The Maxwell bridge works by balancing an unknown inductance against known values of resistance and capacitance to determine its value. It is a specific modification to a Wheatstone bridge circuit.
Understanding the Principle
At its core, the Maxwell bridge operates on the principle of null detection. Like the Wheatstone bridge used for measuring unknown resistance, the Maxwell bridge sets up four "arms" or impedances in a bridge configuration.
- Configuration: The bridge consists of four impedances connected in a loop. A voltage source (often AC for inductance measurements) is connected across two opposite junctions, and a detector (like a galvanometer or AC voltmeter) is connected across the other two junctions.
- Balancing: Known components (resistors, capacitors) are placed in some arms, and the unknown inductance (often represented as an inductor with its series resistance) is placed in another arm.
- Measurement: The values of the known components are adjusted until the detector indicates zero current or voltage difference between its two connection points. This condition is called "balance."
- Calculation: When the bridge is balanced, the ratios of the impedances in the arms are equal. This allows the unknown inductance and its associated resistance to be calculated based on the known, calibrated values.
The Maxwell Bridge Configuration
According to the reference, a Maxwell bridge is specifically defined by the types of calibrated components used:
A Maxwell bridge is a modification to a Wheatstone bridge used to measure an unknown inductance (usually of low Q value) in terms of calibrated resistance and inductance or resistance and capacitance. When the calibrated components are a parallel resistor and capacitor, the bridge is known as a Maxwell bridge.
This highlights two possibilities mentioned in the general definition, but the specific term "Maxwell bridge" often refers to the configuration using calibrated resistance and capacitance. In this common arrangement:
- One arm contains the unknown inductor ($L_x$) in series with its intrinsic resistance ($R_x$).
- A second arm contains a known, variable resistance ($R_2$).
- A third arm contains a known, fixed resistance ($R_3$).
- The fourth arm contains a known, variable resistor ($R_4$) placed in parallel with a known, variable capacitor ($C_4$).
How Balancing Determines the Unknown Inductance
When the bridge is balanced, the following impedance relationship holds:
$Z_1 / Z_2 = Z_3 / Z_4$
Where:
- $Z_1$ = Impedance of the unknown arm ($R_x + j\omega L_x$)
- $Z_2$ = Impedance of the known resistor ($R_2$)
- $Z_3$ = Impedance of the known resistor ($R_3$)
- $Z_4$ = Impedance of the parallel resistor and capacitor ($R_4 || C_4$). The impedance of $Z_4$ is calculated as $\frac{R_4 \times (1/(j\omega C_4))}{R_4 + (1/(j\omega C_4))} = \frac{R_4}{1 + j\omega R_4 C_4}$.
Substituting these into the balance equation and separating the real and imaginary parts allows you to solve for $R_x$ and $L_x$ in terms of the known values ($R_2$, $R_3$, $R_4$, $C_4$).
The balance equations derived are typically:
- $R_x = \frac{R_2 R_3}{R_4}$
- $L_x = R_2 R_3 C_4$
By adjusting $R_4$ and $C_4$ (or $R_2$ in some configurations) until the bridge is balanced (zero signal on the detector), the values of $R_x$ and $L_x$ can be directly calculated from the settings of the known components.
Suitability for Low Q Inductors
The reference specifically notes that the Maxwell bridge is used to measure an unknown inductance (usually of low Q value). The Q factor (Quality Factor) of an inductor is the ratio of its reactance to its resistance ($Q = \omega L / R$). Low Q means the inductor has a significant series resistance compared to its inductive reactance at the operating frequency.
The Maxwell bridge's configuration, particularly the parallel R-C arm, makes it well-suited for accurately measuring inductors with low Q values because the balance equations derived naturally separate the resistive and reactive components. Other bridge types might struggle with the interaction between the resistive and inductive elements in low-Q components.
In summary, the Maxwell bridge works by setting up a balance condition between an unknown inductance (with its series resistance) and calibrated components (specifically a parallel resistor and capacitor in one arm). By adjusting these known components until the bridge is balanced, the unknown inductance and resistance can be precisely determined through simple mathematical relationships. This method is particularly effective for measuring inductors with relatively low Q factors.
