Devices Experiment 1- Inverting and Non-Inverting

University of Technology, Jamaica Faculty of Engineering and Computing

Laboratory Report

Title of Experiment: Non-Inverting , Inverting and Summing Amplifier Experiment #

:1

Instructor

: Mr. Thorpe

Course Name: Electronic Devices and Circuits 2 Programme : BENG Electrical Submitted by: Yanique Gibbs ID No.

: 1206283

Aim: To investigate the operations of an Operational Amplifier through used in configurations of inverting , non-inverting and summing .

Theory: An op-amp is a high-gain direct-coupled amplifier that is often powered by both a positive and a negative supply voltage. This allows the output voltage to swing both above and below ground.

Figure a. Op-amp Note the two inputs label “+” and “-” ; the positive input is called the non-inverting and the negative input the inverting .In many applications , one of the amplifier inputs is grounded , so vo is out of phase with the input if the signal is connected to the non-inverting terminal, and vo is out of phase with the input if the signal is connected to the inverting input. When a small input voltage is applied , this will cause the amplifier to be driven all the way to its extreme positive and negative voltage limits . However resistors are connected to and around the amplifier in such a way that the signal undergoes vastly smaller amplification. The resistors cause gain reduction through signal feedback. Circuits using op-amps and resistors can be configured to perform many useful operations such as inverting , non-inverting and summing .

Figure b. Inverting Amplifier In this very useful application of an op-amp ,the non-inverting input is grounded , Vin , is connected through Rin to the inverting input and feedback resistor Rf is connected between the output and V2. Kirchoff’s node equation at V1 yields V1/ R = 0 Kirchoff’s node equation at V2 yields Vin – V2 / Rin + Vin – V2 / Rf = 0 V1 = V2 yields V1 = V2 = 0 Therefore the closed-loop gain as Vo/Vin = - Rf / Rin The negative sign in the equation indicates an inversion of the output signal with respect to the input as it is 180o out of phase. This is due to the feedback being negative in value.

Figure c. Non-Inverting op-amp Figure c shows another useful application of an op-amp called the non-inverting configuration. The signal Vin is connected directly to the non-inverting input and the resistor R2 is connected from the inverting input to ground. Under the ideal assumption of infinite input impedance , no current flows into the inverting input so Iin =If. Thus , V1 / R2 = Vo – V1/ Rf V0 = A(Vin – V1) V1= Vin – Vo/ A Letting A = infinite , the term Vo/ A foes to zero Thus , V1= Vin Vin / R2 = Vo – Vin / Rf Solving for Vo / Vin / Rf Vo / Vin = 1 + Rf / R2

Figure d. Summing op-amp It is possible to sum several input voltages in one op-amp circuit called a summing amplifier . Figure d shows and inverting op-amp circuit that can be used to sum and scale three input signals. Note that the signals V1 , V2 and V3 are applied through separate resistors R1 , R2 and R3 to the summing junction of the amplifier and that there is a signal feedback resistor Rf . Following the same procedure we used to derive the output of the inverting amplifier having a single input , we obtain for the three-input op-amp. If = I1 + I2 + I3 = - [ V1/Rin + V2/Rin + V3/Rin ] Inverting equation Vo = Rf/Rin * Vin Then Vo = - Rf [ V1/Rin + V2/Rin + V3/Rin ]

Equipment: Solderless Breadboard Cables & Wires 741 C Operational Amplifier Several ¼ - Watt Resistors Decade Capacitor Box (0-10 4-silicon diodes µF ) 9V Battery

Diagrams: Refer to lab manual (Figures 9.5, 9.6 and 9.7). Method: As stated in lab manual (Methods 9.5.1, 9.5.2 and 9.5.3). Results: Table 1 . Showing results for the Non-Inverting Amplifier Ri = 9.73 kΩ

Vi = 0.4 V

Vi (rms) = .141 V

Rf = 32.6 kΩ

Vo = 1.75V

Vo(rms) = 1.23 V

Input and Output voltage in phase

Period (T) = 1ms

Frequency = 1KHz

When Rf is short-circuit Vo equals to Vin. 0.4V = 0.4 V

Diagram 1. Showing Output and Input waveforms in-phase.

When function generator supplies 10 kHz @ 4V peak-to-peak Ri = 9.73 kΩ

Vi = 4 V

Rf = 99.5 kΩ

Vo = 15.5V

Gain = 11.23

% Error in gain = 2.04 %

Slope = 96.96 x 10-6 V/s

Table 2. Showing the results when the gain is 2 (simulated): R1/ kΩ 32.5

Rf/ kΩ 32.6

Vi/v .4

Vi/v(Rms) .141

Vo/v 0.8

Vo/v(Rms) .2828

Vi/v(Rms) 1.41

Vo/v 15.5

Vo/v(Rms) 10.96

Table 3. Showing the results when the gain is 11: R1/ kΩ Rf/ kΩ 9.73 99.5 Frequency= 10kH

Vi/v 4

Slew rate 96.96 x10-6 V/µs: (rise/run) Rise =4.15

Diagram 2. Showing a 4V peak to peak Square wave Input Waveform of Circuit in Figure 9.5 with at 10kHz

Diagram 3. Showing the output and the effect of the slew rate.

