Frequency Response & Transfer Functions

Hello! Welcome back to your course on radioelectronics.

In our last lesson, we mastered the art of analyzing AC circuits at a single, fixed frequency. Using complex impedance and Kirchhoff's laws, you learned to calculate specific phasor voltages and currents. Now, we'll take a significant step forward.

Introduction

Today's Goal: We are shifting our perspective from a static snapshot to a dynamic characterization. Instead of asking "What is the output for a 1 kHz signal?", we will ask, "How does this circuit behave across an entire spectrum of frequencies?" This leads us to the core concepts of Frequency Response and the Transfer Function.

The transfer function is a powerful mathematical construct that acts as a blueprint for a circuit's behavior. It tells us precisely how the circuit will alter the amplitude and phase of any input signal, depending on its frequency. For someone with your background in software and systems, you can think of the transfer function as the defining characteristic of a "signal processing" black box.

By the end of this lesson, you will be able to define what a transfer function is, derive one for a simple circuit, and analyze its magnitude and phase to understand the circuit's frequency response.

Time to complete: Approximately 50 minutes.

1. The Transfer Function: A System-Level View

At its heart, any linear circuit can be viewed as a system that takes an input signal and produces an output signal. The transfer function, denoted $H(j\omega )$, is the crucial link that describes this input-output relationship in the frequency domain.

Let's begin with a formal introduction to this concept.

• Resource: by Michel van Biezen
• Watch: 00:00 - 03:22
• Focus on:
  • The core definition: a frequency-dependent ratio of an output signal to an input signal.
  • The symbol $H(\omega )$ and the idea that its value changes with frequency $\omega$.
  • The different types of transfer functions (voltage gain, current gain, etc.).

As the video explains, the transfer function is formally defined as the ratio of the output phasor to the input phasor. Depending on what you define as your input and output, you can have several types:

The most important takeaway is that $H(j\omega )$ is a complex number. As such, for any given frequency $\omega$, it has both a magnitude and a phase:

• Magnitude $|H(j\omega )|$: This tells you the gain of the circuit. It's the ratio of the output amplitude to the input amplitude. A magnitude of 0.5 means the output signal's amplitude is half of the input's at that frequency.
• Phase Angle $\angle H(j\omega )$: This tells you the phase shift the circuit introduces. A phase angle of -45° means the output signal's waveform lags behind the input signal's waveform by 45 degrees at that frequency.

The collective behavior of $|H(j\omega )|$ and $\angle H(j\omega )$ as you vary $\omega$ is what we call the frequency response.

2. Deriving a Transfer Function

Now that we have the definition, let's make it concrete by deriving the transfer function for a fundamental circuit: the series RC circuit. The method builds directly on what you learned in the last lesson. We'll use the voltage divider rule, but now with complex impedances.

For the circuit below, the input is ${\mathbf{V}}_{in}$ and the output ${\mathbf{V}}_{out}$ is taken across the capacitor.

The voltage divider rule in the frequency domain states:

where $Z_{out}$ is the impedance we're taking the output across, and $Z_{total}$ is the total impedance of the divider.

From this, the transfer function is simply:

Let's watch a clear demonstration of this derivation.

• Resource: by Michel van Biezen
• Watch: 00:34 - 01:55
• Focus on: The step-by-step application of the voltage divider rule using the impedances for the resistor ($R$) and capacitor ($1/j\omega C$).

Let's retrace the key steps from the video:

1. Identify Impedances:

   • The impedance of the component we take the output across is the capacitor: $Z_{out}=Z_C=\frac{1}{j\omega C}$
   • The total impedance is the series combination of the resistor and capacitor: $Z_{total}=Z_R+Z_C=R+\frac{1}{j\omega C}$

2. Apply the Voltage Divider Rule:

   $$H(j\omega )=\frac{Z_C}{Z_R+Z_C}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}$$

3. Simplify the Expression: To get a standard form, multiply the numerator and denominator by $j\omega C$:

   $$H(j\omega )=\frac{1}{j\omega RC+1}=\frac{1}{1+j\omega RC}$$

This final expression is the transfer function for our RC circuit. It elegantly captures the circuit's complete frequency-dependent behavior in a single equation.

