Moving Coil MetersThe design of a voltmeter, ammeter or ohmmeter begins with a currentsensitive element. Though most modern meters have solid state digital readouts, the physics is more readily demonstrated with a moving coil current detector called a galvanometer. Since the modifications of the current sensor are compact, it is practical to have all three functions in a single instrument with multiple ranges of sensitivity. Schematically, a single range "multimeter" might be designed as illustrated.

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VoltmeterA voltmeter measures the change in voltage between two points in an electric circuit and therefore must be connected in parallel with the portion of the circuit on which the measurement is made. By contrast, an ammeter must be connected in series. In analogy with a water circuit, a voltmeter is like a meter designed to measure pressure difference. It is necessary for the voltmeter to have a very high resistance so that it does not have an appreciable affect on the current or voltage associated with the measured circuit. Modern solidstate meters have digital readouts, but the principles of operation can be better appreciated by examining the older moving coil meters based on galvanometer sensors.

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AmmeterAn ammeter is an instrument for measuring the electric current in amperes in a branch of an electric circuit. It must be placed in series with the measured branch, and must have very low resistance to avoid significant alteration of the current it is to measure. By contrast, an voltmeter must be connected in parallel. The analogy with an inline flowmeter in a water circuit can help visualize why an ammeter must have a low resistance, and why connecting an ammeter in parallel can damage the meter. Modern solidstate meters have digital readouts, but the principles of operation can be better appreciated by examining the older moving coil meters based on galvanometer sensors.

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OhmmeterThe standard way to measure resistance in ohms is to supply a constant voltage to the resistance and measure the current through it. That current is of course inversely proportional to the resistance according to Ohm's law, so that you have a nonlinear scale. The current registered by the current sensing element is proportional to 1/R, so that a large current implies a small resistance. Modern solidstate meters have digital readouts, but the principles of operation can be better appreciated by examining the older moving coil meters based on galvanometer sensors.

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Voltmeter/Ammeter MeasurementsThe value of electrical resistance associated with a circuit element or appliance can be determined by measuring the voltage across it with a voltmeter and the current through it with an ammeter and then dividing the measured voltage by the current. This is an application of Ohm's law, but this method works even for nonohmic resistances where the resistance might depend upon the current. At least in those cases it gives you the effective resistance in ohms under that specific combination of voltage and current. 
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