepts from the learning graph:

Transient Response
Steady-State Response
Time Constant
RC Circuit
RC Charging
RC Discharging
RL Circuit
RL Energizing
RL De-energizing
Exponential Response
Initial Conditions
Final Conditions
Natural Response
Forced Response
Complete Response
First-Order Circuits
Step Response

Prerequisites

This chapter builds on concepts from:

Introduction: When Circuits Remember

Up until now, we've analyzed circuits in the comfortable world of steady state—where nothing changes with time. Flip a switch, and we assumed everything instantly reached its final value. But reality doesn't work that way.

When you flip a switch, circuits don't instantly jump to their new state. They transition, following smooth curves that take time to reach their destination. This transition period is the transient response, and understanding it unlocks your ability to design timing circuits, filters, and anything that responds to changing signals.

Here's the magical part: despite the seemingly complex behavior of capacitors and inductors, first-order circuits (one energy storage element) always follow a single pattern—the exponential. Once you recognize this pattern, you can analyze an enormous variety of circuits with the same basic approach.

Think of transient analysis as learning to read a circuit's "body language." The circuit tells you exactly where it's going and how fast it's getting there. You just need to know what to look for.

Ready to watch circuits in motion? Let's flip some switches.

First-Order Circuits: One Storage Element at a Time

A first-order circuit contains exactly one energy storage element (either a capacitor or an inductor) along with resistors and sources. The "first-order" name comes from the first-order differential equation that describes the circuit's behavior.

Why start here? Because first-order circuits are mathematically tractable and appear everywhere in practical applications:

Power supply filtering (RC)
Relay drivers (RL)
Audio coupling circuits (RC)
Timing circuits (RC)
Motor driver protection (RL)

Circuit Type	Energy Storage	Stores	Key Variable
RC Circuit	Capacitor	Electric field energy	Voltage
RL Circuit	Inductor	Magnetic field energy	Current

The Time Constant: Your Circuit's Personality

The time constant (\(\tau\), Greek letter tau) is the single most important parameter in transient analysis. It tells you how fast a circuit responds to changes.

RC Time Constant

\(\tau = RC\)

where:

\(\tau\) is the time constant in seconds (s)
\(R\) is resistance in ohms (Ω)
\(C\) is capacitance in farads (F)

RL Time Constant

\(\tau = \frac{L}{R}\)

where:

\(\tau\) is the time constant in seconds (s)
\(L\) is inductance in henrys (H)
\(R\) is resistance in ohms (Ω)

Notice the difference: RC multiplies, RL divides. This makes physical sense:

RC: More capacitance means more charge to move; more resistance means slower charging. Both increase \(\tau\).
RL: More inductance means more "magnetic inertia"; more resistance means faster energy dissipation. L increases \(\tau\), R decreases it.

What the Time Constant Means

After one time constant (\(t = \tau\)), the circuit has completed approximately 63.2% of its transition. After five time constants (\(t = 5\tau\)), it's essentially complete at 99.3%.

Time	Percent Complete	Percent Remaining
\(0\tau\)	0%	100%
\(1\tau\)	63.2%	36.8%
\(2\tau\)	86.5%	13.5%
\(3\tau\)	95.0%	5.0%
\(4\tau\)	98.2%	1.8%
\(5\tau\)	99.3%	0.7%

The 5τ Rule

Engineers commonly use 5τ as the practical "settling time"—the time needed for a circuit to reach its final value. Technically it never quite gets there, but after 5τ, it's close enough for government work.

Diagram: Time Constant Visualization

Time Constant Visualization MicroSim

Type: microsim Purpose: Demonstrate how the time constant affects the speed of transient response Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain how changing R and C (or L and R) affects the circuit's response speed and relate the time constant to the percentage completion at different times. Instructional Rationale: Interactive adjustment of R and C with real-time waveform updates builds intuition for how component values control response speed. Canvas layout: - Left (400px): Exponential response curve with time markers - Right (200px): Component value controls and calculations Visual elements: - Exponential curve rising from 0 to final value - Vertical dashed lines at τ, 2τ, 3τ, 4τ, 5τ - Horizontal lines at 63.2%, 86.5%, 95%, 98.2%, 99.3% - Current time marker that can be dragged - Percent complete display Interactive controls: - Slider: Resistance (100Ω to 100kΩ) - Slider: Capacitance (1µF to 1000µF) - Display: Calculated τ value - Display: Settling time (5τ) - Draggable time marker shows instantaneous value Default parameters: - R = 1kΩ, C = 100µF - τ = 100ms - 5τ = 500ms Behavior: - Changing R or C recalculates τ and rescales time axis - Curve shape stays the same, time scale changes - Dragging marker shows value and % complete at that time - Highlights the "shape is constant, scale changes" insight Implementation: p5.js with interactive curve and markers