3. Analyzing the Frequency Response

With the transfer function $H(j\omega )=\frac{1}{1+j\omega RC}$ in hand, we can now analyze the circuit's frequency response. This means examining the magnitude and phase at different frequencies.

First, let's find the general expressions for magnitude and phase. Since we have a complex number in the denominator:

• Magnitude: $$|H(j\omega )|=\frac{|1|}{|1+j\omega RC|}=\frac{1}{\sqrt{1^2+(\omega RC{)}^2}}$$
• Phase: $$\angle H(j\omega )=\angle (1)-\angle (1+j\omega RC)=0-\arctan \left(\frac{\omega RC}{1}\right)=-\arctan (\omega RC)$$

The video you just watched confirms these calculations in its final segment (01:55 - 02:49). Now, let's interpret these equations by testing a few key frequency points.

a) At DC ($\omega \to 0$)

• Magnitude: $|H(j0)|=\frac{1}{\sqrt{1+0}}=1$. The gain is 1, meaning the output amplitude equals the input amplitude. The circuit passes DC signals perfectly.
• Phase: $\angle H(j0)=-\arctan (0)=0^{\circ }$. There is no phase shift.
• Intuition: At DC, the capacitor acts as an open circuit. Since no current flows, there's no voltage drop across the resistor, and thus ${\mathbf{V}}_{out}={\mathbf{V}}_{in}$.

b) At Very High Frequencies ($\omega \to \infty$)

• Magnitude: ${\lim }_{\omega \to \infty }|H(j\omega )|={\lim }_{\omega \to \infty }\frac{1}{\sqrt{1+(\omega RC{)}^2}}=0$. The gain approaches zero. The circuit completely blocks signals with very high frequencies.
• Phase: ${\lim }_{\omega \to \infty }\angle H(j\omega )|={\lim }_{\omega \to \infty }-\arctan (\omega RC)=-90^{\circ }$. The output lags the input by 90 degrees.
• Intuition: At infinite frequency, the capacitor's impedance becomes zero, effectively acting as a short circuit to ground. This means ${\mathbf{V}}_{out}$ is pulled to zero.

c) At the "Corner" Frequency (${\omega }_c=1/RC$)

There is a special frequency determined by the circuit's components, R and C. This is called the corner frequency or cutoff frequency. Let's see what happens when $\omega =1/RC$.

• Magnitude: $|H(j{\omega }_c)|=\frac{1}{\sqrt{1+((1/RC)\cdot RC{)}^2}}=\frac{1}{\sqrt{1+1}}=\frac{1}{\sqrt{2}}\approx 0.707$.
• Phase: $\angle H(j{\omega }_c)=-\arctan ((1/RC)\cdot RC)=-\arctan (1)=-45^{\circ }$.

At this specific frequency, the output signal's amplitude is about 70.7% of the input, and it lags by 45 degrees. This $\frac{1}{\sqrt{2}}$ point is also known as the -3dB point, a term you will encounter frequently in radioelectronics and signal processing.

This analysis reveals the circuit's identity: it lets low frequencies pass through but attenuates high frequencies. This is the definition of a low-pass filter.

Conclusion

In this lesson, we made the crucial transition from single-frequency analysis to understanding a circuit's behavior across a whole spectrum.

Key Takeaways:

• The Transfer Function $H(j\omega )$ is the frequency-domain ratio of a circuit's output phasor to its input phasor.
• As a complex quantity, $H(j\omega )$ describes both the gain (magnitude) and phase shift (angle) that the circuit applies to an input signal at a given frequency $\omega$.
• We can derive a circuit's transfer function using standard analysis techniques like the voltage divider rule, applied with complex impedances.
• The Frequency Response is the overall behavior of the transfer function's magnitude and phase across the range of frequencies. Analyzing it reveals the circuit's fundamental purpose, such as filtering.

What's Next?

Our analysis of the RC circuit naturally leads us to our next topic. We will formally explore the Low-Pass RC Filter, defining its cutoff frequency and examining its behavior in more detail. We will then apply the same principles to design and analyze a High-Pass RC Filter, giving you a foundational toolkit for signal shaping.