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title: 'Lecture 45: Use of UJT as relaxation oscillator'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Oscillators
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# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 45: Use of UJT as relaxation oscillator
### Oscillators

<!--
Welcome to Lecture 45 on the UJT Relaxation Oscillator. In the previous lecture, we studied the semiconductor physics and V-I characteristics of the UJT. Today, we put that knowledge into action by analyzing the UJT relaxation oscillator. This simple circuit converts DC power into exponential sawtooth waves and sharp trigger pulses, forming the heart of SCR and TRIAC gate firing controls across industrial power electronics.
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# Lecture Agenda: UJT Relaxation Oscillators

- 1. Circuit Configuration & Terminal Topology of UJT Relaxation Oscillator
- 2. Step-by-Step Operational Mechanics & Waveform Cycles ($V_E, V_{B1}, V_{B2}$)
- 3. Mathematical Derivation of Period ($T$) & Oscillation Frequency ($f_o$)
- 4. Derivation of Upper ($R_{T,max}$) and Lower ($R_{T,min}$) Resistance Design Bounds
- 5. Numerical Design Example & Component Parameter Selection
- 6. Industrial Application: SCR Phase-Control Triggering & Line Synchronization
- 7. Constant Current Charging for Linear Sawtooth Ramp Generation

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Here is our lecture plan. We will begin with the circuit schematic and component roles. Then we will trace the charging and discharging cycles step-by-step, derive the frequency formula, establish the mathematical boundaries for RT, work through a complete design example, examine SCR gate triggering, and explore constant-current linear sweep circuits.
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# Circuit Topology & Component Roles

- Core Circuit Architecture: Timing resistor $R_T$ connected between $V_{CC}$ and Emitter ($E$); timing capacitor $C_T$ connected between Emitter ($E$) and Ground.
- Base Terminal Bias Connections: Base 2 ($B_2$) connected to $V_{CC}$ via temperature compensation resistor $R_2$; Base 1 ($B_1$) connected to Ground via small output load resistor $R_1$ ($50\text{ }\Omega - 100\text{ }\Omega$).
- Output Waveform Terminals: Sawtooth exponential ramp wave taken across capacitor $C_T$ ($V_E$); sharp positive gate trigger spikes taken across $R_1$ ($V_{B1}$); sharp negative spikes taken across $R_2$ ($V_{B2}$).
- Dual-Function Capability: Generates timing sweep signals while simultaneously serving as an isolated, high-current trigger pulse generator for power thyristors.

<!--
Let's examine the schematic of the UJT relaxation oscillator. Capacitor CT and resistor RT form an RC charging path across the DC supply V_CC. Base 2 is tied to V_CC through resistor R2, while Base 1 is connected to ground through a small resistor R1 (typically 50 ohms). The circuit produces two distinct outputs: an exponential sawtooth wave across CT and sharp trigger pulses across R1.
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# Step-by-Step Operational Mechanics & Cycles

- Phase 1: Capacitor Charging (OFF State): Initially $V_E = 0 < V_P$. Diode $D_E$ is reverse biased. $C_T$ charges exponentially from valley voltage $V_V$ toward $V_{CC}$ through $R_T$ with time constant $\tau = R_T C_T$. Emitter current $I_E \approx 0$. Charging equation: $V_E(t) = V_{CC} - (V_{CC} - V_V) e^{-t / R_T C_T}$.
- Phase 2: UJT Firing & Rapid Discharge (ON State): When $V_E(t)$ reaches peak firing voltage $V_P = \eta V_{BB} + V_D$, UJT fires ON. Resistance $R_{B1}$ collapses to $R_{B1,sat} \approx 20\text{ }\Omega$. $C_T$ rapidly discharges through $R_{B1,sat}$ and $R_1$ with small time constant $\tau_{dis} = (R_{B1,sat} + R_1) C_T \ll \tau_{charge}$, producing a sharp voltage spike across $R_1$.
- Phase 3: Turn-OFF & Cycle Reset: When $V_E$ drops down to valley voltage $V_V$ and $I_E < I_V$, UJT turns OFF, restoring high $R_{B1}$ impedance and repeating the cycle.