RC Charging: Building Up the Electric Field

When you connect a capacitor to a voltage source through a resistor, the capacitor charges exponentially.

The Setup

Consider a simple circuit: voltage source \(V_s\), resistor \(R\), and capacitor \(C\) (initially uncharged). At \(t = 0\), a switch closes.

Initial condition: \(V_C(0) = 0\) (capacitor starts uncharged)

Final condition: \(V_C(\infty) = V_s\) (capacitor charges to source voltage)

The Solution

RC Charging Voltage

\(V_C(t) = V_s(1 - e^{-t/\tau})\)

where:

\(V_C(t)\) is the capacitor voltage at time \(t\)
\(V_s\) is the source voltage
\(\tau = RC\) is the time constant
\(e\) is Euler's number (≈ 2.718)

RC Charging Current

\(I(t) = \frac{V_s}{R}e^{-t/\tau}\)

where:

\(I(t)\) is the current at time \(t\)
\(V_s/R\) is the initial current (maximum)
The current decays exponentially as the capacitor charges

Let's trace the physics:

At \(t = 0\): Capacitor is uncharged (\(V_C = 0\)), so full source voltage appears across R. Maximum current flows: \(I_0 = V_s/R\)
As time passes: Capacitor voltage rises, reducing voltage across R, reducing current
At \(t = \infty\): Capacitor fully charged (\(V_C = V_s\)), no voltage across R, zero current

The current creates the charging—but the more the capacitor charges, the less current flows. This self-limiting behavior creates the characteristic exponential curve.

Diagram: RC Charging Circuit

RC Charging MicroSim

Type: microsim Purpose: Visualize RC charging with simultaneous voltage and current waveforms Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain the complementary relationship between capacitor voltage (rising) and circuit current (falling) during charging, and relate both to the time constant. Instructional Rationale: Animated charge flow synchronized with voltage/current graphs connects the abstract exponential math to physical electron movement. Data Visibility Requirements: - Stage 1: Show circuit with switch open, initial conditions - Stage 2: Close switch, show initial current surge - Stage 3: Display voltage rising, current falling exponentially - Stage 4: Mark τ points on both curves - Stage 5: Show final steady state (V=Vs, I=0) Canvas layout: - Top (250px): Circuit schematic with animated current flow - Bottom (250px): Dual graphs (Vc and I vs time) Visual elements: - Battery, switch, resistor, capacitor in loop - Animated electrons flowing (speed proportional to current) - Capacitor with charge level indicator - Two synchronized graphs: Vc rising, I falling - Time markers at τ intervals Interactive controls: - Slider: Source voltage (1V to 20V) - Slider: Resistance (1kΩ to 100kΩ) - Slider: Capacitance (1µF to 100µF) - Button: "Close Switch" starts animation - Button: "Reset" returns to initial state - Display: τ, I_initial, current values Default parameters: - Vs = 10V, R = 10kΩ, C = 10µF - τ = 100ms Behavior: - Switch closure triggers animation - Electron animation slows as current decreases - Graphs build in real-time - Pausing shows instantaneous values - Can restart with different parameters Implementation: p5.js with particle animation and synchronized graphing

RC Discharging: Releasing the Energy

Now consider a charged capacitor (initial voltage \(V_0\)) connected across a resistor. The capacitor discharges through the resistor.

