Resistors in Series

1. Definition and Characteristics of Series Circuits

Definition and Characteristics of Series Circuits

In a series circuit, resistors are connected end-to-end such that there is only a single path for current to flow. The defining characteristic of this configuration is that the same current I passes through each resistor sequentially, while the total voltage Vtotal is distributed across them. This behavior arises from Kirchhoff’s Voltage Law (KVL), which states that the sum of potential differences around any closed loop must equal zero.

Mathematical Derivation of Equivalent Resistance

For n resistors connected in series, the total resistance Req is the sum of individual resistances. This can be derived from Ohm’s Law (V = IR) and KVL:

$$ V_{total} = V_1 + V_2 + \cdots + V_n $$

Substituting Ohm’s Law for each voltage drop:

$$ IR_{eq} = IR_1 + IR_2 + \cdots + IR_n $$

Since current I is constant in a series circuit, it cancels out:

$$ R_{eq} = R_1 + R_2 + \cdots + R_n $$

Key Characteristics

Practical Implications

Series configurations are ubiquitous in voltage dividers, current-limiting circuits, and sensor networks. For example, precision resistor chains in analog-to-digital converters rely on series connections to generate reference voltages. In high-power applications, series resistors distribute thermal dissipation, though tolerance mismatches can lead to uneven power sharing.

Visualization

A series circuit with three resistors (R1, R2, R3) shows a direct line from the voltage source through each component, with no branching paths. The equivalent resistance Req is the straight arithmetic sum of the three values.

Series Circuit with Three Resistors A schematic diagram showing three resistors (R1, R2, R3) connected in series with a voltage source, illustrating current flow and voltage distribution. V_total R1 V1 R2 V2 R3 V3 I
Diagram Description: The diagram would show three resistors connected end-to-end with a single current path, visually demonstrating the series configuration and voltage distribution.

1.2 Key Properties of Resistors in Series

Equivalent Resistance

When resistors are connected in series, the total or equivalent resistance (Req) is the sum of the individual resistances. This arises because the current must flow sequentially through each resistor, encountering their resistances additively. Mathematically:

$$ R_{eq} = R_1 + R_2 + R_3 + \dots + R_N $$

For example, three resistors of 10 Ω, 20 Ω, and 30 Ω in series yield an equivalent resistance of 60 Ω. This property holds regardless of the resistor values or their order in the series chain.

Current Uniformity

In a series configuration, the same current flows through every resistor. This is a direct consequence of Kirchhoff's Current Law (KCL), which states that charge cannot accumulate at any node in a steady-state DC circuit. Thus:

$$ I_{total} = I_1 = I_2 = I_3 = \dots = I_N $$

This uniformity is critical in applications like current sensing, where series resistors ensure identical current flow through measurement shunts.

Voltage Division

The voltage across each resistor in series is proportional to its resistance. The total voltage (Vtotal) divides according to:

$$ V_n = I \cdot R_n = \left( \frac{V_{total}}{R_{eq}} \right) R_n $$

This forms the basis of voltage divider circuits, widely used in signal conditioning and reference voltage generation. For instance, in a 12 V circuit with two series resistors (R1 = 2 kΩ, R2 = 4 kΩ), the voltages across R1 and R2 are 4 V and 8 V, respectively.

Power Dissipation

Power dissipation in each resistor is given by P = I²R. Since current is uniform, higher-value resistors dissipate more power. The total power is the sum of individual dissipations:

$$ P_{total} = I^2 R_{eq} = P_1 + P_2 + \dots + P_N $$

This property is crucial in designing circuits with power constraints, ensuring no single resistor exceeds its thermal limits.

Practical Implications

Frequency-Dependent Behavior

At high frequencies, parasitic capacitance and inductance introduce impedance (Z), altering the purely resistive behavior. The equivalent impedance becomes:

$$ Z_{eq} = \sum_{k=1}^N (R_k + j\omega L_k + \frac{1}{j\omega C_k}) $$

where Lk and Ck are parasitic elements. This is significant in RF circuits and transmission line analysis.