Table 4. Showing results for the Inverting Amplifier Gain of -10

Gain of -2.2

Gain of -1

Ri

.986 kΩ

.986 kΩ

.986 kΩ

Rf

9.72 kΩ

2.2 kΩ

.986 kΩ

Vi

0.4 V

0.4 V

0.4 V

Vi (rms)

.141 V

.141 V

.141 V

Vo

3.94 V

845 mV

382 mV

Vo (rms)

1.39 V

0.299 V

0.135 V

Uncertainty in Gain

1.4 %

1.36 %

0%

Diagram 4. Showing the input and output waveforms produced from the inverting op-amp in order to obtain a voltage gain of -1

Diagram 5. Showing the input and output waveforms produced from the inverting amplifier having a voltage gain of -2.2 and is out of phase by 180 o.

Table 5. Showing results for Multiple Inputs

the Inverting Amplifier with Inverting Amplifier with Multiple Inputs 1.2 kΩ R1 R2

17.80 kΩ

Rf

3.9 kΩ

V1

0.4 V

V2

9.08 V

Vo

1.6V

V0 (rms)

0.5656

Diagram 4

Calculation and Uncertainties: Sample Calculation for a non-inverting amplifier ( (

) )

Sample Calculation for a inverting amplifier Vo = - (Rf / R) * Vin Vo = - (9.72 kΩ / .986 kΩ ) * 0.4 Vo = 3.94 V

Sample Calculation for a inverting amplifier with multiple inputs. VO1= -(Rf/R1)*Vin → -(3.9kΩ/1.2kΩ)*.4v → 1.3v VO2= -(Rf/R2)*Vin → -(3.9kΩ/17.80kΩ)*.4v → .0876v Vo= VO1 + VO2 Vo = 1.30876V Theoretical Gain Vo = 1.6 V % Error in Vo = (1.6 – 1.30876)*100%/ 1.6 = 18.20%

Analysis of Results: 1. The derived theoretical input-output relationship for a non-inverting amplifier. Ii = If Vi / Ri = (Vo - V- )

/

Rf

But Vo = (V+ - V- ) A Hence Vi / Ri = (Vi

+

- Vi- ) A – V- / Rf

Vo = A (V+ - Vi )

Vi = Vi+ - Vo /A Vi = Vi+ Since Vi- = Vi+ , then we can replace Vi- with Vi + . Hence Vi- /R can be written as Vi+ / R. Hence Vi +/ Ri = (Vo - Vi+) / Rf Vi+ * Rf = (Vo – Vi+) / R1 Vi+ * Rf = Vo

*

R1 – Vi+ * R1

Vi+ * Rf + Vi+ * R1

=

Vo * R1

Vi (Rf + R1 ) = Vo * R1 Therefore Rf + R1 = Vo * R1 / Vi Therefore Vo / Vi = Rf + Ri / Ri Vo / Vi = 1 + Rf / Ri

Gains for circuit depicted in Figure 9.5 of Lab sheet: Rf = 32.6 kΩ Ri = 9.73 kΩ Theoretical Gain AV = (1 + (32.6 / 9.73)) = 4.35 Practical Gain AV = 1.85/0.4 = 4.35 % Error in Gain = (4.35 – 4.35)*100%/4.61 = 0%

Gains for circuit depicted in Figure 9.5 of Lab sheet with Rf short circuited: Rf = 0 kΩ (Short Circuit) Ri = 9.73 kΩ Theoretical Gain AV = (1 + (0 / 9.73)) = 1

Practical Gain AV = 0.4/0.4 = 1 % Error in Vo = (0.4 – 0.4)*100%/0.4 = 0 %

To get a gain of 2 for circuit depicted in Figure 9.5 of Lab sheet: Theoretical Gain AV = (1 + 1) = 2 Rf/Ri = 1 so Rf = Ri Rf = 32.6 kΩ Ri = 32.5 kΩ Practical Gain AV = 0.801/0.4 = 2.0025 % Error in Gain= (2.0025 – 2.00)*100%/ 2.00= 0.125%

To get a gain of 11 for circuit depicted in Figure 9.5 of Lab sheet: Theoretical Gain AV = (1 + 10) = 11 Rf/Ri = 10 so Rf = 10*Ri Rf = 32.6kΩ Ri = 3.260kΩ Practical Gain AV = 4.4 /0.4 = 11 % Error in Gain = (11 – 11)*100%/11 = 0%

To get a gain of 11 @10kHz 4V peak-peak for circuit depicted in Figure 9.5 of Lab sheet: Theoretical Gain AV = (1 + 10) = 11 Rf/Ri = 10 so Rf = 10*Ri Rf = 99.5 kΩ Ri = 9.73 kΩ Practical Gain AV = 44.9 /4 = 11.22 % Error in Gain = (11.22 – 11)*100%/11 = 1.96%

2. The derived theoretical input-output relationship for a inverting amplifier. If + Iin = 0 Iin = - If Vin / Rin =

-Vo/ Rf

Vo = -Rf * Vin / Rin Vo/Vin = - Rf/ Rin

3.

