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When the contact resistance in a power connection goes up, how does it affect the power dissipation in the circuit?
June 25, 2007
The full text of this question, comes from Jack Smith, managing editor of Plant Engineering : “On multiple occasions, industry experts have told me that the reason a marginal connection in an ac power circuit overheats is that the slight increase in resistance somehow causes the current through the connection to rise exponentially. Is this true and, if so, how does this effect work? If it doesn’t, why does the connection get hot?”
The effect—a marginal increase in the contact resistance in a power circuit causing the connection to heat up—is real. The explanation is wrong. The current does not rise at all, let alone exponentially. It, in fact, must fall. To see why, look at a simple power-delivery circuit made up of a connection with a contact resistance RC in series with a resistive load RL.
Generally, power-delivery circuits are constant-voltage sources. They are, therefore, low impedance sources, which is why I haven’t included a source impedance circuit element. The analysis, however, doesn’t change when the source impedance is included.
The power delivery circuit includes a fixed load resistance and a variable connector contact resistance.
The explanation says that the current will increase exponentially with increasing RC when RC << RL. To see if this is true, we solve Ohm’s law for the current and differentiate with respect to contact resistance:
V = I (RC + RL)
I = V (RC + RL) -1
dI / dRC = -V (RC + RL) -2
Since the voltage and both resistances are positive-definite quantities, the derivative is always negative. Thus, the current always decreases with increasing contact resistance.
If the current always decreases, why does the connection get hot?
To see that effect, remember that we are only concerned with the power dissipated in the connection:
PC = I 2 RC
PC = V 2 RC (RC + RL) -2
Differentiating with respect to RC gives:
dPC / dRC = V 2 (RC + RL) -2 [1 – 2 RC (RC + RL) -1]
Using the assumption that RC << RL finally gives:
dPC / dRC = (V / RL) 2,
which is both positive definite and nonlinear. It is, in fact, the square of what the current would be without contact resistance.
So, in the end, the reason that the connection heats up is that while the total power dissipated remains virtually the same (because the connector resistance is small compared to the total resistance), the power dissipated in the connector rises as the square of the load current.
Posted by on June 25, 2007 | Comments (0)