Series Resistor Circuit with Voltage Division A schematic diagram showing a series resistor circuit with a battery and three resistors, illustrating voltage division and uniform current flow. V_total R1 V1 R2 V2 R3 V3 I
Diagram Description: A diagram would visually demonstrate the voltage division across resistors in series and the uniform current flow, which are spatial concepts.

1.3 Common Applications of Series Resistor Configurations

Voltage Division and Attenuation

Series resistors are fundamental in constructing voltage dividers, which scale down an input voltage to a desired level. The output voltage \( V_{out} \) across resistor \( R_2 \) in a two-resistor divider is given by:

$$ V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2} $$

This principle is widely used in sensor interfacing, where high voltages must be attenuated to match the input range of analog-to-digital converters (ADCs). For precision applications, resistors with low temperature coefficients (e.g., 25 ppm/°C) are selected to minimize drift.

Current Limiting and LED Driving

In LED circuits, series resistors enforce safe operating currents by exploiting Ohm’s Law:

$$ R = \frac{V_{supply} - V_{LED}}{I_{LED}} $$

For example, a 5V supply driving a red LED (\( V_{LED} \approx 1.8V \)) at 20 mA requires:

$$ R = \frac{5V - 1.8V}{0.02A} = 160 \Omega $$

This configuration prevents thermal runaway and ensures consistent brightness. In high-power applications, resistors must dissipate power \( P = I^2R \) without exceeding their ratings.

Impedance Matching in RF Circuits

Series resistors combine with parallel components to create ladder networks for impedance matching. In transmission lines, the characteristic impedance \( Z_0 \) is matched using resistive terminations to minimize reflections. For instance, a 50 Ω source driving a 75 Ω load may use a series 25 Ω resistor, though this introduces 3 dB attenuation:

$$ Z_{total} = Z_{source} + R_{series} = 75 \Omega $$

Pull-Up/Pull-Down Configurations

Digital logic circuits use series resistors with pull-up/down networks to define default states while limiting current during signal transitions. In I²C buses, pull-up resistors \( R_{pu} \) set the logic high level while controlling rise time \( \tau \):

$$ \tau = R_{pu} \cdot C_{bus} $$

Typical values range from 1 kΩ to 10 kΩ, balancing speed and power consumption.

Inrush Current Suppression

Power supplies employ negative temperature coefficient (NTC) thermistors in series with mains inputs to limit inrush current. The thermistor’s high initial resistance drops as it heats, minimizing energy loss during steady-state operation. For capacitive loads, the peak current \( I_{peak} \) is constrained by:

$$ I_{peak} \leq \frac{V_{max}}{R_{series}} $$

Precision Measurement Circuits

Kelvin-Varley dividers use cascaded series resistors to achieve voltage division with ppm-level accuracy. In four-wire resistance measurements, series resistors isolate the current injection path from voltage sensing lines, eliminating lead resistance errors.

Series Resistor Network
Series Resistor Applications Three side-by-side circuits demonstrating series resistor applications: a voltage divider, an LED driver, and an impedance matching network. Voltage Divider R1 R2 Vout Vin LED Driver R I_LED Vsupply LED Impedance Matching Z0 Z_load
Diagram Description: A diagram would physically show the voltage divider circuit with labeled resistors and input/output voltages, and the LED current limiting circuit with component values.