Initial condition: \(V_C(0) = V_0\) (capacitor starts charged)

Final condition: \(V_C(\infty) = 0\) (capacitor fully discharged)

The Solution

RC Discharging Voltage

\(V_C(t) = V_0 \cdot e^{-t/\tau}\)

where:

\(V_C(t)\) is the capacitor voltage at time \(t\)
\(V_0\) is the initial voltage
\(\tau = RC\) is the time constant

RC Discharging Current

\(I(t) = -\frac{V_0}{R}e^{-t/\tau}\)

where:

The negative sign indicates current flows opposite to charging direction
Magnitude decays exponentially

The discharging curve is a pure exponential decay—the simplest transient response. The capacitor's stored energy dissipates as heat in the resistor.

Energy Accounting

During discharge, the capacitor's initial energy (\(\frac{1}{2}CV_0^2\)) is entirely converted to heat in the resistor. You can verify this by integrating \(I^2R\) over time. The universe's energy bookkeeping always balances.

Diagram: RC Discharging

RC Discharging MicroSim

Type: microsim Purpose: Visualize RC discharge and energy dissipation Bloom Level: Apply (L3) Bloom Verb: calculate Learning Objective: Students will calculate the voltage at any time during discharge and determine how long it takes to reach a specified voltage level. Canvas layout: - Top: Circuit with discharging capacitor - Bottom: Voltage decay curve with energy calculation Visual elements: - Capacitor with decreasing charge indicator - Resistor with heat glow (intensity = power) - Exponential decay curve - Energy remaining vs energy dissipated bar chart Interactive controls: - Slider: Initial voltage (1V to 20V) - Slider: Resistance (1kΩ to 100kΩ) - Slider: Capacitance (1µF to 100µF) - Button: "Start Discharge" - Input: Target voltage, calculates time to reach - Display: Current V, I, E_remaining, E_dissipated Default parameters: - V0 = 10V, R = 10kΩ, C = 10µF - τ = 100ms Behavior: - Real-time discharge animation - Heat glow on resistor fades as current decreases - Energy bar chart shows transfer from C to R (heat) - Can input target voltage and get time Implementation: p5.js with energy visualization

The Universal Form: Step Response

Both charging and discharging (and indeed all first-order transients) follow a universal pattern. The step response describes how any first-order circuit responds to a sudden change in input.

General Step Response Formula

\(x(t) = x(\infty) + [x(0) - x(\infty)]e^{-t/\tau}\)

where:

\(x(t)\) is the variable of interest (voltage or current) at time \(t\)
\(x(0)\) is the initial value (at \(t = 0^+\), just after the switch)
\(x(\infty)\) is the final value (steady state)
\(\tau\) is the time constant

This single formula handles any first-order transient. Just plug in:

Initial value: What is \(x\) just after the switch event?
Final value: What will \(x\) be after a long time?
Time constant: What is \(\tau\) for this circuit?

Scenario	\(x(0)\)	\(x(\infty)\)	Response Type
RC Charging	0	\(V_s\)	Rising exponential
RC Discharging	\(V_0\)	0	Decaying exponential
Partial charge	\(V_1\)	\(V_2\)	Rising (if \(V_2 > V_1\)) or falling

RL Circuits: Magnetic Energy Storage

RL circuits follow the same exponential pattern, but with current as the key variable (since inductors store energy in current, not voltage).

RL Energizing

Connect a voltage source through a resistor to an inductor (initially with zero current). At \(t = 0\), close the switch.

Initial condition: \(I_L(0) = 0\) (no initial current)

Final condition: \(I_L(\infty) = V_s/R\) (determined by Ohm's Law—inductor becomes a short circuit at DC)

RL Energizing Current

\(I_L(t) = \frac{V_s}{R}(1 - e^{-t/\tau})\)

where:

\(I_L(t)\) is the inductor current at time \(t\)
\(V_s/R\) is the final current
\(\tau = L/R\) is the time constant

RL Energizing Voltage

\(V_L(t) = V_s \cdot e^{-t/\tau}\)

where:

\(V_L(t)\) is the voltage across the inductor
\(V_s\) is the initial inductor voltage (full source voltage at \(t = 0\))

The physics:

At \(t = 0\): Inductor current is zero (can't change instantaneously), so inductor appears as an open circuit. Full source voltage appears across L.
As time passes: Current builds up, inductor voltage drops, more voltage appears across R
At \(t = \infty\): Inductor carries full current (\(V_s/R\)), inductor voltage is zero (just a wire)

RL De-energizing

An inductor carrying current \(I_0\) is suddenly disconnected from its source and allowed to discharge through a resistor.

