When wiring inductance in series, the total inductance is the simple arithmetic sum of the individual components ($L_{TOTAL} = L_1 + L_2 + ... + L_n$), provided there is zero mutual magnetic coupling between them. While parallel configurations divide current, a series string forces the exact same AC and DC current through every coil. This makes it the mandatory topology for applications like differential mode chokes, multi-stage LC filtering, and current-limiting ballasts.
The Series Inductor Topology: Nodes, Current, and Voltage
To understand how a series inductor string behaves dynamically, we must map the circuit nodes and apply Faraday’s and Kirchhoff’s laws. Consider a basic two-inductor string driven by a voltage source:
- Node A (Input): The connection point between the voltage source (or AC signal) and the first terminal of Inductor 1 (L1).
- Node B (Junction): The electrical tie between the second terminal of L1 and the first terminal of Inductor 2 (L2). This node is critical for measuring intermediate voltage drops.
- Node C (Output/Return): The second terminal of L2, connecting to the load or ground reference.
In a series configuration, the current ($I$) is identical through L1 and L2 at any given instant. However, the voltage across each inductor divides proportionally to its inductance value during transient changes. According to the fundamental inductor equation $V = L(di/dt)$, if L1 is 10mH and L2 is 5mH, L1 will develop twice the voltage drop of L2 during a current ramp. For deep-dive theory on how these voltage transients interact, the Electronics Tutorials guide on inductors in series provides excellent mathematical proofs for transient response.
At high frequencies (above 1MHz), the parasitic parallel capacitance (EPC) of each inductor begins to form a resonant tank. In a series string, these parasitic capacitances combine in series, slightly raising the overall self-resonant frequency (SRF) of the network compared to a single, larger inductor.
Series vs. Parallel: Why Choose Inductance in Series?
Why wire inductors in series instead of parallel? Parallel inductors are rarely used in practical power or signal design because they suffer from severe current-hogging. Due to manufacturing tolerances, the DC Resistance (DCR) of two parallel inductors will never match perfectly. The inductor with the lower DCR will carry a disproportionate share of the DC current, leading to premature thermal saturation. A series string guarantees equal current sharing by definition.
Furthermore, placing inductance in series allows designers to achieve high total inductance values while maintaining a high saturation current ($I_{SAT}$) rating, which is often impossible to find in a single, physically small component.
Failure Mode Contrast: What Breaks at the Extremes?
Understanding how a series string fails is critical for designing protective circuits. Below is the behavior matrix detailing what happens when an element changes state or fails catastrophically.
| Parameter | Adding an Inductor in Series | Shorting One Inductor (e.g., L1) | Opening One Inductor (e.g., L1) |
|---|---|---|---|
| Total Inductance ($L_T$) | Increases ($L_1 + L_2$) | Decreases to remaining $L$ (e.g., $L_2$) | Infinite (Circuit Broken) |
| DC Resistance (DCR) | Increases ($R_1 + R_2$) | Decreases to remaining DCR | Infinite (Open Circuit) |
| Saturation Current ($I_{SAT}$) | Limited by the lowest $I_{SAT}$ in string | Limited by remaining component | Zero Current Flows |
| Node B Voltage | Divides based on $L$ ratio | Drops to ~0V (grounded via short) | Floats to Source Voltage (Node A) |
If L1 shorts internally (a common failure mode for enamel-coated wire when subjected to voltage spikes), the total inductance drops. In an LC filter, this shifts the cutoff frequency higher, potentially allowing destructive high-frequency noise to reach the load. If L1 opens, the circuit halts entirely, and Node B floats up to the source voltage, which can arc across the physical gap of the broken winding if the source voltage is high enough.
Design Walkthrough: Building a 10kHz LC Low-Pass Filter
Let’s apply this topology to a real-world scenario: designing a 10kHz LC low-pass filter to clean up the PWM-driven output of a microcontroller DAC. We need a total inductance of 10mH. Instead of sourcing a single 10mH inductor—which would likely have a high DCR and low saturation current in a through-hole package—we will wire two 5mH inductors in series.
Component Selection:
- L1 & L2: Bourns 78FR50K-RC (5mH radial inductor). Specs: 1.8A $I_{SAT}$, 2.2Ω DCR. Cost: ~$1.15 each. Total $L = 10mH$. Total DCR = 4.4Ω.
