This signal conditional circuit outputs the sum of five 76Ohm output sensors in the 10mV range and the voltage gains are 10,15,20,25,30. Could someone please explain to me why do we need a resistor at the non-inverting terminal of the summer amplifier and why do we need an inverting amplifier connected to the summing amplifier? I think it is because the output voltage of the summer opamp is negative but I do not know if that's the reason to add an inverting opamp to the circuit.
Could someone please explain to me why do we need a resistor at the non-inverting terminal of the summer amplifier
The resistors R7 and R10 are there to remove a voltage offset at the opamp input terminals due to the 'common mode input bias current.' This is a dc current drawn (equally) by both terminals. The resistor is chosen to have the same value as the equivalent resistance seen by the inverting input terminal (R1-R6 of the summing amplifier, in this example). By doing so, an equivalent current drawn by the input terminals leads to the same voltage drop across the input resistor networks for each terminal, eliminating a voltage offset at the input terminals that would otherwise result.
In many cases where the opamp has very low input bias currents, and where high precision is not required of the circuit, this resistor is not necessary, and the input terminal could be directly connected to ground.
and why do we need an inverting amplifier connected to the summing amplifier?
The output of the summing amplifier has the opposite polarity as the weighted sum of the input voltages. If the rest of the circuit that uses the weighted-sum voltage requires it to have the same polarity as the weighted sum voltage, then the inverting amplifier restores the polarity so that the output is not negated. Sometimes the rest of the circuit could be designed to use the negated voltage instead, in which case it would not be necessary to have the inverting amplifier follow the summer.
A frequent usage scenario would be to use the weighted sum voltage as input to an ADC that only accepts positive voltages, which would motivate using this inverting amplifier so that the output of this sub-circuit has positive polarity for a set of positive input voltages.
Op amps are not always ideal op amps, with infinite input impedance.
The resistor in the (+) input will, with the op amp input's bias current, develop a small voltage drop to ground. That bias current, and thus the small voltage drop, is highly temperature-dependent. The series resistor compensates for the (-) input bias current and input resistance of the sources' output impedances and summing resistors, and feedback resistor.
In short, it's compensation for thermal effects due to the op amp's input bias currents. This is usually not needed for FET-input ampifiers, but a resistive imbalance in (for instance) a venerable uA741 amplifier will dominate the voltage offset errors at about 4 kOhm.
As for the inverting amplifier, it might be for gain, or simply to ensure that the output signal has the same polarity as the input. The inverting input summing junction amplifier has near-zero crosstalk between the multiple input channels, but always inverts the sense of the inputs.