The Kickback Problem

Unlike capacitors, inductors resist changes in current. If you try to suddenly interrupt inductor current (like opening a switch), the inductor will generate whatever voltage is needed to maintain current flow. This can be hundreds or thousands of volts from a small battery! This "inductive kick" can arc across switch contacts, damage semiconductors, and cause serious harm. Always provide a current path for inductors.

RL De-energizing Current

\(I_L(t) = I_0 \cdot e^{-t/\tau}\)

where:

\(I_L(t)\) is the inductor current at time \(t\)
\(I_0\) is the initial current
\(\tau = L/R\) is the time constant

Diagram: RL Circuit Response

RL Circuit Response MicroSim

Type: microsim Purpose: Compare RL energizing and de-energizing transients Bloom Level: Understand (L2) Bloom Verb: compare Learning Objective: Students will compare RL and RC circuit behavior, recognizing that RL current behaves like RC voltage (and vice versa). Instructional Rationale: Side-by-side RL and RC comparison highlights the duality between the circuit types. Canvas layout: - Top (250px): RL circuit with magnetic field visualization - Bottom (250px): Current and voltage waveforms Visual elements: - Inductor with animated magnetic field lines - Field intensity proportional to current - Dual graphs: IL and VL vs time - Flyback diode option for de-energizing Interactive controls: - Slider: Source voltage (1V to 20V) - Slider: Resistance (10Ω to 1kΩ) - Slider: Inductance (1mH to 100mH) - Button: "Energize" / "De-energize" - Toggle: Include flyback diode - Display: τ, I_final, current values Default parameters: - Vs = 12V, R = 100Ω, L = 10mH - τ = 100µs Behavior: - Magnetic field animation strengthens with current - Energizing: current rises, voltage falls - De-energizing: current falls, shows voltage spike without diode - With flyback diode, spike is clamped Implementation: p5.js with magnetic field visualization

Initial and Final Conditions: The Key to Every Problem

The secret to transient analysis is finding the initial conditions (values at \(t = 0^+\)) and final conditions (values at \(t = \infty\)). Once you have these, the exponential formula does the rest.

Finding Initial Conditions (\(t = 0^+\))

At the instant just after a switching event:

Capacitor voltage cannot change instantaneously: \(V_C(0^+) = V_C(0^-)\)
Inductor current cannot change instantaneously: \(I_L(0^+) = I_L(0^-)\)

These are the "memory" of the circuit—the energy stored just before the switch.

Finding Final Conditions (\(t = \infty\))

After a long time (steady state):

Capacitor acts as an open circuit: No current through it
Inductor acts as a short circuit: No voltage across it (just a wire with resistance)

Analyze the circuit with these replacements to find final values.

The Step-by-Step Method

Find initial conditions: Analyze circuit just before switch (capacitor = some voltage, inductor = some current)
Apply continuity: These values persist at \(t = 0^+\)
Find final conditions: Analyze circuit at \(t = \infty\) (capacitor = open, inductor = short)
Calculate time constant: \(\tau = RC\) or \(\tau = L/R\) for the circuit after switching
Apply the formula: \(x(t) = x(\infty) + [x(0) - x(\infty)]e^{-t/\tau}\)

Diagram: Initial and Final Conditions

Initial and Final Conditions MicroSim

Type: microsim Purpose: Practice finding initial and final conditions for various switching scenarios Bloom Level: Apply (L3) Bloom Verb: solve Learning Objective: Students will solve for initial and final conditions given a circuit with a switch, then predict the transient response. Instructional Rationale: Step-by-step guided analysis with visual circuit transformation (C→open, L→short) builds systematic problem-solving skills. Data Visibility Requirements: - Stage 1: Show original circuit with switch position - Stage 2: Analyze circuit for t < 0 to find pre-switch values - Stage 3: Apply continuity rules to get t = 0+ values - Stage 4: Show equivalent circuit at t = ∞ - Stage 5: Display complete solution with formula Canvas layout: - Left (350px): Circuit that transforms to show analysis steps - Right (250px): Solution panel Visual elements: - Circuit with switch (toggleable) - Capacitor/inductor highlighted for condition analysis - Equivalent circuits at t=0+ and t=∞ - Formula with calculated values filled in Interactive controls: - Button: "Step Through Analysis" - Multiple circuit presets to practice - Display: Initial value, final value, τ - Show/hide solution option Example circuits: 1. Simple RC charging from zero 2. RC with initial charge 3. RL energizing 4. Thevenin equivalent with RC Behavior: - Each step highlights relevant circuit portion - Shows capacitor becoming open / inductor becoming short - Formula populates with specific values - Final curve displayed Implementation: p5.js with step-through circuit analysis

