Use this Series Parallel Circuit Calculator to quickly determine the total resistance, total current, and total power dissipation for a simple series-parallel circuit consisting of a series resistor ($R_S$) and two parallel resistors ($R_{P1}$ and $R_{P2}$) connected to a DC Voltage source ($V$).
Series Parallel Circuit Calculator
Total Circuit Resistance ($R_{Total}$):
- Enter values and click “Calculate” to see the steps.
Series Parallel Circuit Calculator Formula
The total resistance ($R_{Total}$) for a simple series-parallel circuit (one series resistor $R_S$ connected to two parallel resistors $R_{P1}$ and $R_{P2}$) is calculated in two steps:
Step 1: Calculate Parallel Resistance ($R_{Parallel}$)
$$R_{Parallel} = \frac{1}{\frac{1}{R_{P1}} + \frac{1}{R_{P2}}}$$
Step 2: Calculate Total Resistance ($R_{Total}$)
$$R_{Total} = R_S + R_{Parallel}$$
Step 3: Calculate Total Current ($I_{Total}$) and Power ($P_{Total}$)
$$I_{Total} = \frac{V_{Total}}{R_{Total}}$$
$$P_{Total} = V_{Total} \cdot I_{Total}$$
Formula Sources: All About Circuits, Wikipedia (Circuit Theory)
Variables
- $V_{Total}$ (Volts): The total electromotive force (voltage) supplied by the source (e.g., battery).
- $R_S$ (Ohms): The resistance value of the resistor connected in series with the parallel group.
- $R_{P1}$ (Ohms): The resistance value of the first resistor in the parallel branch.
- $R_{P2}$ (Ohms): The resistance value of the second resistor in the parallel branch.
- $R_{Total}$ (Ohms): The equivalent resistance of the entire circuit.
- $I_{Total}$ (Amperes): The total current flowing from the voltage source.
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What is a Series-Parallel Circuit?
A series-parallel circuit is a complex circuit that combines both series and parallel components. In a series connection, all components are connected end-to-end, forming a single path for current flow, meaning the current is the same through each component, but the voltage drops across each. In a parallel connection, components are connected across the same two points, creating multiple paths for current flow, meaning the voltage is the same across each branch, but the total current is the sum of branch currents.
These hybrid circuits are common in real-world applications, from household wiring to complex electronic devices. To analyze them, one must systematically reduce the complex circuit into simpler, equivalent circuits. The parallel portions are solved first to find their equivalent resistance, and this equivalent is then treated as a single series component with the remaining series resistors.
How to Calculate a Series-Parallel Circuit (Example)
Let’s use the calculator’s setup: $V=12V$, $R_S=10\Omega$, $R_{P1}=20\Omega$, and $R_{P2}=30\Omega$.
- Find the equivalent resistance of the parallel branch ($R_{Parallel}$): $$\frac{1}{R_{Parallel}} = \frac{1}{20\Omega} + \frac{1}{30\Omega} = \frac{3+2}{60} = \frac{5}{60}$$ $$R_{Parallel} = \frac{60}{5} = 12\Omega$$
- Find the total circuit resistance ($R_{Total}$): $$R_{Total} = R_S + R_{Parallel} = 10\Omega + 12\Omega = 22\Omega$$
- Find the total circuit current ($I_{Total}$): $$I_{Total} = \frac{V}{R_{Total}} = \frac{12V}{22\Omega} \approx 0.545\text{ Amperes}$$
- Find the total circuit power ($P_{Total}$): $$P_{Total} = V \cdot I_{Total} = 12V \cdot 0.545A \approx 6.545\text{ Watts}$$
Frequently Asked Questions (FAQ)
Why is $R_{Total}$ always greater than $R_S$ in this type of circuit?
Because the total resistance is the sum of the series resistor ($R_S$) and the equivalent parallel resistance ($R_{Parallel}$). Since all individual resistances must be positive, adding $R_{Parallel}$ (which is also positive) to $R_S$ will always result in a larger total resistance.
What happens if one of the parallel resistors is zero?
If a resistor has $0\Omega$ (a short circuit), the total parallel resistance ($R_{Parallel}$) becomes $0\Omega$. This effectively shorts out the parallel branch. The $R_{Total}$ would then only be $R_S$. The current ($I_{Total}$) would be very high ($V/R_S$), potentially causing damage. The calculator prevents $0$ or negative inputs to avoid non-physical results.
Can this calculator handle more than two parallel resistors?
This version is structured for two parallel resistors ($R_{P1}$ and $R_{P2}$). To handle $N$ parallel resistors, the parallel resistance formula would extend: $$\frac{1}{R_{Parallel}} = \frac{1}{R_{P1}} + \frac{1}{R_{P2}} + \dots + \frac{1}{R_{PN}}$$
How does Kirchhoff’s laws apply to this circuit?
Kirchhoff’s Current Law (KCL) is used for the parallel junction (current entering equals current leaving). Kirchhoff’s Voltage Law (KVL) is used for the series loop (the sum of voltage drops around the loop equals the total source voltage).