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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
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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
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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
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