2. Deriving the Equivalent Resistance Formula

2.1 Deriving the Equivalent Resistance Formula

Consider a series circuit with N resistors connected end-to-end, where the current I flows uniformly through each resistor. Let the individual resistances be R1, R2, ..., RN. Kirchhoff's voltage law dictates that the total voltage Vtotal across the series combination equals the sum of voltage drops across each resistor:

$$ V_{total} = V_1 + V_2 + \cdots + V_N $$

Applying Ohm's law (V = IR) to each resistor yields:

$$ V_{total} = IR_1 + IR_2 + \cdots + IR_N $$

The current I is common to all terms and can be factored out:

$$ V_{total} = I(R_1 + R_2 + \cdots + R_N) $$

By definition, the equivalent resistance Req satisfies Vtotal = IReq. Comparing this with the previous equation gives:

$$ IR_{eq} = I(R_1 + R_2 + \cdots + R_N) $$

Since I ≠ 0, it cancels out, leaving the fundamental result:

$$ R_{eq} = R_1 + R_2 + \cdots + R_N $$

Practical Implications

This derivation has immediate consequences for circuit design:

Historical Context

Gustav Kirchhoff formulated his circuit laws in 1845, providing the theoretical foundation for this derivation. Series resistor networks became essential in early telegraph systems, where precise voltage division was critical for signal transmission over long distances.

Experimental Verification

The formula can be confirmed experimentally by:

  1. Measuring individual resistances with a multimeter
  2. Connecting them in series and measuring Req
  3. Comparing the result with the calculated sum

Modern digital multimeters typically show agreement within 0.1% for precision resistors, validating the theoretical prediction.

2.2 Voltage Division in Series Circuits

In a series resistor network, the total applied voltage divides proportionally across each resistor based on its resistance relative to the total resistance. This principle, known as the voltage divider rule, is fundamental in circuit analysis and design.

Mathematical Derivation

Consider a series circuit with N resistors \( R_1, R_2, \dots, R_N \) connected to a voltage source \( V_{in} \). The current \( I \) flowing through the circuit is uniform and given by Ohm's Law:

$$ I = \frac{V_{in}}{R_{total}} $$

where \( R_{total} = R_1 + R_2 + \dots + R_N \). The voltage drop \( V_k \) across the \( k \)-th resistor is:

$$ V_k = I R_k = V_{in} \frac{R_k}{R_{total}} $$

This equation forms the basis of the voltage divider rule, showing that the voltage across each resistor is proportional to its fraction of the total resistance.

Practical Implications

Voltage division is widely used in:

A common implementation involves two resistors \( R_1 \) and \( R_2 \), producing an output voltage \( V_{out} \) at their junction:

$$ V_{out} = V_{in} \frac{R_2}{R_1 + R_2} $$

Non-Ideal Considerations

In real-world applications, the voltage divider's accuracy is affected by:

Extended Case: Multiple Voltage Dividers

For cascaded dividers (e.g., \( V_{in} \rightarrow R_1 \rightarrow R_2 \rightarrow \dots \rightarrow GND \)), the voltage at node \( k \) becomes:

$$ V_k = V_{in} \frac{\sum_{i=k+1}^N R_i}{R_{total}} $$

This generalizes the two-resistor case and is useful in ladder networks like R-2R DACs.

Series Resistor Voltage Divider A schematic diagram of a series resistor voltage divider circuit, showing a voltage source connected to resistors in series with labeled voltage drops and current direction. R1 R2 R3 Vin GND V1 V2 V3 I
Diagram Description: The diagram would physically show a series circuit with labeled resistors, voltage source, and voltage drops across each resistor to visualize proportional division.

2.3 Current Behavior in Series Configurations

In a series resistor network, the current remains uniform across all components due to the absence of alternative pathways. Kirchhoff's Current Law (KCL) dictates that the algebraic sum of currents entering and exiting a node must be zero, but in a series circuit, no branching nodes exist. Consequently, the same current flows through each resistor, irrespective of individual resistances.

