R-2R Ladder Lab Activity
Student Name: _____________________________
Student Name: _____________________________
1 CIRCUIT SCHEMATIC
Below is the circuit under analysis:
X
A
+
R2
1k
R1
2k
24V
R4
1k
B
C
R6
1k
R7
1k
R5
2k
R3
2k
-
2 CIRCUIT THEORY / CIRCUIT OPERATION
To understand how this circuit works, one must understand the following: what is an R2R ladder circuit, the need to distinguish between a series connection and a parallel
connection in a series-parallel circuit, and the different formulas necessary to predict the
operation of the circuit. To help apprehend this, a diagram of an R-2R ladder circuit is
placed below:
R-2R Ladder Circuit
R
R
+
2R
2R
Vs
2R
2R
-
To begin with, an R-2R ladder circuit is a series-parallel circuit—which includes both
series and parallel connections of components—that contains only two resistor values:
resistors of a certain value (shown as R in the above diagram) and resistors of twice that
previous value (shown as 2R in the above diagram).
An R-2R circuit is created in such a way that no matter what value for R is used and no
matter what value the voltage source (shown as Vs in the above diagram) is, the total
current in the circuit is divided in half at each junction (point where the circuit splits into
two paths). This phenomenon is explained later on in this section.
It is assumed that current is flowing from positive (+) to negative (-) as per conventional
current flow and that the current is always flowing towards the ground (shown on the
negative side of the voltage source).
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Next, before performing the calculations to predict the behavior of this circuit, one must
comprehend the difference between a series and parallel connection. The key difference
between these two connections is how the components are connected at their terminals.
First off, a series connection is a one in which two components are connected at one
terminal only.
Consequently, a parallel connection is a connection in which two components are
connected at both terminals. Because of this difference, formulas to calculate resistance,
current, and voltage in parallel circuits are different than those for series circuits.
Examples of components in series as well as parallel connections are shown below:
Components in Series
Components in Parallel
An R-2R ladder circuit is a series-parallel circuit, meaning that it includes a combination
of both series and parallel connections in its circuitry although it may be difficult to
perceive at first. To better see the series and parallel connections in a series-parallel
circuit, it is helpful redraw the circuit in question. For example, the R-2R ladder circuit
under analysis in this lab is shown in its regular format below:
R4
1k
R2
1k
+
24V
R6
1k
R5
2k
R3
2k
R1
2k
R7
1k
-
It is quite difficult to actually see the various series and parallel connections in this
circuit. To help one easily identify the series and parallel connections in this circuit, the
redrawn model is shown below:
R6
1k
R4
1k
R2
1k
R3
2k
R1
2k
+
24V
-
2
R5
2k
R7
1k
From this simpler redrawn model, one can easily identify that the only two resistors
directly in series to one another are R6 and R7. That series combination is connected in
parallel with R5. This parallel combination is then connected in series with R4 and so on
and so forth.
Upon differentiating between a series and parallel connection, formulas for deriving the
circuit’s voltage, current, and resistance are used.
Unlike series circuits, analysis of parallel circuits encompasses a totally different
mindset. Although certain aspects, such as the implementation of Ohm’s Law (which can
be used with any type of circuit), are similar, formulas for calculating voltage, current,
and resistance in a parallel circuit are much different from those of a series circuit.
Furthermore, the rules of current and voltage for a parallel circuit are the exact opposite
of those in a series circuit. In a series circuit, voltage is divided proportionally among the
various resistors of a circuit, but in a parallel circuit, it is the current that is divided
proportionally among the resistors. This phenomenon is called the duality principle.
Conversely, although current is the same throughout the entirety of a series circuit, it is
the voltage that is unchanging throughout a parallel circuit. So, with respect to voltage
and current, a parallel circuit is the exact opposite of a series circuit.
The formulas for calculating total resistance in a parallel circuit are shown below. When
looking over these formulas, keep in mind that the total resistance in a series circuit is
found by simply adding up all the resistor values.
