1. Theremin Oscillator Circuit
EGRE 101 – Fall 2018

2. Capacitors q=Cv Conceptual drawing of capacitor:
Dielectric sandwiched between
two parallel plates.
q=Cv
Symbol of Capacitor
C: Characteristic parameter of the capacitor
called the “Capacitance”. Measured in units
of Farads (F).

3. Capacitor i-u Relationship
For a voltage, v(t), across the capacitor, the current, i(t), through the capacitor is given by:
Denotes change in 𝜐!
The above relationship means that, unless
𝜐 𝑡 varies with time, 𝑖 𝑡 will be zero!!
The parameter C is the capacitance, and is a
characteristic of the capacitor.

4. Inductors Denotes change in 𝑖 𝑡 ! L: Inductance
Value depends on physical parameters
such as coil area, core material, etc.
Inductance will be given as a value
expressed in units of H (Henry)
The 𝜐−𝑖 relationship of the inductor
indicates that, for a constant current, 𝑖,
the induced voltage, 𝜐, is zero.
Denotes change in 𝑖 𝑡 !

5. The inverter The output is the inverse of the input.
Input=1=>Output=0
Feed output to input, so input is now 0
Input=0=>Output=1
Feed output to input, so input is now 1 …
This causes a repeating sequence of 0’s and 1’s to appear at the output

6. The circuit that generates the oscillating signal

7. The circuit

8. inverter 𝑉 𝑖𝑛 𝑉 𝑜𝑢𝑡 𝑉 𝑏𝑖𝑎𝑠 is the bias voltage of the inverter
𝑉 𝑇 is the threshold voltage of the inverter
The change in voltage levels occurs with a time delay ∆𝑡 caused by capacitive effects in the inverter circuit.
Truth Table
Voltage
𝑽 𝒊𝒏
𝑽 𝒐𝒖𝒕 1
𝑽 𝒊𝒏
𝑽 𝒐𝒖𝒕
≤ 𝑉 𝑇
𝑉 𝑏𝑖𝑎𝑠
> 𝑉 𝑇

9. inverter 𝑉 𝑖𝑛 𝑉 𝑜𝑢𝑡 𝑉 𝑏𝑖𝑎𝑠 =+5𝑉 𝑉 𝑇 =+2.5𝑉 Voltage 𝑽 𝒊𝒏 (V) 𝑽 𝒐𝒖𝒕 (V)5
0.9
3.1 7 -2 -7
Truth Table
𝑽 𝒊𝒏
𝑽 𝒐𝒖𝒕 1

10. Assume Dt = 0
𝑉 𝑏𝑖𝑎𝑠 = ?
𝑉 𝑇 = ?

11. A Ring Oscillator What if the output looks just like the input?
Can we then use the output to “drive” the input?
Even number of inverters -> STABLE circuit, voltages stay constant -> a MEMORY ELEMENT
Odd number of inverters -> INSTABLE circuit, voltages oscillate -> an OSCILLATOR
(WHY? Will be studied later in Signals and Systems and Circuits classes.)

12. V1 V2 V3
Time (in units of Δt)

13. V1 V2 V3
Time (in units of Δt)

14. Ring Oscillator Frequency
3 inverters:
Period 𝑇=2 3∆𝑡 =6∆𝑡
Frequency f= 1 𝑇 = 1 6∆𝑡
N inverters:
Period 𝑇=2𝑁∆𝑡
Frequency f= 1 𝑇 = 1 2𝑁∆𝑡

15. Assume Dt = 10ms and we want a maximum oscillation frequency of 1kHz.
What is the minimum number of inverters needed?
3 inverters:
Period 𝑇=2 3∆𝑡 =6∆𝑡
Frequency f= 1 𝑇 = 1 6∆𝑡
N inverters:
Period 𝑇=2𝑁∆𝑡
Frequency f= 1 𝑇 = 1 2𝑁∆𝑡

16. Theremin – multiplier/MIXER circuit (XNOR)
EGRE 101 – Fall 2018

17. sine Wave vS square wave
Why use a square?
Circuits are much simpler.
Digital circuits: cheap, robust

18. Digital Circuits: Logic Levels
Bits are very useful abstractions allowing computer engineers to design very complex digital systems.
Voltages vs Bits:
“Low” voltage: “0”
“High” voltage: “1”