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The operation consists of three repeating phases: 1) Charging Phase: CT charges through RT toward V_CC until its voltage reaches peak firing threshold V_P; 2) Discharge Phase: The instant V_E reaches V_P, the UJT fires ON, collapsing R_B1 to 20 ohms and dumping CT's charge through R1 in microseconds; 3) Reset Phase: As V_E drops below valley voltage V_V, current falls below I_V, turning the UJT OFF to restart the cycle.
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# Derivation of Oscillation Frequency (f_o)

- Charging Period Approximation: Because discharge time constant $\tau_{dis} \ll \tau_{charge}$, total oscillation period $T \approx t_{charge}$.
- Setting Boundary Voltage Conditions: $V_E(T) = V_P = \eta V_{CC} + V_D$. Assuming initial voltage $V_V \approx 0$ and $V_D \ll V_{CC}$:
- Voltage Equation: $\eta V_{CC} = V_{CC} (1 - e^{-T / R_T C_T}) \Rightarrow 1 - \eta = e^{-T / R_T C_T}$.
- Taking Natural Logarithm: $\frac{T}{R_T C_T} = \ln\left(\frac{1}{1 - \eta}\right) \Rightarrow T = R_T C_T \ln\left(\frac{1}{1 - \eta}\right)$.
- Oscillation Frequency ($f_o$): $f_o = \frac{1}{T} = \frac{1}{R_T C_T \ln\left(\frac{1}{1 - \eta}\right)}$.
- Standoff Ratio Simplification: For typical UJT with $\eta = 0.632$, $\ln\left(\frac{1}{1 - 0.632}\right) = \ln(2.718) = 1.00 \Rightarrow f_o \approx \frac{1}{R_T C_T}$.

<!--
Let's perform the mathematical derivation for oscillation frequency. Setting the charging equation equal to peak firing voltage V_P = eta * V_CC yields 1 - eta = exp(-T / RT CT). Taking the natural logarithm gives period T = RT CT ln(1 / (1 - eta)). Notice that if the UJT's standoff ratio eta equals 0.632, ln(1 / (1 - 0.632)) equals exactly 1, simplifying the frequency equation to f_o = 1 / (RT CT).
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# Design Limits & Bounds for Timing Resistor (R_T)

- Upper Bound ($R_{T,max}$ for Successful Turn-ON): Current supplied through $R_T$ when $V_E = V_P$ must exceed Peak Point Current $I_P$ to trigger negative resistance: $R_{T,max} = \frac{V_{CC} - V_P}{I_P}$. If $R_T > R_{T,max}$, capacitor $C_T$ fails to reach $V_P$ and UJT never fires.
- Lower Bound ($R_{T,min}$ for Successful Turn-OFF): Current supplied through $R_T$ when $V_E = V_V$ must be less than Valley Point Current $I_V$ to allow UJT to turn OFF: $R_{T,min} = \frac{V_{CC} - V_V}{I_V}$. If $R_T < R_{T,min}$, UJT remains latched ON continuously.
- Permissible Resistance Window: $\frac{V_{CC} - V_P}{I_P} > R_T > \frac{V_{CC} - V_V}{I_V}$.
- Design Selection Rule: Select $R_T$ safely near the geometric mean of $R_{T,min}$ and $R_{T,max}$ to guarantee reliable oscillation.