Condition
Formula
2 or more parallel resistors
n
1
1
=∑
RT i =1 Ri
2 parallel resistors
Resistors of equal value
RT =
R1 × R2
R1 + R2
RT =
R
n
These formulas are—for the most part—self-explanatory. For 2 or more resistors in
parallel, the reciprocal of the total resistance is equal to the sum of the reciprocals of the
individual resistances in parallel. This, obviously, is more easily understood through the
formula.
The first formula for calculating total resistance can be very tedious and time-consuming.
Thankfully, there is another formula for the case that there are only two parallel resistors.
This formula states that the total resistance (RT) in a parallel circuit with two resistors is
equal to the product of the two resistor values (R1 × R2) divided by the sum of those same
resistor values (R1 + R2)
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In the special case that all resistors in a parallel circuit are equal, a special equal-value
formula can be used. This formula states that the total resistance is equal to the value of
one of the resistors in the parallel circuit (R) divided by the total number of parallel
resistors (n).
It is important to remember that when resistors are connected in parallel, the total
resistance is always less than the smallest resistor in that parallel combination.
Finding the total voltage in a parallel circuit is much easier than finding the resistance.
The voltage throughout a parallel circuit remains the same for all components. As stated
above, this is similar to current in a series circuit. The formula below sums up this point:
VT = Vi
Because parallel circuits divide current proportionally—like voltage in a series circuit—
there is a current-divider rule (CDR) for parallel circuits similar to the voltage-divider
rule (VDR) for series circuits shown on the following page:
I Ri =
I T RT
Ri
This formula states that the current across a certain resistor in parallel (IRi) is equal to the
product of the circuit’s total current (IT) and its total resistance (RT) divided by the
resistor of interest (Ri).
Furthermore, Kirchhoff’s Current Law (KCL)—similar to Kirchhoff’s Voltage Law
(KVL)—states that the sum of all the currents entering a junction is equal to the sum of
all the currents leaving the same junction. Kirchhoff’s Current Law is summarized below:
n
I T = ∑ I Ri
i =1
An example of Kirchhoff’s Current Law is demonstrated in the diagram below:
Iin(A)
Iout(A)
Iout(B)
Iout(C)
Iin(B)
I in ( A) + I in ( B ) = I out ( A) + I out ( B ) + I out (C )
This diagram shows that the two currents entering the junction, Iin(A) and Iin(B), are equal
to the three currents exiting the junction, Iout(A), Iout(B), and Iout(C). This is demonstrated in
the formula shown below the diagram. Note that the arrow in the diagram indicates the
direction in which the current is flowing.
Finally, if two of the following is known of a series-parallel circuit: total current, total
voltage, or total resistance, one can use Ohm’s Law to find out the missing third variable.
Although the above formulas can only be used for a parallel circuit, Ohm’s Law can be
used for either a series or parallel circuit. The formulas for Ohm’s Law are summarized
below:
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1. Voltage : V = IR [Volts → V ]
V
2. Current : I =
[ Amps → A]
R
V
3. Resistance : R =
[Ohms → Ω]
I
With Ohm’s Law, it is imperative to remember that current is electron flow, voltage is
force that causes electron flow, and resistance is the force that opposed electron flow.
Ohm’s Law demonstrates the relationship between voltage, current, and resistance.
Although these ideas seem very simple, they are fundamental principles that are very
important for this lab as well as many other labs.
3 EXPERIMENT STEPS
The following steps are performed in order to execute this experiment:
1) Calculate the total resistance for this R-2R ladder circuit.
2) Using PSPICE, verify the total resistance found in step 1.
3) Calculations necessary to predict circuit behavior are to be performed.
Calculations are performed so to find the voltage and current at points A, B, C.
4) The designated circuit is simulated using PSPICE.
5) Compare the calculated values with the simulation results using a table format.
6) Build the circuit on a protoboard.
7) Measure the voltage and current at points A, B, C.
8) Compare the measured values to the calculated values and the simulation results
using a table format.
9) Based on your observations, what are the voltage and current values at points A,
B, C if the 24 volts supply is lowered to 12 volts?
10) Based on your observations, what are the voltage and current values at points A,
B, C if the R-2R ladder is modified with R = 2k Ω?
Show your instructor the result for each step.
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