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AC Applications of the Wheatstone Bridge

### Applying AC to the Basic Wheatstone Bridge  When we introduced the Wheatstone Bridge, we showed the input as some sort of dc source, as indicated in the schematic diagram to the left. A galvanometer was used to indicated when the bridge was balanced. However, there is no reason we can't feed the bridge with an ac source, as shown to the right. The bridge itself will behave in the same way.

We do have a requirement to replace the galvanometer with some other indicator or circuit. The galvanometer is a dc device, and will simply vibrate if ac is applied to it. This is not a major problem; if precision is required we can connect the output terminals to an instrumentation amplifier, possibly followed by a precision full-wave rectifier or half-wave rectifier circuit, depending on the requirements of the parameter being measured. If less precision is required, simpler versions of these circuits can be used.

Applying ac to the bridge gives us some additional flexibility: we are no longer limited to resistors and resistive devices as components in the bridge. We can now include capacitors and inductors. This allows us to make the balance of the bridge depend on frequency and/or phase angle as well as impedance. The range of possible applications expands greatly.

The basic equation for the bridge is similar: when the bridge is completely balanced, Z1/Z2 = Z3/Z4. However, there is now an additional element. Each impedance Z may involve any combination of resistance, inductance, and capacitance, and the balanced condition for the bridge includes matching phase shifts between the two halves.

### Measuring Capacitance  The Wheatstone Bridge can be modified in a couple of ways to measure capacitance. The circuit on the left uses a variable capacitance to balance the unknown capacitance, CX. When the bridge is balanced, the relationship R1/XC1 = R2/XCx will hold true. We can simplify this relationship by noting that XC = 1/jωC on both sides of the bridge. Therefore R1(jωC1) = R2(jωCX). But the same signal is applied to both legs of the bridge circuit, so ω is the same (and of course j is also the same) on both sides of the equation. Therefore the expression for balance reduces to: R1C1 = R2CX, or CX = C1(R1/R2).

We can just as easily use a fixed standard capacitor for C1 and make one of the resistors variable. This will be less expensive and will allow us to switch different standard capacitors in and out as C1, to give us reasonable capacitance ranges.

An alternate approach is to use the circuit shown to the right. This gives us a resistive voltage divider and a capacitive voltage divider, but the behavior of the bridge when balanced is the same, and again CX = C1(R1/R2).

Note that both versions of this circuit operate accurately at any frequency, since the same frequency appears on both halves, and therefore cancels out. This is very convenient for ac line-powered equipment, because the needed ac source can be a low-voltage secondary winding on the power transformer, and the circuit will work just as well for 50 Hz European power sources as for 60 Hz American power sources.

### Measuring Inductance Theoretically, we could simply replace the capacitors in the above circuits with inductors, and thus have an equivalent circuit to measure inductances. Unfortunately, three problems appear when we try this:

1. There is always the possibility of some magnetic coupling between the unknown inductance and the reference inductance. This will of course skew the results, and we can never be entirely certain of the extent of coupling.
2. The winding that makes up the inductance will typically have a significant amout of resistance as well. This makes the inductive impedance a complex value, and unless the two inductors have related resistances, the bridge will never balance properly.
3. Standard inductances are hard to make and are correspondingly expensive. Even slight variations in winding placement will have a noticeable effect on the exact inductance of the component, and physical vibration may cause that placement to change slightly.

A more practical way is to use the circuit shown to the right. This circuit is known as the Maxwell-Wien Bridge, or sometimes just as the Maxwell Bridge. Here, the unknown inductance is separated into two separate parts: inductance (LX) and resistance (RX). These will be balanced separately by a variable standard capacitance (C1) and a variable standard resistance (R1).

Because there are two separate components that must be balanced simultaneously, this bridge circuit is a bit more complicated to adjust. However, once the bridge is correctly balanced, the unknown inductance can be computed as:

RX = R2R3/R1
LX = R2R3C1

Keep in mind that component values must be expressed in Ohms, Farads, and Henries, or you can easily run into problems with powers of ten in your results. Also note that R2 and/or R3 can be switched in decade ranges to allow for different orders of magnitude for the unknown inductance.

### Measuring Frequency Unlike the bridge circuits discussed above, the Wien Bridge, shown to the left, is sensitive to the frequency of the ac source. Therefore, it can be used to measure the frequency of that source. Typically it is used only to measure audio frequencies when used this way. This circuit gets its name from its developer, Max Wien.

In the most general terms, when the variable components have been adjusted for complete balance, the following two equations hold true:

 ω² = 1 RXR3CXC3 CX = R2 – R3 C3 R1 RX

It is not necessary for C3 and CX to be the same, although they certainly can be. Likewise, R3 and RX may be the same, but need not be. Regardless of their values, at some frequency ω the reactance of the R3C3 segment will be an integer multiple of the reactance of the RXCX segment. If the ratio of R2/R1 is the same integer, the bridge will be balanced. Then the above equations can be used to determine either CX or ω, assuming the other factor is known.

In most applications, C3 = CX and R3 = RX. In this case, R2 = 2R1, with slight variations to allow for the fact that real-world components aren't quite identical. The bridge still balances at only one frequency, as noted above.

The fact that the Wien Bridge can only be balanced at a single frequency can also be used to set the operating frequency of a circuit designed to generate an ac signal at some desired frequency. The resulting circuit is known as a Wien Bridge oscillator.

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