An electrical circuit that contains a combination of both series and parallel components is known as a series-parallel, or mixed, circuit. These configurations are common in electronics, representing a more complex arrangement than the simplest single-path or branched circuits. Current flowing from a source, such as a battery, will encounter sections where it follows a single path and others where it splits along multiple branches. Determining the overall opposition to current flow, known as the total resistance (\(R_T\)), requires a systematic approach. This methodology involves breaking down the complex circuit into smaller, manageable segments.
Review of Series and Parallel Resistance Calculations
Calculating the total resistance in a mixed circuit relies on the two fundamental rules governing simple series and parallel connections. In a series circuit, resistors are connected end-to-end, forming a single pathway for the current. Since the current passes sequentially through every component, the total opposition is cumulative. The total series resistance (\(R_s\)) is found by adding the individual resistance values: \(R_s = R_1 + R_2 + R_3 + \dots\).
Parallel circuits are significantly different because the current has multiple paths to travel. When resistors are connected in parallel, they share two common connection points. The total resistance (\(R_p\)) is always less than the value of the smallest individual resistor because providing alternate paths effectively increases the overall area for current flow. The general formula for parallel resistance involves the sum of the reciprocals of the individual resistances: \(1/R_p = 1/R_1 + 1/R_2 + 1/R_3 + \dots\).
For the specific case of only two resistors in parallel, the “product-over-sum” rule can be used as a shortcut. This simplified formula states that the total resistance is the product of the two resistances divided by their sum: \(R_p = (R_1 \times R_2) / (R_1 + R_2)\). Both the reciprocal and product-over-sum formulas are applied repeatedly to solve for the total resistance in a mixed circuit.
The Reduction Strategy for Mixed Circuits
The primary method for analyzing a series-parallel circuit is called circuit reduction or simplification, which systematically converts the complex network into a single equivalent resistance. This process involves a sequential application of the series and parallel resistance formulas to isolated segments of the circuit. The overall goal is to reduce the circuit diagram step-by-step until only one resistance value remains.
The initial step is to identify the innermost segments that are purely in series or purely in parallel. These are the smallest groups of resistors that can be clearly isolated as one of the two basic configurations. A purely series segment forces current through each resistor sequentially, while a purely parallel segment offers multiple paths between two distinct connection nodes.
Once a segment is identified, calculate its equivalent resistance, often denoted as \(R_{EQ}\), using the appropriate formula. This calculation uses either the simple sum for a series connection or the reciprocal/product-over-sum formula for a parallel connection. This calculation yields a single resistance value that represents the combined opposition of that entire segment.
The circuit must then be conceptually redrawn, replacing the entire isolated segment with the single \(R_{EQ}\) value just calculated. This simplification reduces the complexity of the circuit, transforming it into a new, simpler mixed circuit. The process repeats: identify the next simplest series or parallel segment, calculate its \(R_{EQ}\), and redraw the circuit. This iterative process continues until the entire network has been reduced to a single equivalent resistor, which is the total resistance (\(R_T\)).
Step-by-Step Example Calculation
To illustrate the reduction strategy, consider a mixed circuit where resistor \(R_1\) is connected in series with a parallel segment consisting of \(R_2\) and \(R_3\). Let the specific values be \(R_1 = 1.00 \Omega\), \(R_2 = 6.00 \Omega\), and \(R_3 = 13.0 \Omega\). The goal is to find the total resistance \(R_T\).
The first step is to isolate and calculate the resistance of the parallel segment (\(R_2\) and \(R_3\)) using the product-over-sum rule. The calculation is \(R_p = (R_2 \times R_3) / (R_2 + R_3)\), which becomes \((6.00 \Omega \times 13.0 \Omega) / (6.00 \Omega + 13.0 \Omega)\).
Completing the arithmetic yields \(R_p = 78.0 \Omega^2 / 19.0 \Omega\). The equivalent resistance for the parallel segment is \(R_p \approx 4.11 \Omega\). This value is now used to conceptually redraw the original circuit as a simple series circuit.
The simplified circuit now consists of the original series resistor, \(R_1\), connected in series with the newly calculated equivalent resistance, \(R_p\). The final step is to apply the series addition rule to these two remaining components: \(R_T = R_1 + R_p\).
Substituting the values, \(R_T = 1.00 \Omega + 4.11 \Omega\), resulting in \(R_T = 5.11 \Omega\). This value, measured in Ohms (\(\Omega\)), represents the circuit’s total opposition to current flow.