<!--
Can we choose any value for timing resistor RT? Absolutely not! RT must be strictly bounded between two limits. For the UJT to fire ON, RT must be small enough to supply at least peak current I_P at voltage V_P. Conversely, for the UJT to turn OFF, RT must be large enough so charging current at V_V remains below valley current I_V. Choosing RT outside this window results in circuit latchup or total failure to trigger.
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# UJT Oscillator Complete Design Example

- Problem Statement: Design a UJT relaxation oscillator operating at $f_o = 400\text{ Hz}$ with $C_T = 0.1\text{ }\mu\text{F}$. Given UJT parameters $\eta = 0.65$, $V_P = 13.7\text{ V}$, $V_V = 2.0\text{ V}$, $I_P = 5.0\text{ }\mu\text{A}$, $I_V = 4.0\text{ mA}$, and supply $V_{CC} = 20.0\text{ V}$, calculate required timing resistor $R_T$ and verify design bounds.
- Step 1: Calculate Target Oscillation Period ($T$): $T = \frac{1}{f_o} = \frac{1}{400\text{ Hz}} = 2.5\text{ ms} = 2.5 \times 10^{-3}\text{ s}$.
- Step 2: Calculate Required Timing Resistor ($R_T$): $T = R_T C_T \ln\left(\frac{1}{1 - \eta}\right) \Rightarrow 2.5 \times 10^{-3} = R_T (0.1 \times 10^{-6}) \ln\left(\frac{1}{1 - 0.65}\right) = R_T (10^{-7}) \ln(2.857) = R_T (10^{-7}) (1.0498) \Rightarrow R_T = \frac{2.5 \times 10^{-3}}{1.0498 \times 10^{-7}} \approx 23.81\text{ k}\Omega$ (standard $24\text{ k}\Omega$).
- Step 3: Calculate Upper Resistance Limit ($R_{T,max}$): $R_{T,max} = \frac{V_{CC} - V_P}{I_P} = \frac{20.0\text{ V} - 13.7\text{ V}}{5.0 \times 10^{-6}\text{ A}} = \frac{6.3\text{ V}}{5.0 \times 10^{-6}\text{ A}} = 1.26\text{ M}\Omega$.
- Step 4: Calculate Lower Resistance Limit ($R_{T,min}$): $R_{T,min} = \frac{V_{CC} - V_V}{I_V} = \frac{20.0\text{ V} - 2.0\text{ V}}{4.0 \times 10^{-3}\text{ A}} = \frac{18.0\text{ V}}{4.0 \times 10^{-3}\text{ A}} = 4.5\text{ k}\Omega$.
- Step 5: Verify Design Validity: $4.5\text{ k}\Omega < 23.81\text{ k}\Omega < 1.26\text{ M}\Omega$ (Design is fully valid and stable).

<!--
Here is a complete numerical design walkthrough. To obtain a 400 Hz output with a 0.1 uF capacitor and eta = 0.65, we calculate RT = 23.81 k-ohms. Next, we evaluate the boundary conditions: RT,max is 1.26 M-ohms and RT,min is 4.5 k-ohms. Since 23.81 k-ohms lies safely within this range, our oscillator design is fully valid.
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# Output Pulse Characteristics across Base-1 (R_1)

- Peak Trigger Voltage ($V_{p1}$): Peak pulse voltage appearing across $R_1$ during discharge: $V_{p1} \approx \frac{R_1}{R_1 + R_{B1,sat}} (V_P - V_V)$.
- Pulse Duration ($\left.t_p\right|_{50\%}$): Determined by $C_T$ discharge time constant: $t_p \approx (R_1 + R_{B1,sat}) C_T$. Typically narrow ($1.0\text{ }\mu\text{s} - 20.0\text{ }\mu\text{s}$).
- Resistor Selection Guidelines: $R_1$ must be kept small ($20\text{ }\Omega - 100\text{ }\Omega$) to minimize quiescent DC voltage drop during OFF state while producing sufficient pulse amplitude ($1.0\text{ V} - 3.0\text{ V}$) to trigger SCR gates.
- Base-2 Resistor Selection: $R_2 \approx \frac{10000}{\eta V_{CC}}$ to stabilize peak firing voltage $V_P$ across ambient temperature variations.

