Lesson 6: Series-Parallel DC Circuits

Learning Objectives Apply the rules for analyzing series and parallel circuits to a series-parallel circuit. Compute the total resistance in a series-parallel circuit. Analyze series-parallel circuits for current through and voltage across each component. Analyze the power dissipated by each element in a series parallel circuit and calculate the total circuit power.

Series - Parallel Circuits Topology: Branch: Part of a circuit that can be simplified into two terminals (2 nodes). A single element such as voltage source or resistor. Node: Point of connection between two or more branches.

Series - Parallel Circuits To analyze a circuit: You need to be able to identify which elements are in series and which elements are in parallel: (a) RT = R1+(R2||R3||R4)+R5 RT = R1||(R2+(R3||R4))

Reduce and Return Approach Rules for Analysis Same current occurs through all series elements. Same voltage occurs across all parallel elements. KVL and KCL apply for all circuits, whether they are series, parallel, or series-parallel. Redraw complicated circuits showing the source at the left-hand side. Label all nodes. Solve the problem… Reduce and Return Approach

Reduce and Return Approach Series and parallel elements from (a) must be combined to establish the reduced circuit of (b). Then series elements are combined to form the simplest of configurations in (c). The source current can now be determined using Ohm’s law, and we can proceed back through the network as shown in (d). The voltage V2 can be determined and then the original network can be redrawn, as shown in (e). Since V2 is now known, the voltage divider rule can be used to find the desired voltage V4. Because of the similarities between the networks of (a) and (e), and between (b) and (d), the networks drawn during the reduction phase are often used for the return path.

Hints… Develop a strategy: Best to begin analysis with components most distant from the source. Simplify recognizable combinations of components. Determine equivalent resistance (RT). Solve for the total current. Label polarities of voltage drops on all components. Calculate how currents and voltages split between elements in a circuit. Verify your answer by taking a different approach (when feasible).

The Series-Parallel Network Given the series-parallel circuit below, how would you analyze the circuit?

The Series-Parallel Network FIRST: Identify elements in series and/or parallel: R2 (20Ω), R3 (30Ω), and R4 (8Ω) are in parallel. This parallel combination is in series with R1 (2Ω) and R5 (6Ω).

The Series-Parallel Network SECOND: Simplify and redraw the circuit from calculation(s) from step 1.

The Series-Parallel Network Given the series-parallel circuit below, how would you analyze the circuit?

The Series-Parallel Network Step 1: Understand the circuit: In this circuit R3 and R4 are in parallel. Combination is in series with R2. Entire combination is in parallel with R1.

The Series-Parallel Network Step 2: Redraw the circuit from step 1. Now combine the series elements R34 and R2 for the resultant R234.

The Series-Parallel Network Step 3: Redraw the circuit from step 2. Now calculate the parallel resistance from elements R1 and R234 for the resultant RT.

Example Problem 1 Determine the Rbc of this network: Series (100+50 = 150) 150Ω 160Ω 40Ω 37.5Ω Series (37.5+40 = 77.5) 52.2Ω 160Ω 77.5Ω Rbc = 72.2Ω Series (20+52.2 = 72.2)

Example Problem 2 Determine IT, I1, I2, Vad. Substitute the parallel equivalent resistance for resistors R2 and R3 16Ω 24Ω 12V Add R’ to R1 to get RT 40Ω 12V RT First, simplify the circuit (deconstruct): After the circuit is simplified, now solve for IT (from Ohm’s Law): Now solve for I1 and I2 using CDR: Now we can solve for Vad (using Ohm’s Law): You might be wondering what happened here; why doesn’t Vad = E? Don’t forget about the voltage drop that occurs with R1! Notice the use of Req here. Since we want I1 and I2, we need to figure out the fraction of IT flowing through the parallel legs. R1 is not part of the calculations of I1 and I2 except in the sense that it used to verify through KCL. Don’t worry, KVL still holds: E = V1+Vad=7.2V+4.8V = 12V

Again, don’t worry, KVL still holds: Example Problem 3 Determine IT, I1, I2, Vad. 1. Deconstruct the circuit: a) Realize the R3 and R4 are in series thus an equivalent resistance of 30Ω can replace these resistors. R34 = R3 + R4 = 5Ω+25Ω = 30Ω b) Now find the parallel resistance of the R34 and R2 resistors (AKA Req): c) Complete deconstruction by adding the R1 to R2||34 RT = R1 + R2||34 = 2.5Ω+7.5Ω = 10Ω 2. Find IT: 3. Reconstruct the circuit: Again, don’t worry, KVL still holds: E = V1+Vad=6V+18V = 24V a) Find I1 and I2 using CDR: b) Now find Vad using Ohm’s Law:

Solution Steps Determine equivalent resistance RT. Solve for the total current IT. Label polarities of voltage drops on all components. Calculate how currents and voltages split between elements in a circuit. Verify your answer by taking a different approach (when feasible).

Common Mistakes Applying VDR E in the VDR is the voltage across JUST the series elements. Va is the voltage ‘left over’ after the voltage drop across R1 (Va = V2= 40V = Vbd). Rx is the resistor for which you want to determine the voltage drop. RT refers to the combination of all resistors in the circuit (90Ω). Req refers to the combination of the resistors that are in series (below this is R3+R4 = 40Ω). Req’ refers to the combination of the resistors that are in parallel (below that is (R3+R4)//R2 = 30Ω) that you know the total voltage across. =V2 + V2 = 40V _

Power Calculations Again, to calculate the power dissipated by each resistor, use either: V I, I2 R, or V2/R Total power consumed in a Series-Parallel Circuit is the sum of the individual powers: PT = P1+P2+P3+…+PN

Example Problem 4 Determine the voltage drop across the R4 resistor (Vcd) using the VDR. Determine power dissipated by each resistor and verify total power = sum of all power dissipated. To confirm total power (PT) calculated, it is the summation of power consumed in the circuit: PT=P1+P2+P3+…+PN Element power consumed is VI, I2R, or V2/R Verify: PT=P1+P2+P3+P4=106.7+13.33+10+30=160W

Common Mistakes Applying CDR REQ refers to the combination of the resistors that are in parallel that you know the total current through. Not using all impedances in the branch.

Example Problem 5 Determine I2 using the CDR. Find Va, Vbc

Example Problem 6 Determine IT, I1, Va, Vad.

QUESTIONS?