Mathematical Derivation

Consider a series circuit with resistors R1, R2, ..., Rn connected to a voltage source V. The total resistance Rtotal is the sum of individual resistances:

$$ R_{total} = R_1 + R_2 + \cdots + R_n $$

By Ohm's Law, the current I through the circuit is:

$$ I = \frac{V}{R_{total}} $$

Since the current is identical at every point in the series loop, the voltage drop Vi across each resistor Ri is:

$$ V_i = I \cdot R_i $$

Practical Implications

This behavior has critical implications in circuit design:

Experimental Validation

In a lab setting, an ammeter placed at any point in a series circuit will yield identical readings. For example, a circuit with R1 = 100 Ω, R2 = 200 Ω, and V = 9 V produces:

$$ I = \frac{9 \text{V}}{100 + 200 \ \Omega} = 30 \text{mA} $$

Measurements at R1 and R2 will confirm this current value, empirically validating the theory.

3. Power Dissipation Across Series Resistors

Power Dissipation Across Series Resistors

In a series resistor network, the total power dissipated is the sum of the individual power dissipations across each resistor. This arises from the conservation of energy, where the electrical energy supplied by the source is entirely converted into heat by the resistors. The power dissipated by a resistor is given by Joule's first law:

$$ P = I^2 R $$

For resistors in series, the current I is identical through each resistor. Thus, the power dissipated by the i-th resistor in the series is:

$$ P_i = I^2 R_i $$

The total power dissipation Ptotal is the sum of the individual dissipations:

$$ P_{total} = \sum_{i=1}^n P_i = I^2 \sum_{i=1}^n R_i = I^2 R_{total} $$

where Rtotal is the equivalent series resistance. This confirms that the total power dissipation can also be calculated using the total resistance and the common series current.

Voltage-Dependent Power Dissipation

Alternatively, power dissipation can be expressed in terms of the voltage drop Vi across each resistor. Since Vi = I Ri, the power dissipated by the i-th resistor becomes:

$$ P_i = \frac{V_i^2}{R_i} $$

The total power is then:

$$ P_{total} = \sum_{i=1}^n \frac{V_i^2}{R_i} $$

However, this form is less commonly used in series circuits because the voltage division complicates the calculation compared to the current-based approach.

Practical Implications

In high-power applications, such as voltage dividers or current-limiting circuits, power dissipation must be carefully managed to prevent overheating. Resistors in series share the total power dissipation proportionally to their resistance values. For instance, a larger resistor in the series will dissipate more power than a smaller one for the same current.

Consider a series circuit with resistors R1 = 100 Ω and R2 = 200 Ω carrying a current of 50 mA:

$$ P_1 = (0.05)^2 \times 100 = 0.25 \text{ W} $$ $$ P_2 = (0.05)^2 \times 200 = 0.5 \text{ W} $$ $$ P_{total} = 0.25 + 0.5 = 0.75 \text{ W} $$

This example highlights how power dissipation scales with resistance in a series configuration.

Thermal Considerations

Exceeding a resistor's power rating can lead to thermal runaway or failure. Engineers must ensure that each resistor's power dissipation remains within its specified limits. For precision circuits, temperature coefficients must also be considered, as resistance values may drift with heating.

In summary, power dissipation in series resistors is straightforward to compute using the common current, but practical designs must account for thermal effects and component tolerances.

3.2 Tolerance and Stability in Series Networks

Tolerance in Series Resistor Networks

When resistors are connected in series, their individual tolerances combine to affect the overall precision of the total resistance. The tolerance of a resistor defines the permissible deviation from its nominal value, typically expressed as a percentage (e.g., ±1%, ±5%). For a series network of N resistors, the worst-case tolerance Ttotal can be derived from the root-sum-square (RSS) method, which accounts for statistical variations:

$$ T_{total} = \sqrt{T_1^2 + T_2^2 + \dots + T_N^2} $$

where Ti represents the tolerance of the i-th resistor. For identical tolerances (T1 = T2 = \dots = TN = T), this simplifies to:

$$ T_{total} = T \sqrt{N} $$

In precision applications, this statistical approach is preferred over arithmetic summation, as it provides a more realistic estimate of the combined tolerance.

