Theremin Oscillator Circuit EGRE 101 – Fall 2018

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).

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.

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 !

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

The circuit that generates the oscillating signal

The circuit

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

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

Assume Dt = 0 𝑉 𝑏𝑖𝑎𝑠 = ? 𝑉 𝑇 = ?

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.)

V1 V2 V3 Time (in units of Δt)

V1 V2 V3 Time (in units of Δt)

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

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

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

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

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”

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

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

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=?

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

Digital Circuits: Multiplication vs XNOR

Digital Circuits: Multiplication vs XNOR

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 2 . 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.

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 2 . 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))

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 2 . 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.

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 2 . 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.