<!--
Let's examine the trigger pulse generated across Base 1 resistor R1. When the UJT fires, CT rapidly discharges through R_B1,sat and R1. The peak voltage spike across R1 equals R1 / (R1 + R_B1,sat) * (V_P - V_V). Choosing R1 between 50 and 100 ohms delivers a sharp 2-volt pulse capable of reliably triggering heavy-duty SCR gates.
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# Application: SCR Phase-Control Triggering

- Synchronized Firing Topology: UJT relaxation oscillator powered by Zener-clamped full-wave rectified AC voltage ($V_{zener}$).
- Zero-Crossing Synchronization: At the end of each AC half-cycle, Zener voltage drops to zero, discharging $C_T$ to zero and synchronizing oscillator timing with line frequency ($50\text{ Hz} / 60\text{ Hz}$).
- Firing Angle Control ($\alpha$): Adjusting variable timing resistor $R_T$ alters capacitor charging rate, sweeping SCR trigger firing angle $\alpha$ smoothly from $0^\circ$ to $180^\circ$.
- Industrial Applications: Light dimmers, motor speed control, electric heater regulators, and controlled AC-to-DC converters.

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One of the most famous applications of the UJT relaxation oscillator is SCR phase control. By powering the oscillator with a Zener-clamped rectified AC supply, the timing cycle resets to zero at every AC zero-crossing. Adjusting potentiometer RT changes the charging slope, allowing smooth control of the SCR firing angle alpha from 0 to 180 degrees.
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# SCR Firing Angle Calculation Example

- Problem Statement: A UJT relaxation oscillator synchronized to a $50\text{ Hz}$ AC line ($T_{half} = 10.0\text{ ms}$) triggers an SCR. Given $C_T = 0.22\text{ }\mu\text{F}$ and $\eta = 0.632$, calculate timing resistor $R_T$ required for firing angles: (a) $\alpha = 45^\circ$, (b) $\alpha = 90^\circ$.
- Step 1: Calculate Delay Time for $\alpha = 45^\circ$: $t_{d1} = \frac{45^\circ}{180^\circ} \times 10.0\text{ ms} = 2.5\text{ ms}$.
- Step 2: Solve for $R_{T1}$ at $\alpha = 45^\circ$: Since $\eta = 0.632$, $\ln\left(\frac{1}{1-0.632}\right) = 1.00 \Rightarrow t_{d1} = R_{T1} C_T \Rightarrow R_{T1} = \frac{2.5 \times 10^{-3}\text{ s}}{0.22 \times 10^{-6}\text{ F}} \approx 11.36\text{ k}\Omega$.
- Step 3: Calculate Delay Time for $\alpha = 90^\circ$: $t_{d2} = \frac{90^\circ}{180^\circ} \times 10.0\text{ ms} = 5.0\text{ ms}$.
- Step 4: Solve for $R_{T2}$ at $\alpha = 90^\circ$: $R_{T2} = \frac{5.0 \times 10^{-3}\text{ s}}{0.22 \times 10^{-6}\text{ F}} \approx 22.73\text{ k}\Omega$.

<!--
Let's calculate timing resistance for an SCR trigger circuit operating on a 50 Hz AC line. Each half-cycle lasts 10 ms. For a 45-degree firing angle, delay time is 2.5 ms, requiring RT = 11.36 k-ohms. For a 90-degree firing angle, delay time is 5.0 ms, requiring RT = 22.73 k-ohms.
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# Linear Sawtooth Generation via Constant Current

- Non-Linearity Problem: Standard passive $R_T C_T$ charging produces an exponential curve ($V_E(t) = V_{CC}(1 - e^{-t/\tau})$), causing sweep non-linearity.
- Constant Current Charging Solution: Replace passive timing resistor $R_T$ with a PNP transistor constant current source ($I_C = \frac{V_{CC} - V_B - V_{BE}}{R_E}$).
- Linear Voltage Ramp Equation: Capacitor charges linearly according to $V_E(t) = \frac{I_C}{C_T} \times t$.
- Linear Period Formula: $T = \frac{C_T (V_P - V_V)}{I_C} = \frac{C_T (\eta V_{CC} + V_D - V_V)}{I_C}$.
- Applications: Oscilloscope Cathode Ray Tube (CRT) time-base sweep generators and linear ramp analog-to-digital converters.