Thermal Stability and Temperature Coefficients

The stability of a series resistor network under varying temperatures depends on the temperature coefficient of resistance (TCR) of individual components. TCR, expressed in ppm/°C (parts per million per degree Celsius), quantifies the linear change in resistance with temperature. For a series combination, the effective TCR (\alpha_{total}) is a weighted average:

$$ \alpha_{total} = \frac{R_1 \alpha_1 + R_2 \alpha_2 + \dots + R_N \alpha_N}{R_1 + R_2 + \dots + R_N} $$

where Ri and \alphai are the resistance and TCR of the i-th resistor, respectively. If resistors have matched TCRs (\alpha_1 = \alpha_2 = \dots = \alpha_N), the network's TCR remains unchanged.

Practical Implications in Circuit Design

In high-precision analog circuits, such as voltage dividers or feedback networks, resistor tolerance and thermal stability directly impact performance. For example:

To mitigate tolerance stacking, designers often use resistor arrays (e.g., thin-film networks) where individual elements share matched characteristics due to monolithic fabrication.

Noise Considerations

Thermal noise (Johnson-Nyquist noise) in series resistors adds quadratically, as noise power is uncorrelated:

$$ V_{n,total} = \sqrt{4k_B T (R_1 + R_2 + \dots + R_N) \Delta f} $$

where kB is Boltzmann's constant, T is absolute temperature, and \Delta f is bandwidth. High-value series resistors exacerbate noise, necessitating careful selection in low-noise amplifiers.

Case Study: Precision Voltage Divider

Consider a series network of two resistors (R1 = 9 kΩ ±0.1%, R2 = 1 kΩ ±0.1%) forming a 10:1 divider. The worst-case tolerance of the ratio is:

$$ T_{ratio} = \sqrt{0.1^2 + 0.1^2} \approx 0.141\% $$

This demonstrates how even matched tolerances degrade the precision of the divider ratio. Using resistors with TCR tracking (e.g., ±5 ppm/°C) further ensures minimal drift across temperature.

3.3 Troubleshooting Common Series Circuit Issues

Identifying Open Circuits in Series Configurations

An open circuit in a series connection disrupts current flow entirely, as the equivalent resistance becomes infinite. The voltage drop across the open component equals the full supply voltage due to Ohm's Law:

$$ V_{open} = I_{total} \times R_{open} = 0 \times \infty \Rightarrow V_{supply} $$

Diagnostically, measure voltage across each resistor while powered. A resistor showing full supply voltage indicates an open downstream. For example, in a 12V circuit with three 1kΩ resistors, an open at R2 would yield:

$$ V_{R2} = 12V,\quad V_{R1} = 0V,\quad V_{R3} = 0V $$

Unexpected Voltage Division Anomalies

Deviations from theoretical voltage division often stem from:

Current Measurement Discrepancies

Series circuits should exhibit identical current at all nodes. Observed variations indicate:

Power Dissipation Miscalculations

Incorrect power ratings often cause thermal runaway. The actual dissipation in resistor Ri is:

$$ P_i = I^2R_i = \left( \frac{V_{total}}{\sum R} \right)^2 R_i $$

A common mistake is neglecting derating requirements: military standard MIL-HDBK-217F specifies 50% derating for reliable operation. For example, a 1/4W resistor in series with 10mA current requires:

$$ R_{min} = \frac{P}{I^2} = \frac{0.125W}{(0.01A)^2} = 1.25k\Omega $$

Transient Response Issues

Series resistor-inductor networks exhibit time constant Ï„ = L/R. When troubleshooting:

Noise and Thermal Considerations

Johnson-Nyquist noise in series resistors adds quadratically:

$$ V_{n,rms} = \sqrt{4k_BT \Delta f (R_1 + R_2 + \cdots + R_n)} $$

For cryogenic applications, metal-film resistors are preferred due to their lower 1/f noise corner frequency (~10Hz vs carbon's ~1kHz).