- C1: Kemet C315C271K2G5TA (270nF, 200V, C0G/NP0 dielectric). Cost: ~$0.18. (NP0 is critical here to prevent capacitance drift with applied voltage).
The Math:
The cutoff frequency formula is $f_c = \frac{1}{2\pi\sqrt{LC}}$.
Plugging in our values: $f_c = \frac{1}{2\pi\sqrt{0.01 \cdot 270 \times 10^{-9}}} \approx 9.68 \text{ kHz}$.
By splitting the 10mH requirement into two 5mH series components, we keep the physical footprint manageable on a breadboard or perfboard, and we distribute the $I^2R$ heat losses across two separate magnetic cores. For more on calculating LC filter stages for power and signal chains, refer to the All About Circuits chapter on inductor networks.
Breadboard Testing: Step-by-Step Verification
Before committing this filter to a soldered PCB, verify the frequency response on a solderless breadboard. Breadboards introduce roughly 2pF to 5pF of stray capacitance per node, which is negligible for a 10kHz filter but matters at RF.
- Insert the Inductors: Place L1 and L2 into the breadboard. Ensure they are physically separated by at least 1 inch to prevent mutual magnetic coupling (which would alter the total inductance).
- Create Node B: Insert a short jumper wire connecting the adjacent terminal of L1 to the adjacent terminal of L2.
- Add the Capacitor: Connect C1 from the free terminal of L2 (Node C) to the breadboard ground rail.
- Inject the Signal: Connect the BNC-to-alligator clip from your function generator to Node A (free terminal of L1). Set the generator to output a 1Vpp sine wave with a 0.5V DC offset.
- Probe with Oscilloscope: Connect Channel 1 to Node A (input) and Channel 2 to Node C (output). Set both channels to 500mV/div and trigger on Channel 1.
- Sweep and Measure: Sweep the function generator frequency from 1kHz to 100kHz. At 1kHz, Ch1 and Ch2 should be nearly identical in amplitude. As you pass 9.68kHz, Ch2 should begin to attenuate. By 100kHz (two decades up), the signal on Ch2 should be attenuated by roughly -40dB (a factor of 100), confirming a standard 2nd-order low-pass roll-off.
If you are testing this series string with a square wave or DC switching source, always place a flyback diode (e.g., 1N4148) in reverse bias across the entire series string (cathode to Node A, anode to Ground). When the source turns off, the collapsing magnetic field in the 10mH string will generate a massive negative voltage spike ($V = -L \cdot di/dt$) that can destroy your function generator's output stage.
Frequently Asked Questions
Does mutual coupling affect inductance in series?
Yes, drastically. The formula $L_{TOTAL} = L_1 + L_2$ assumes the magnetic flux of one inductor does not intersect the other. If you place two inductors physically close together and their magnetic fields interact, mutual inductance ($M$) comes into play. The total inductance becomes $L_{TOTAL} = L_1 + L_2 \pm 2M$. If the fields aid each other, the total inductance increases; if they oppose, it decreases. In practical PCB layout and breadboarding, always rotate series inductors 90 degrees relative to one another or space them widely to force $M$ to zero.
What happens to the Q factor when wiring inductors in series?
The Quality factor ($Q$) of an inductor at a specific frequency is defined as $Q = \frac{\omega L}{R}$, where $R$ is the DCR. When you place two identical inductors in series, both the total inductance ($2L$) and the total resistance ($2R$) double. Mathematically, the 2s cancel out, meaning the overall Q factor of the series string remains roughly identical to the Q factor of a single component. However, if you mix a high-Q inductor with a low-Q (high DCR) inductor, the high-DCR component will drag down the overall Q of the network, increasing insertion loss in filter applications.
Can I mix different inductor values in a series string?
Electrically, yes. A 2mH inductor in series with an 8mH inductor yields 10mH. However, from a power handling perspective, the entire string is bottlenecked by the weakest link. If the 2mH inductor has a saturation current ($I_{SAT}$) of 500mA, and the 8mH inductor has an $I_{SAT}$ of 2A, the entire series string will saturate at 500mA. Once the smaller inductor saturates, its permeability drops to that of air, its inductance collapses to near zero, and the circuit effectively becomes a single 8mH inductor with a small resistive wire in series. Always match or exceed the $I_{SAT}$ rating of the largest inductor in the string.
