3.2: RC Circuits

Learning Objectives

• Qualitatively describe and plot the voltage vs. time and current vs. time for elements in an RC circuit that is charging and discharging.
• Set up and solve a differential equation for charge vs. time in an RC circuit that is either charging or discharging.
• Derive equations for voltage and current as functions of time for different circuit elements in charging and discharging RC circuits.

Model

A capacitor is initially uncharged and connected to a resistor and a voltage source as shown in the diagram. The graph for potential difference across each circuit element is partially filled in. The origin is time \(t_0\) and each subsequent “tick” on the x-axis represents times \(t_1\), \(t_2\), \(t_3\), \(t_4\). The graph of current vs. time below is for the current through the resistor at those same clock-times.

Explore

1. Construct a voltage equation using the Kirchhoff Voltage Rule for this circuit and put (+) / (−) above each \(\Delta V\) term.
2. Use the equation to solve for the voltage drop across the resistor \((\Delta V_R)\) at \(t_0\) and \(t_1\).

   At \(t_0\), \(\Delta V_R=\)

   At \(t_1\), \(\Delta V_R=\)

3. Plot those voltage drop values for the resistor on the graph.

Critical Thinking & Concept Invention

1. How should the current through the resistor compare (\(>\), \(<\), \(=\)) at times \(t_0\) and \(t_1\)?

   \(I_0\) ______ \(I_1\)

2. Based on your answer to the previous question, how should the amount of charge that is added to the capacitor from \(t_1\) to \(t_2\) compare (\(>\), \(<\), \(=\)) to the amount of charge added to the capacitor from \(t_0\) to \(t_1\)?

   \(\Delta Q_{1\rightarrow2}\) ______ \(\Delta Q_{0\rightarrow1}\)

3. Based on your answer to the previous question, discuss how much the voltage drop across the capacitor should change compared to the previous two values. Plot an approximate value for the voltage drop across the capacitor for time \(t_2\), then plot the voltage drop for the resistor at \(t_2\).
4. Plot the voltage difference for the voltage source, resistor, and capacitor for \(t_3\) and \(t_4\).
5. Use the voltage drops you plotted for the resistor to qualitatively plot the current through the resistor at each clock-time.
6. Describe the way voltage changes or does not change for each circuit element while a capacitor is charging. Explain why this occurs.
7. Describe the way current changes or does not change through a circuit while a capacitor is charging. Explain why this occurs.

Apply

A capacitor is initially charged to a voltage of \(\Delta V_C=\varepsilon\) and is connected to a resistor as shown in the diagram. Construct a voltage equation, then qualitatively plot the potential difference across the capacitor and resistor as time goes on, then sketch a curve through the points for each graph.

Model

Explore

1. Draw a Kirchhoff Path in the direction of the current.
2. Construct a Voltage Equation using Kirchhoff’s Voltage Rule for the circuit and put a (+) or (−) sign over each \(\Delta V\) term.
3. Substitute into the equation the equations for \(\Delta V_R\) and \(\Delta V_C\) with appropriate sign (+/−) in front of them.

Critical Thinking & Concept Invention

1. Discuss as a group: Does the current represent the rate charge is entering or leaving the capacitor?
2. If \(Q\) represents the charge of the capacitor, which of the following should be used for \(I\)? Circle one.

\(I=\frac{dQ}{dt}\)

OR

\(I=-\frac{dQ}{dt}\)

3. Solve the voltage equation:
   • Rewrite your voltage equation with your selection above in place of \(I\).
   • Separate Variables: Perform algebra necessary to group the charge quantities (\(Q\) and \(dQ\)) on one side of the equation and the time quantities (\(t\) and/or \(dt\)) on the other side. If there is a minus sign, place it on the time side.
   • Integrate both sides (\(Q_0\) to \(Q\) and \(0\) to \(t\)).
   • Perform necessary algebra and logarithmic identities to get an equation for:

   \(Q=\)

Summary

The charge on a discharging capacitor is an exponential decay that depends on the initial charge (\(Q_0\)) and the values for the resistance (\(R\)) and capacitance (\(C\)) of the circuit.

The product \(RC\) is called the time constant (\(\tau\)) for the capacitive circuit.

\(Q(t)=\)

\(\tau=RC\)

Graphs of charge, voltage, or current for capacitive circuits are typically scaled so that the x-axis is integer multiples of the time constant.

Model

Explore

1. Draw a Kirchhoff Path in the direction of the current.
2. Construct a Voltage Equation using Kirchhoff’s Voltage Rule for the circuit and put a (+) or (−) sign over each \(\Delta V\) term.
3. Substitute into the equation the equations for \(\Delta V_{source}\), \(\Delta V_R\), and \(\Delta V_C\) with appropriate sign (+/−) in front of them.

Critical Thinking & Concept Invention

1. Discuss as a group: Does the current represent the rate charge is entering or leaving the capacitor?
2. If \(Q\) represents the charge of the capacitor, which of the following should be used for \(I\)? Circle one.

\(I=\frac{dQ}{dt}\)

OR

\(I=-\frac{dQ}{dt}\)

3. Solve the voltage equation:
   • Rewrite your voltage equation with your selection above in place of \(I\).
   • Separate Variables: Perform algebra necessary to group the charge quantities (\(Q\) and \(dQ\)) on one side of the equation and the time quantities (\(t\) and/or \(dt\)) on the other side. If there is a minus sign, place it on the time side.
   • Integrate both sides (\(Q_0\) to \(Q\) and \(0\) to \(t\)).
   • Perform necessary algebra and logarithmic identities to get an equation for:

   \(Q=\)

   • Factor out any common factors from terms.
4. Remind each other how the current relates to charge and perform the necessary calculus operation to your \(Q(t)\) equation to generate an \(I(t)\) equation.
5. Generate a \(\Delta V(t)\) equation for both the resistor and capacitor using your equations for \(I(t)\) and \(Q(t)\).

\(\Delta V_R(t)=\)

\(\Delta V_C(t)=\)

6. Put all of your \(\Delta V(t)\) terms back into your original voltage equation to check your results. Does Kirchhoff’s Voltage Rule still hold true?

Summary

Circuits with capacitors and inductors are not steady circuits in that current and voltages change with time throughout the circuit.

Many textbooks use lower-case variables (\(i\), \(q\), \(v\)) for time-dependent current, charge, and voltage. This document uses the same variable symbols with explicit time dependence shown using \((t)\), but both conventions are considered correct.

Kirchhoff’s Voltage Rule (and an extension of it) is still true for each moment in time, even though the voltages and current might change with time.

We solve these circuits by setting up the necessary voltage and current equations, then solving the differential equation by relating current and charge.

Memorizing equations can be a bad idea when dealing with complicated circuits that vary with time. It is always best to start with the fundamental concepts:

• Kirchhoff’s Voltage Rule
• Kirchhoff’s Current Rule

Then generate specific equations for the circuit you are analyzing.