<!--
Standard passive RC charging yields an exponential ramp. For precision applications like oscilloscope time-base sweeps, we need a perfectly linear sawtooth ramp. By replacing resistor RT with a PNP transistor constant current source IC, the capacitor charges at a constant rate dV/dt = IC / CT, producing a perfectly linear ramp voltage.
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# Comparison: UJT Relaxation vs. Op-Amp Astable

- Waveform Outputs: UJT (exponential/linear sawtooth across $C_T$ + narrow trigger spikes across $R_1$); Op-Amp Astable (square wave output + exponential triangle across $C$).
- Component Count & Simplicity: UJT requires only 1 discrete 3-terminal transistor + 3 resistors + 1 capacitor; Op-Amp astable requires multi-transistor IC package.
- High Peak Current Discharge: UJT discharges $C_T$ through collapsed channel ($R_{B1,sat} \approx 20\text{ }\Omega$), delivering heavy peak current pulses ($> 1\text{ A}$) ideal for SCR triggering; Op-Amp output current is internally limited ($\sim 20-40\text{ mA}$).
- Frequency Range: UJT (up to $\sim 500\text{ kHz}$); Op-Amp Astable (up to several MHz).

<!--
Comparing a UJT relaxation oscillator to an Op-Amp astable multivibrator highlights key operational differences. The UJT oscillator excels in generating high-current trigger pulses (> 1 Amp peak) because CT discharges directly through its collapsed 20-ohm channel. Op-amps cannot deliver such high peak currents due to internal short-circuit protection.
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# Summary of UJT Relaxation Oscillator

- UJT Relaxation Oscillator converts DC supply into exponential sawtooth ramps ($V_E$) and sharp positive gate pulses ($V_{B1}$).
- Oscillation period $T = R_T C_T \ln\left(\frac{1}{1 - \eta}\right) \approx R_T C_T$ (when $\eta \approx 0.632$).
- Timing resistance must be bounded within $\frac{V_{CC} - V_P}{I_P} > R_T > \frac{V_{CC} - V_V}{I_V}$ to ensure turn-ON and turn-OFF.
- Extensively applied in SCR/TRIAC phase control, linear time-base ramp generators, and pulse timing delays.

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In summary, the UJT relaxation oscillator is a foundational circuit in power and industrial electronics. Its operation relies on exponential charging of CT followed by rapid conductivity-modulation discharge through R1. Precise timing resistance bounds guarantee reliable oscillation, while PNP current sources enable linear sweep ramps.
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# Formula Reference & Key Equations

- Oscillation Period: $T = R_T C_T \ln\left(\frac{1}{1 - \eta}\right)$.
- Oscillation Frequency: $f_o = \frac{1}{T} = \frac{1}{R_T C_T \ln(1 / (1 - \eta))}$.
- Timing Resistance Bounds: $\frac{V_{CC} - V_P}{I_P} > R_T > \frac{V_{CC} - V_V}{I_V}$.
- Base-1 Peak Pulse Voltage: $V_{p1} \approx \frac{R_1}{R_1 + R_{B1,sat}} (V_P - V_V)$.
- Linear Ramp Period (Current Source $I_C$): $T = \frac{C_T (V_P - V_V)}{I_C}$.

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This final slide summarizes all core mathematical formulas for UJT relaxation oscillators and SCR firing angle calculations. Use this as your reference sheet for assignments and laboratory design work.
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