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Tutorial: Mechanic – electrician Topic: Electronics II. class Operational Amplifiers: Non-Inverting Amplifiers – Part 2 – Derivations Prepared by: Ing. Jaroslav Bernkopf AVOP-ELEKTRO-Ber-005 Projekt Anglicky v odborných předmětech, CZ.1.07/1.3.09/04.0002 je spolufinancován Evropským sociálním fondem a státním rozpočtem České republiky. Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain Op Amp Circuit Analysis: Non-Inverting Amplifier https://www.youtube.com/watch?v=_CFR2gViJqo Operational Amplifiers 2 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain Let‘s apply a voltage V1 of +1 V to the input of the amplifier in the figure. The voltage difference between the inputs V+ and V- is zero. This implies that the voltage on the input V- is the same as the input voltage, i.e. +1 V. This voltage will cause a current to flow through the resistor R1. What will this current be? V+ 3 Vin V1 = +1 V 2 V- + 1 Vout R2 10k R1 1k +1 V Operational Amplifiers 3 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain The voltage across the resistor R1 is 1 V. Using the Ohm‘s law we can calculate the current through the resistor R1: 𝐼𝑅1 V+ 3 Vin V1 = +1 V 2 V- 1𝑉 = = 1𝑚𝐴 1𝑘 + 1 Vout R2 10k R1 1k +1 V IR1 = 1 mA Operational Amplifiers VR1 = 1 V 4 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain Can this current flow from the input V-? No, it can‘t, because as we know • the input resistance of an op amp is infinite • the input current of an op amp is zero So the current of the resistor R1 has to flow through R2. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 Vout R2 10k IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 1k VR1 = 1 V 5 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain The current of 1 mA will create a voltage drop across the resistor R2. We can calculate it using the Ohm‘s law: VR2 = IR2 * R2 VR2 = 1mA * 10k VR2 = 10 V V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 Vout R2 10k VR2 = 10 V 1k VR1 = 1 V IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 6 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain What is the value and the polarity of the output voltage? The upper end of R1 is positive with respect to the ground. We can draw a small plus sign to the upper end of R1, a minus sign to its lower end. The current through R2 is the same as the current through R1: It flows from above to below. We can draw the same plus and minus signs to the ends of R2. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V VR1 = 1 V 7 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain The output voltage consists of the sum of two individual resistor voltage drops. These voltage drops are in series, like two batteries. Their voltages add together. The total output voltage V2 is V2 = VR1 + VR2 V2 = 1 V + 10 V V2 = 11 V The output voltage is positive and its value is +11 V. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V V2 = +11 V VR1 = 1 V 8 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain What is the voltage gain? As you can see, the amplifier makes • eleven volts out of one volt • positive output voltage out of positive input voltage Its voltage gain must be +11. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V V2 = +11 V VR1 = 1 V 9 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain Look at the voltages and at the values of the resistors: 𝑉2 10 𝑉 + 1 𝑉 = = 10 + 1 𝑉1 1𝑉 1 𝑉 …1 𝑘 10 𝑉 … 10 𝑘 10 𝑉 + 1 𝑉 … 10 𝑘 + 1 𝑘 10 𝑘 + 1 𝑘 = 10 + 1 1𝑘 Do the values look so similar just by accident? No, it is no accident. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V V2 = +11 V VR1 = 1 V 10 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain The voltage gain is determined by the voltages: 𝑽𝟐 𝑨𝒗 = 𝑽𝟏 The voltages are determined by the values of the resistors. Therefore, the voltage gain is determined by the resistors: 𝑨𝒗 = 𝑹𝟐 + 𝑹𝟏 𝑹𝟏 𝑨𝒗 = 𝑹𝟐 +𝟏 𝑹𝟏 V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V V2 = +11 V VR1 = 1 V 11 Non-Inverting Amplifiers – Part 2 - Derivations Voltage Gain This is the equation we are looking for: 𝑹𝟐 𝑨𝒗 = +𝟏 𝑹𝟏 Now you can forget it. Because in future you will always be able to derive it again. V+ 3 Vin V1 = +1 V 2 V- +1 V + 1 - Vout + R2 IR2 = 1 mA R1 IR1 = 1 mA Operational Amplifiers 10k + 1k - VR2 = 10 V V2 = +11 V VR1 = 1 V 12 Non-Inverting Amplifiers – Part 2 - Derivations Task Construct the first stage of a preamplifier for record players. The preamplifier should have the following features: • Input resistance Rin = 47kΩ • Voltage gain Av = 34 Use the conception shown in the figure below. Vin V+ 3 2 V- + 1 Vout R2 10k R1 1k Operational Amplifiers 13 Non-Inverting Amplifiers – Part 2 - Derivations Solution The input resistance Rin of a non-inverting amplifier is infinite. The desired input resistance Rin of the preamplifier is 47 kΩ. Rin = ∞ Vin V+ 3 2 V- Ri = ∞ + 1 Vout R2 10k R1 1k Operational Amplifiers 14 Non-Inverting Amplifiers – Part 2 - Derivations Solution We must add another resistor R3 = 47k in parallel to achieve the desired input resistance Rin = 47kΩ. Rin = 47k V+ 5 Vin 6 V- R3 + 7 Vout - 47k Operational Amplifiers R2 10k R1 1k 15 Non-Inverting Amplifiers – Part 2 - Derivations Solution The values of R1 and R2 are not given. Let‘s choose the resistance of R1 = 1kΩ. Then we will calculate the value of R2. V+ 5 Vin 6 V- R3 + 7 Vout - 47k Operational Amplifiers R2 10k R1 1k 1k 16 Non-Inverting Amplifiers – Part 2 - Derivations Solution The equation for the voltage gain is: 𝑅 𝐴𝑣 = 𝑅2 + 1 1 Let‘s express R2: 𝑅2 = (𝐴𝑣 − 1) ∗ 𝑅1 Let‘s substitute the known values for Av and R1: 𝑅2 = 34 − 1 ∗ 1𝑘 𝑅2 = 33𝑘 Rin = 47k V+ 5 Vin 6 V- + 7 Vout R2 10k 33k 1k R1 1k 1k The resistance of R2 will be 33 kΩ. R3 47k Operational Amplifiers 17 Non-Inverting Amplifiers – Part 2 - Derivations References http://www.wikipedia.com http://www.thefreedictionary.com Operational Amplifiers 18