The derived theoretical input-output relationship for a inverting amplifier with multiple inputs. It = I1 + I2 + I3 +…. In But It + If = 0 Hence It = - If Also, I1 = V1/R1 , I2 = V2/R2 , In= Vn/Rn And If = Vo / Rf Therefore from It = - If , we can write these in terms of voltage and resistance So, It = - If ; V1 / R1 + V2/ R2 + ….. Vn/ Rn = Vo / Rf If

R 1 ≠ R2 ≠ R3

Then Vo = - Rf (V1 / R1 + V2/ R2 + ….. Vn/ Rn ) If R1 = R2 = R3 , then : V1 / R1 + V2/ R2 + ….. Vn/ Rn = - Vo / Rf Rf / R1 (V1 + V2 + ….. Vn ) = Vo

Discussion : The theory and the practical application for a non-inverting amplifier coincide as the input and the output signals were found to 90 degrees out of-phase ; as noted in Diagram 1. Also all output signals from the non-inverting amplifier had the same frequency as the input signals. From table 1. an input of .4V was applied to the circuit and a output of 1.75 V . It can be clearly seen that the signal was amplified. However when the feedback resistor was short out of the circuit , the voltage applied was equal to the measured at Vo. It can be noted that the feedback takes part of the amplified output , such that the gain is constrained much more by the feedback network and less by the open loop gain. In this case however the amplifier acts as an buffer and only passes the input signal to the output. The gains were measured by using the formula Vout = Vin(1+ (Rf/Ri ) as highlighted in the theory. According to the equation , an increase in Vin with brig rise to an increase in the output voltage ; as they are proportional . However they will be a slight decrease in Vo because as the frequency increases , the gain decreases , thus resulting in a change in Vo. The gain of the amplifier was found to be 4.35 with a percentage error of zero. To determine the gain for the other non-inverting circuits the same equation was applied. The uncertainties for the gain of 2 and 11 were found to be o and 1.96 %. The slew rate of the of the circuit was found to be 96.96 x 10-6 V/s . This was done by taking the frequency and then tused to calculate the time period. The difference between the peak and the trough were measured and then divided by the time period calculated. The slew rate was however smaller that the accepted slew rate of 0.5V/ µs . This meant that the change between the input and output signal was very slow. As the faster the slew rate the faster the change in output signal. The discrepancy in reading may have been due to the ±16V which was supplied to the circuit as the amplifier was unable to handle the amplitude of the output. From the theory is was noted that the inverting amplifier input and output signals will be out of phase by 180 degrees and producing a negative gain which coincides with the experiment done. The gain was calculated by using the equation Vo/ Vi = -Rf/Ri as derived in the theory above. To obtain a gain of -10 , resistors .986 kΩ and 9.72 kΩ were chosen for R1 and Rf respectively. These values were substituted into the equation to produce the gain of -10. When applied to the circuit the uncertainty in reading were calculated to be 1.4 %. To obtain a gain of -2.2, the

resistors Rf and Ri were used as 2.2 kΩ and 0.986 kΩ respectively. The uncertainty in the gain was found to be 1.36 %. To obtain a voltage gain of -1, the resistors Rf and Ri were used as .986 kΩ and .986 kΩ respectively. The uncertainty was calculated to be zero . The output waveforms for the inverting circuit were observed to be is out of phase with the input signals by 180o. The summing amplifier is similar that of and inverting amplifier but just with multiple inputs. To measure the gain ,the method of superposition had to be introduced . Which is done by measuring output voltage and signal at a time . After obtaining the voltages V01 and V02 , the gain of each input signal was calculated by equation -Rf/Ri and then summed . The theoretical output voltage was determined by the use of the equation Vo = (G1V1 + G2V2). The outputs obtained were out of phase with the input, this is so because a summing amplifier is an inverting amplifier, as was earlier stated.

Conclusion : The experiment was proven to be true as the experiment values coincide with that of experimental values; however slight discrepancies in values were noted. It was seen that the opamps with a combination resistors can be configured to perform many useful operations such as inverting , non-inverting and summing .

References: 

A.V.Bakshi. (2009). Electrical Networks. India: Technical Publications Pune. Peter Y. Yu, M. C. (2007). Fundamentals of ectification and smoothingr: Physics and Applications. Germany: Springer.



Boylestad, Robert; Nashelsky, Louis (1998). Electronic Devices and Circuit Theory, Prentice Hall



Edward Hughes, Ian McKenzie Smith, et al, Hughes Electrical and Electronic Technology 10th Edition,

http://www.electronics-tutorials.ws/diode/diode_1.html