19. Digital Circuits: Logic Gates
Truth tables IN
NOT 1
One input:
NOT
Two inputs:
AND/NAND
OR/NOR
XOR/XNOR
(exclusive OR)
(“EQUAL”)
IN1
IN2
AND 1
IN1
IN2
XNOR 1

20. Truth Tables IN NOT 1 IN1 IN2 AND 1 IN1 IN2 OR 1 IN1 IN2 XOR 1 IN1 IN21
IN1
IN2
AND 1
IN1
IN2 OR 1
IN1
IN2
XOR 1
IN1
IN2
NAND 1
IN1
IN2
NOR 1
IN1
IN2
XNOR 1

21. Example: Logic Function
1. A=0, B=1, C=1, Y=?
2. A=1, B=1, C=1, Y=?
3. A=1, B=0, C=0, Y=?

22. Digital Circuits: Multiplication vs XNOR
Approximating sine waves with square waves -> Can use XNOR instead of multiplications

23. Digital Circuits: Multiplication vs XNOR

24. Digital Circuits: Multiplication vs XNOR

25. Product of two sine signals
𝒔𝒊𝒈𝒏𝒂𝒍 𝟏: 𝒔𝒊𝒏 𝟐𝝅 𝒇 𝟏 𝒕
𝒔𝒊𝒈𝒏𝒂𝒍 𝟐: 𝒔𝒊𝒏 𝟐𝝅 𝒇 𝟐 𝒕
𝒔𝒊𝒏 𝟐𝝅 𝒇 𝟏 𝒕 ∙𝒔𝒊𝒏 𝟐𝝅 𝒇 𝟐 𝒕
The signal 𝑠𝑖𝑛 2𝜋 𝑓 1 𝑡 ∙𝑠𝑖𝑛 2𝜋 𝑓 2 𝑡 is what is generated by the mixer. This signal contains (review trigonometric identities – or if you haven’t studied trigonometry yet, at least pay a lot of attention to them when you do study them) a component at frequency 𝑓 1 + 𝑓 2 and one at 𝑓 1 − 𝑓 A low-pass filter will cut off the 𝑓 1 + 𝑓 2 component, which is at very high frequencies, and will allow the 𝑓 1 − 𝑓 2 component to pass through to the speaker.

26. equals… sum of cosine signals
The signal 𝑠𝑖𝑛 2𝜋 𝑓 1 𝑡 ∙𝑠𝑖𝑛 2𝜋 𝑓 2 𝑡 is what is generated by the mixer. This signal contains (review trigonometric identities – or if you haven’t studied trigonometry yet, at least pay a lot of attention to them when you do study them) a component at frequency 𝑓 1 + 𝑓 2 and one at 𝑓 1 − 𝑓 A low-pass filter will cut off the 𝑓 1 + 𝑓 2 component, which is at very high frequencies, and will allow the 𝑓 1 − 𝑓 2 component to pass through to the speaker.
𝒔𝒊𝒈𝒏𝒂𝒍 𝟏:0.5 cos(2pt (f2 + f1))
𝒔𝒊𝒈𝒏𝒂𝒍 𝟐:0.5 cos(2pt (f2 − f1))

27. XNOR OUTPUT = f2-f1 signal
The signal 𝑠𝑖𝑛 2𝜋 𝑓 1 𝑡 ∙𝑠𝑖𝑛 2𝜋 𝑓 2 𝑡 is what is generated by the mixer. This signal contains (review trigonometric identities – or if you haven’t studied trigonometry yet, at least pay a lot of attention to them when you do study them) a component at frequency 𝑓 1 + 𝑓 2 and one at 𝑓 1 − 𝑓 A low-pass filter will cut off the 𝑓 1 + 𝑓 2 component, which is at very high frequencies, and will allow the 𝑓 1 − 𝑓 2 component to pass through to the speaker.

28. XNOR OUTPUT
The signal 𝑠𝑖𝑛 2𝜋 𝑓 1 𝑡 ∙𝑠𝑖𝑛 2𝜋 𝑓 2 𝑡 is what is generated by the mixer. This signal contains (review trigonometric identities – or if you haven’t studied trigonometry yet, at least pay a lot of attention to them when you do study them) a component at frequency 𝑓 1 + 𝑓 2 and one at 𝑓 1 − 𝑓 A low-pass filter will cut off the 𝑓 1 + 𝑓 2 component, which is at very high frequencies, and will allow the 𝑓 1 − 𝑓 2 component to pass through to the speaker.