Natural and Forced Response

The complete response of a first-order circuit can be viewed as the sum of two components:

Natural Response

The natural response (also called homogeneous or transient response) is what the circuit does on its own, without any input. It's the exponential decay determined by the circuit's time constant:

\(x_n(t) = Ae^{-t/\tau}\)

The natural response always decays to zero. It represents the circuit "forgetting" its initial condition.

Forced Response

The forced response (also called particular or steady-state response) is what the circuit does due to the input forcing function. For a DC source, it's just the final DC value:

\(x_f(t) = x(\infty)\)

Complete Response

The complete response is the sum:

Complete Response

\(x(t) = x_n(t) + x_f(t) = Ae^{-t/\tau} + x(\infty)\)

where:

\(A\) is determined by initial conditions
The natural response decays away
The forced response is what remains

Using initial conditions: \(x(0) = A + x(\infty)\), so \(A = x(0) - x(\infty)\).

This gives us back our universal formula:

\(x(t) = x(\infty) + [x(0) - x(\infty)]e^{-t/\tau}\)

Why This Matters

The natural/forced decomposition becomes essential when inputs are sinusoidal (AC analysis). The forced response to a sinusoid is another sinusoid at the same frequency—that's the AC steady state we'll study soon.

Practical Examples

Example 1: Camera Flash Charging

A camera flash uses a 100 µF capacitor charged to 300V. The charging circuit has 10 kΩ effective resistance.

Time constant: \(\tau = RC = (10 \times 10^3)(100 \times 10^{-6}) = 1\) second

Time to 99% charge: \(5\tau = 5\) seconds

This explains why camera flashes take several seconds to recharge between shots. Want faster recycling? Use lower R (more expensive charger circuit) or smaller C (weaker flash).

Example 2: RC Timing Circuit

You need a time delay of 2 seconds. You have a 10 µF capacitor and want to trigger when voltage reaches 63.2% of final value.

Required τ: 2 seconds (since 63.2% occurs at \(t = \tau\))

Required R: \(R = \tau/C = 2/(10 \times 10^{-6}) = 200\) kΩ

Example 3: Relay Protection

A relay coil has L = 50 mH and R = 100 Ω (coil resistance). When carrying 100 mA, the switch opens.

Without protection: The inductor tries to maintain 100 mA. With nowhere to go, voltage spikes to whatever breaks down first—potentially hundreds of volts.

With flyback diode: Current freewheels through the diode. Time constant: \(\tau = L/R = 50 \times 10^{-3}/100 = 0.5\) ms. Current decays exponentially, dissipating energy safely.

Example 4: Audio Coupling Capacitor

An audio signal must pass through a 0.1 µF coupling capacitor into a 10 kΩ load. What's the low-frequency cutoff?

Time constant: \(\tau = RC = (10 \times 10^3)(0.1 \times 10^{-6}) = 1\) ms

Cutoff frequency: \(f_c = \frac{1}{2\pi\tau} = \frac{1}{2\pi(0.001)} = 159\) Hz

Frequencies below 159 Hz will be attenuated—this might cut into the bass range. For better bass response, increase C or use a higher impedance load.