4. Non-Ideal Resistor Effects in Series

4.1 Non-Ideal Resistor Effects in Series

When analyzing resistors in series, ideal behavior assumes purely resistive elements with no parasitic effects. However, real-world resistors exhibit non-ideal characteristics that become significant in precision circuits, high-frequency applications, or high-power systems. These effects include parasitic inductance, capacitance, temperature dependence, and voltage coefficient of resistance.

Parasitic Inductance and Capacitance

All physical resistors possess distributed inductance (L) due to their conductive path and capacitance (C) between terminals. The impedance Z of a non-ideal resistor becomes frequency-dependent:

$$ Z = R + j\omega L + \frac{1}{j\omega C} $$

where ω is the angular frequency. For a series resistor network, the total parasitic inductance sums linearly, while capacitance combines reciprocally:

$$ L_{total} = \sum_{i=1}^n L_i $$ $$ \frac{1}{C_{total}} = \sum_{i=1}^n \frac{1}{C_i} $$

This becomes critical in RF circuits where the self-resonant frequency (fSRF) of the resistor network may fall within the operating bandwidth.

Temperature Effects

Resistor temperature dependence is characterized by the temperature coefficient of resistance (TCR), typically expressed in ppm/°C. For series-connected resistors with different TCR values, the net TCR becomes a weighted average:

$$ TCR_{total} = \frac{\sum (R_i \cdot TCR_i)}{\sum R_i} $$

Carbon composition resistors exhibit ≈500 ppm/°C, while precision metal foil resistors achieve <1 ppm/°C. In power applications, Joule heating creates thermal gradients that further modify resistance values.

Voltage Coefficient

At high voltages (>50% of rated voltage), many resistors show non-linear behavior described by the voltage coefficient (VCR):

$$ R(V) = R_0 (1 + VCR \cdot V) $$

Thick-film resistors typically have VCR values of -50 to -200 ppm/V, while bulk metal foil resistors maintain <0.1 ppm/V. In series networks, voltage division causes each resistor to operate at different points along its VCR curve.

Noise Contributions

Resistors generate thermal (Johnson-Nyquist) noise and, in some types, excess current noise. For series-connected resistors, noise voltages add quadratically:

$$ v_{n,total} = \sqrt{4kT \Delta f \sum R_i + \sum (v_{n,excess,i}^2)} $$

where k is Boltzmann's constant and Δf is the bandwidth. Carbon composition resistors exhibit the highest excess noise (≈-12 dB), while wirewound types are the quietest.

Practical Mitigation Strategies

These non-ideal behaviors explain why simple series resistance addition often fails to predict actual circuit performance in sensitive measurement systems or high-speed applications. Modern network analyzers can characterize these parasitic effects up to 110 GHz, enabling more accurate modeling.

Non-Ideal Resistor Impedance Model in Series A schematic diagram showing the frequency-dependent impedance model of a non-ideal resistor with parasitic L and C components, and how these combine in series. R L C Z(ω) Frequency Response f f_SRF ΣL_i 1/Σ(1/C_i) Series Network
Diagram Description: The diagram would show the frequency-dependent impedance model of a non-ideal resistor with parasitic L and C components, and how these combine in series.

Temperature Dependence of Series Resistances

The resistance of a material is inherently temperature-dependent due to variations in charge carrier mobility and lattice vibrations. For resistors connected in series, the cumulative temperature effect must be analyzed to predict circuit behavior under thermal variations.

Mathematical Modeling of Temperature Dependence

The resistance of an individual resistor at temperature T can be expressed using the linear approximation:

$$ R(T) = R_0 \left[1 + \alpha (T - T_0)\right] $$

where:

For a series combination of N resistors, the total resistance becomes:

$$ R_{total}(T) = \sum_{i=1}^{N} R_{0,i} \left[1 + \alpha_i (T - T_0)\right] $$

Case 1: Identical TCR (αi = α)

When all resistors share the same TCR, the temperature dependence simplifies to:

$$ R_{total}(T) = R_{total,0} \left[1 + \alpha (T - T_0)\right] $$

where Rtotal,0 = ΣR0,i. This shows that the entire network scales uniformly with temperature.

