Kirchhoff's Rules

Objectives: To analyze resistive, direct current circuits with Kirchhoff’s rules and Ohm's law.

Objectives: To analyze resistive, direct current circuits with Kirchhoff’s rules and Ohm's law.

Junction Rule The sum of the currents entering any junction �must equal the sum of the currents leaving that junction -A statement of Conservation of Charge�

� Loop Rule The sum of the potential differences across all �the elements around any closed circuit loop must be zero -A statement of Conservation of Energy�

Junction RuleThe sum of the currents entering any junction must equal the sum of the currents leaving that junction.

The algebraic sum of the changes in potential across all of the elements around any closed circuit loop must be zero.

A junction is any point in a circuit where the current has a choice about which way to go. The first rule, also known as the point rule, is a statement of conservation of charge. If current splits at a junction in a circuit, the sum of the currents leaving the junction must be the same as the current entering the junction.

Junction Rule: Σ Iin = Σ Iout

• I1 =I2 + I3• From Conservation ofCharge• Diagram (b) shows a mechanical analog

� In general, the number oftimes the junction rule can be used is one fewer than the number of junction points in the circuit

The second rule, also known as the LOOP RULE, is a statement of conservation of energy. Recall that although charge is not "used up" as current flows through resistors in a circuit, potential is. As current flows through each resistor of a resistive circuit the potential drops. The sum of the potential drops must be the same as the applied potential.

Loop Rule:∑ ΔV = 0

closed loop

• Traveling around the loop froma to b• In (a), the resistor is traversed in the direction of the current, the potential across the resistor is – IR• In (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is + IR

LoopRule,final• In (c), the source of emf is traversed in the direction of the emf (from – to +), and the change in the electric potential is +ε

• In (d), the source of emf is traversed in the direction opposite of the emf (from + to-), and the change in the electric potential is -ε

Loop Equations from Kirchhoff’s Rules

• The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation

• You need as many independent equations as you have unknowns

Resistor-Capacitor Circuits

A Resistor-Capacitor(RC) Circuit

is one where you have a capacitor and resistor in the same

circuit.

Behavior of an RC Circuit

Figure 3.Schematic of an RC circuit. The components in the dotted box are analogous to a square-wave generator with outputs at points and . The switch continuously moves between points and creating a square wave as shown in Figure 4a.

Suppose we connect a battery, with voltage, , across a resistor and capacitor in series as shown by Figure 3. This is commonly known as an RC circuit and is used often in electronic timing circuits. When the switch (S) is moved to position 1, the battery is connected to the circuit and a time-varying current I (t) begins flowing through the circuit as the capacitor charges. When the switch is then moved to position 2, the battery is taken out of the circuit and the capacitor discharges through the resistor. If the switch is moved alternately between positions 1 and 2 , the voltage across points A and B can be plotted and would resemble Figure 4.

Figure 4. A voltage pattern known as a square wave. Moving the switch in Figure 3 alternatively between positions 1 and 2 can produce this voltage pattern. When the switch is in position 1, the input voltage is the peak voltage is Vo . When the switch is moves to position 2 , the input voltage drops to zero. A function generator more commonly produces square-wave voltages.

This voltage pattern is known as a square wave, for obvious reasons, and is commonly produced by a function generator. The function generator is capable of producing voltages that behave

like a sine, square or saw-tooth functions. Additionally, the frequency of the wave may be varied with the function generator.

The dotted-box in Figure 3 may be thought of as a function generator with points A and B as outputs.

We will use a two-channel oscilloscope to monitor the important voltages throughout the experiment. An oscilloscope is an invaluable tool for testing electronic circuits by measuring voltages over time, and Figure 5 shows the schematic for

monitoring an RC circuit with an oscilloscope. As shown in the figure below, the input voltage from the square-wave generator is

monitored by channel one (CH 1) and the voltage across the capacitor is monitored by channel two (CH 2).

Figure 5. The RC circuit diagram. The oscilloscope's Channel 1 monitors the

function generator while Channel 2 monitors the voltage drop across the

capacitor.

The capacitor responds to the square-wave voltage input by going through a process of charging and discharging. It is shown below that during the charging cycle, the voltage across the capacitor . When the switch is in position , the square-wave generator outputs a zero voltage and the capacitor discharges. It can also be shown that during the discharging cycle, the voltage across the capacitor is

Circuit designers must be careful to ensure that the period of the square wave gives sufficient time for the capacitor to fully charge and discharge. It can be shown3 that, as a general rule of thumb, the time necessary for the capacitor of an RC circuit to nearly completely charge to Vo, or discharge to zero, is 4RC .