Diagram: Practical RC/RL Applications

RC/RL Applications MicroSim

Type: microsim Purpose: Explore practical applications of RC and RL timing Bloom Level: Apply (L3) Bloom Verb: design Learning Objective: Students will design RC or RL circuits to achieve specific timing requirements. Canvas layout: - Top: Application selector with circuit schematic - Bottom: Parameter adjustment and timing verification Applications included: 1. Camera flash charging time 2. Simple timer circuit 3. Relay protection 4. Audio coupling capacitor Interactive controls: - Dropdown: Select application - Sliders: Adjust R, C, or L values - Display: Calculated timing parameters - Target input: Enter desired time/frequency Behavior: - Each application shows relevant circuit - Adjusting components shows effect on timing - Can input target and calculate required values - Shows practical component value ranges Implementation: p5.js with application-specific visualizations

The Exponential Response: A Deeper Look

The exponential function \(e^{-t/\tau}\) appears throughout physics and engineering. Here's why it's natural for circuits:

The Differential Equation

For an RC circuit, Kirchhoff's Voltage Law gives:

\(V_s = V_R + V_C = RC\frac{dV_C}{dt} + V_C\)

Rearranging: \(\frac{dV_C}{dt} + \frac{V_C}{\tau} = \frac{V_s}{\tau}\)

This is a first-order linear differential equation. Its solution is exponential because the rate of change is proportional to how far the variable is from its final value.

Think about it: when the capacitor is far from fully charged, current (and thus charging rate) is high. As it approaches full charge, the rate slows. This "rate proportional to remaining distance" relationship always produces exponentials.

Graphical Interpretation

The exponential has a beautiful geometric property: if you draw a tangent line at any point on the curve, it intercepts the final value at exactly one time constant later.

This means:

The initial slope, if maintained, would reach the final value at \(t = \tau\)
At any point, the current rate of change would complete the remaining change in one more τ

Diagram: Exponential Properties

Exponential Properties MicroSim

Type: microsim Purpose: Visualize mathematical properties of the exponential response Bloom Level: Analyze (L4) Bloom Verb: examine Learning Objective: Students will examine the exponential function's properties, including the tangent-line interpretation and the constant-ratio property. Canvas layout: - Main area: Exponential curve with interactive tangent line Visual elements: - Rising or falling exponential curve - Draggable point on curve - Tangent line at that point - Shows intercept at t + τ - Annotations of mathematical properties Interactive controls: - Drag point along curve - Toggle: Rising vs falling exponential - Display: Slope at current point - Display: Value, time to final Behavior: - Tangent line always intercepts final value at t + τ - Moving point shows this property holds everywhere - Displays the derivative relationship Implementation: p5.js with draggable curve analysis

Summary and Key Takeaways

You've mastered the fundamentals of transient analysis. Here's what you now understand:

The Big Picture:

First-order circuits have one energy storage element (C or L)
They respond to changes with exponential transients
The time constant τ determines response speed

Key Formulas:

Formula	Application
\(\tau = RC\)	RC time constant
\(\tau = L/R\)	RL time constant
\(x(t) = x(\infty) + [x(0) - x(\infty)]e^{-t/\tau}\)	Universal step response
\(V_C(t) = V_s(1 - e^{-t/\tau})\)	RC charging
\(V_C(t) = V_0 e^{-t/\tau}\)	RC discharging

Problem-Solving Method:

Find initial conditions (use continuity: \(V_C\) and \(I_L\) can't jump)
Find final conditions (C → open, L → short at DC)
Calculate time constant
Apply the universal formula

Practical Wisdom:

After 5τ, the transient is essentially complete
Larger τ means slower response
Inductors resist current changes (dangerous voltage spikes!)
Always provide current paths for inductors

What's Coming:

In the next chapter, we'll add a second energy storage element and discover something remarkable: circuits that oscillate, ring, and resonate. The world gets more interesting with two components.

Self-Check: Can You Answer These?

An RC circuit has R = 47 kΩ and C = 10 µF. Calculate the time constant. How long until the circuit reaches 95% of its final value?
A capacitor charged to 12V discharges through a 100 kΩ resistor. If C = 1 µF, what is the voltage after 50 ms?
An RL circuit has L = 20 mH and R = 40 Ω. Calculate the time constant. If the final current is 300 mA, what is the current after 0.5 ms?
A circuit switches at t = 0. Before the switch, the capacitor has 8V across it. After a long time, it will have 2V. If τ = 5 ms, write the complete expression for Vc(t).
You need a timing circuit with a 3-second delay (to 63.2%). If you have a 100 µF capacitor, what resistance do you need?