Case 2: Mixed TCR Values

For resistors with differing TCRs (αi ≠ αj), the temperature behavior becomes nonlinear. The effective TCR of the network is a weighted average:

$$ \alpha_{eff} = \frac{\sum_{i=1}^{N} R_{0,i} \alpha_i}{\sum_{i=1}^{N} R_{0,i}} $$

This relationship is particularly important in precision circuits where resistor matching is critical.

Practical Implications

In high-precision analog circuits (e.g., instrumentation amplifiers, voltage references), designers must:

Modern resistor networks often specify tracking TCR - the maximum difference in TCR between components - which is more critical than absolute TCR for matched pairs.

Advanced Considerations

At cryogenic temperatures or extreme heating (>150°C), the linear TCR model breaks down. The Callendar-Van Dusen equation provides better accuracy:

$$ R(T) = R_0 \left[1 + A(T-T_0) + B(T-T_0)^2 + C(T-T_0)^3\right] $$

where A, B, and C are material-specific coefficients. Platinum resistors used in precision temperature sensors typically follow this relationship.

In high-frequency applications, skin effect and dielectric losses introduce additional temperature-dependent resistance components that must be considered in series resistance calculations.

4.3 Frequency Response Considerations

Parasitic Effects in Series Resistors

While resistors are primarily considered frequency-independent components, parasitic inductance and capacitance become significant at high frequencies. A series resistor exhibits a non-ideal impedance Z(ω) given by:

$$ Z(ω) = R + jωL + \frac{1}{jωC} $$

Here, L represents the parasitic series inductance (typically 1–10 nH for axial resistors), and C models the inter-turn or lead capacitance (0.1–5 pF). The frequency where the reactive terms dominate is determined by the self-resonant frequency (SRF):

$$ \text{SRF} = \frac{1}{2Ï€\sqrt{LC}} $$

Implications for High-Speed Circuits

In RF or high-speed digital applications, parasitic effects cause:

Practical Mitigation Strategies

To minimize parasitic impacts:

Case Study: 50Ω Termination at 10 GHz

A 50Ω carbon-film resistor with L = 5 nH and C = 0.5 pF exhibits an SRF of:

$$ \text{SRF} = \frac{1}{2Ï€\sqrt{5 \times 10^{-9} \times 0.5 \times 10^{-12}}} \approx 3.18 \text{ GHz} $$

Beyond this frequency, the impedance rises sharply, making it unsuitable for precise terminations. A high-frequency thin-film resistor (SRF > 20 GHz) would be preferable.

Thermal Noise and Frequency Dependence

Johnson-Nyquist noise (4kTRB) is frequency-independent, but parasitic capacitance filters higher-frequency noise components. The effective noise bandwidth B becomes:

$$ B = \int_0^\infty \frac{1}{1 + (ωRC)^2} \, dω = \frac{π}{2RC} $$
Impedance vs. Frequency for Series Resistor with Parasitics Bode plot showing impedance magnitude versus frequency, illustrating resistive, inductive, and capacitive regions with labeled SRF point. Frequency (log scale) Impedance Magnitude (log scale) R (Flat Region) Inductive Rise (ωL) Capacitive Fall (1/ωC) SRF Resistive Dominance Inductive Dominance Capacitive Dominance
Diagram Description: The section discusses impedance behavior across frequencies and resonance effects, which are best visualized with a frequency response curve.

5. Essential Textbooks on Circuit Theory

5.1 Essential Textbooks on Circuit Theory

5.2 Research Papers on Resistor Networks

5.3 Online Resources and Interactive Tools