Here it should be noted that the product RC is known as the time constant,t, and has units of time4. The time constant is the characteristic time of the charging and discharging behavior of an RC circuit and represents the time it takes the current to decrease to of its initial value, whether the capacitor is charging or discharging. Over the period of one t, the voltage across the charging capacitor increases by a factor Conversely the voltage across the discharging capacitor decreases by a factor of over the same period, Put another way, in 1t the voltage across a charging capacitor grows to 63.2% of its maximum voltage,Vo , and in 1t the voltage across a discharging capacitor shrinks to 36.8% of Vo .

Figure 6a. The square wave that drives the RC circuit. When the switch in Figure 3 is in position , the input voltage is the peak voltage is . When the switch is moves to position , the input voltage drops to zero. In this experiment, this input voltage is read by the oscilloscope's CH 1.

Figure 6b. The voltage drop across the capacitor of Figure 3 as read by the oscilloscope's CH 2. The capacitor alternately charges toward and discharges toward zero according to the input voltage shown in Figure 6a. Here, the frequency (and therefore period) of the input square wave voltage is exactly such that the capacitor is allowed to fully charge and discharge. The time constant, , is equivalent to , and is defined by Equations 11 or 14.

The Charging Process

An RC Circuit: Charging

Circuits with resistors and batteries have time-independent solutions: the current doesn't change as time goes by. Adding one or more capacitors changes this. The solution is then time-dependent: the current is a function of time.

Consider a series RC circuit with a battery, resistor, and capacitor in series. The capacitor is initially uncharged, but starts to charge when the switch is closed. Initially the potential difference across the resistor is the battery emf, but that steadily drops (as does the current) as the potential difference across the capacitor increases.

Applying Kirchoff's loop rule:

e - IR - Q/C = 0

As Q increases I decreases, but Q changes because there is a current I. As the current decreases Q changes

more slowly.

I = dQ/dt, so the equation can be written:

e - R (dQ/dt) - Q/C = 0

This is a differential equation that can be solved for Q as a function of time. The solution (derived

in the text) is:

Q(t) = Qo [ 1 - e-t/t ]

where Qo = C e and the time constant t = RC.

Differentiating this expression to get the current as a function of time gives:

I(t) = (Qo/RC) e-t/t = Io e-t/t

where Io = e/R is the maximum current possible in the circuit.

The time constant t = RC determines how quickly the capacitor charges. If RC is small the

capacitor charges quickly; if RC is large the capacitor charges more slowly.

TIME CURRENT0 Io

1*t Io/e = 0.368 Io

2*t Io/e2 = 0.135 Io

3*t Io/e3 = 0.050 Io

The DisCharging

Process

What happens if the capacitor is now fully charged and is then discharged through the resistor? Now the potential difference across the resistor is the capacitor voltage, but that decreases (as does the current) as time goes by.

Applying Kirchoff's loop rule:

-IR - Q/C = 0

I = dQ/dt, so the equation can be written:

R (dQ/dt) = -Q/C

This is a differential equation that can be solved for Q as a function of time. The

solution is:

Q(t) = Qo e-t/t

where Qo is the initial charge on the capacitor and the time constant t = RC.

Differentiating this expression to get the current as a function of time gives:

I(t) = -(Qo/RC) e-t/t = -Io e-t/t

where Io = Qo/RC

Note that, except for the minus sign, this is the same expression for current we had when the capacitor was charging. The

minus sign simply indicates that the charge flows in the opposite direction.

Here the time constant t = RC determines how quickly the capacitor discharges. If RC is small the capacitor discharges quickly; if RC is large the capacitor discharges more

slowly.

SOURCES: physics.wustl.edu/introphys/Phys117_118/Lab_Manual/.../

RC.pdf www.physics.isu.edu/~hackmart/spl2kir.pdf www.phy-astr.gsu.edu/cymbalyuk/Lecture20.pdf www-physics.ucsd.edu/students/courses/

fall2010/.../21B_temp.pdf www.cbooth.staff.shef.ac.uk/phy101E&M/Kirchhoff.pdf bowlesphysics.com/images/AP_Physics_C_-

_RC_Circuits.pdfphysics.wustl.edu/introphys/Phys117_118/Lab_Manual/.../RC.pdf

http://webphysics.davidson.edu/physlet_resources/bu_semester2/c11_RC.html

http://www.clemson.edu/ces/phoenix/labs/223/rc/index.html